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Elementary Algebra
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Title: Elementary Algebra
Physical Description: Book
Language: en-US
Creator: Ellis, Wade
Burzynski, Denny
Publisher: Connexions, Rice University
Place of Publication: Houston, TX
Publication Date: 2008
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Subjects / Keywords: mathematics, algebra, algebraic techniques, algebraic notation, OGT+ isbn: 9781616100308
Algebra, Equations (Mathematics), Mathematics
Mathematics / Algebra
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Abstract: Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques. Includes glossary,sample sets and practice sets, section and review exercises, and solutions. Contents: 1) Arithmetic Review. 2) Basic Properties of Real Numbers. 3) Basic Operations with Real Numbers. 4) Algebraic Expressions and Equations. 5) Solving Linear Equations and Inequalities. 6) Factoring Polynomials. 7) Graphing Linear Equations and Inequalities in One and Two Variables. 8) Rational Expressions. 9) Roots, Radicals, and Square Root Equations. 10) Quadratic Equations. 11) Systems of Linear Equations.
General Note: Expositive
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General Note: Denny Burzynski, Wade Ellis
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General Note: http://cnx.org/content/col10614/latest/
General Note: MAT 012 - DEVELOPMENTAL ARITHMETIC WITH ALGEBRA, MAT 020 - COLLEGE PREP MATH I, MAT 024 - BASIC ALGEBRA, MAT 033 - INTERMEDIATE ALGEBRA
General Note: http://florida.theorangegrove.org/og/file/a9ddf3ff-a17d-7503-d845-24d16a38f278/1/ElementaryAlgebra.pdf
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Source Institution: University Press of Florida
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Rights Management: Copyright ©2008 Wade Ellis, Denny Burzynski. This selection and arrangement of content is licensed under the Creative Commons Attribution License: http://creativecommons.org/licenses/by/2.0/
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ElementaryAlgebra By: WadeEllis DennyBurzynski

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ElementaryAlgebra By: WadeEllis DennyBurzynski Online: < http://cnx.org/content/col10614/1.3/ > CONNEXIONS RiceUniversity,Houston,Texas

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2008WadeEllis,DennyBurzynski ThisselectionandarrangementofcontentislicensedundertheCreativeCommonsAttributionLicense: http://creativecommons.org/licenses/by/2.0/

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TableofContents Preface ...............................................................................................1 Acknowledgments ...................................................................................5 1ArithmeticReview 1.1 Objectives...................................................................................7 1.2 Factors,Products,andExponents...........................................................7 1.3 PrimeFactorization........................................................................11 1.4 TheLeastCommonMultiple...............................................................14 1.5 EquivalentFractions........................................................................18 1.6 OperationswithFractions..................................................................22 1.7 DecimalFractions..........................................................................28 1.8 Percent.....................................................................................36 Solutions........................................................................................41 2BasicPropertiesofRealNumbers 2.1 Objectives..................................................................................47 2.2 SymbolsandNotations.....................................................................48 2.3 TheRealNumberLineandtheRealNumbers..............................................55 2.4 PropertiesoftheRealNumbers............................................................63 2.5 Exponents..................................................................................71 2.6 RulesofExponents.........................................................................79 2.7 ThePowerRulesforExponents............................................................88 2.8 SummaryofKeyConcepts.................................................................96 2.9 ExerciseSupplement.......................................................................97 2.10 ProciencyExam........................................................................104 Solutions.......................................................................................106 3BasicOperationswithRealNumbers 3.1 Objectives.................................................................................121 3.2 SignedNumbers...........................................................................122 3.3 AbsoluteValue............................................................................127 3.4 AdditionofSignedNumbers...............................................................132 3.5 SubtractionofSignedNumbers............................................................139 3.6 MultiplicationandDivisionofSignedNumbers............................................144 3.7 NegativeExponents.......................................................................153 3.8 ScienticNotation.........................................................................163 3.9 SummaryofKeyConcepts................................................................170 3.10 ExerciseSupplement.....................................................................172 3.11 ProciencyExam........................................................................176 Solutions.......................................................................................179 4AlgebraicExpressionsandEquations 4.1 Objectives.................................................................................197 4.2 AlgebraicExpressions.....................................................................198 4.3 Equations.................................................................................206 4.4 ClassicationofExpressionsandEquations................................................214 4.5 CombiningPolynomialsUsingAdditionandSubtraction...................................221 4.6 CombiningPolynomialsUsingMultiplication..............................................227 4.7 SpecialBinomialProducts.................................................................239 4.8 TerminologyAssociatedwithEquations...................................................246 4.9 SummaryofKeyConcepts................................................................249

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iv 4.10 ExerciseSupplement.....................................................................250 4.11 ProciencyExam........................................................................257 Solutions.......................................................................................259 5SolvingLinearEquationsandInequalities 5.1 Objectives.................................................................................277 5.2 SolvingEquations.........................................................................278 5.3 SolvingEquationsoftheFormax=bandx/a=b...........................................284 5.4 FurtherTechniquesinEquationSolving...................................................290 5.5 ApplicationI-TranslatingfromVerbaltoMatheticalExpressions.........................298 5.6 ApplicationII-SolvingProblems.........................................................304 5.7 LinearinequalitiesinOneVariable........................................................313 5.8 LinearEquationsinTwoVariables.........................................................322 5.9 SummaryofKeyConcepts................................................................328 5.10 ExerciseSupplement.....................................................................330 5.11 ProciencyExam........................................................................335 Solutions.......................................................................................338 6FactoringPolynomials 6.1 Objectives.................................................................................353 6.2 FindingthefactorsofaMonomial.........................................................354 6.3 FactoringaMonomialfromaPolynomial..................................................357 6.4 TheGreatestCommonFactor.............................................................362 6.5 FactoringbyGrouping....................................................................367 6.6 FactoringTwoSpecialProducts...........................................................370 6.7 FactoringTrinomialswithLeadingCoecient1...........................................378 6.8 FactoringTrinomialswithLeadingCoecientOtherThan1...............................383 6.9 SummaryofKeyConcepts................................................................393 6.10 ExerciseSupplement.....................................................................394 6.11 ProciencyExam........................................................................398 Solutions.......................................................................................400 7GraphingLinearEquationsandInequalitiesinOneandTwoVariables 7.1 Objectives.................................................................................411 7.2 GraphingLinearEquationsandInequalitiesinOneVariable...............................412 7.3 PlottingPointsinthePlane...............................................................419 7.4 GraphingLinearEquationsinTwoVariables..............................................428 7.5 TheSlope-InterceptFormofaLine........................................................448 7.6 GraphingEquationsinSlope-InterceptForm..............................................464 7.7 FindingtheEquationofaLine............................................................476 7.8 GraphingLinearInequalitiesinTwoVariables.............................................485 7.9 SummaryofKeyConcepts................................................................497 7.10 ExerciseSupplement.....................................................................498 7.11 ProciencyExam........................................................................508 Solutions.......................................................................................513 8RationalExpressions 8.1 Objectives.................................................................................545 8.2 RationalExpressions......................................................................546 8.3 ReducingRationalExpressions............................................................553 8.4 MultiplyingandDividingRationalExpressions............................................561 8.5 BuildingRationalExpressionsandtheLCD...............................................568 8.6 AddingandSubtractingRationalExpressions.............................................581

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v 8.7 RationalEquations........................................................................591 8.8 Applications...............................................................................600 8.9 ComplexRationalExpressions.............................................................610 8.10 DividingPolynomials....................................................................618 8.11 SummaryofKeyConcepts...............................................................628 8.12 ExerciseSupplement.....................................................................629 8.13 ProciencyExam........................................................................636 Solutions.......................................................................................638 9Roots,Radicals,andSquareRootEquations 9.1 Objectives.................................................................................659 9.2 SquareRootExpressions..................................................................660 9.3 SimplifyingSquareRootExpressions......................................................669 9.4 MultiplicationofSquareRootExpressions.................................................677 9.5 DivisionofSquareRootExpressions.......................................................684 9.6 AdditionandSubtractionofSquareRootExpressions.....................................691 9.7 SquareRootEquationswithApplications..................................................699 9.8 SummaryofKeyConcepts................................................................704 9.9 ExerciseSupplement......................................................................706 9.10 ProciencyExam........................................................................710 Solutions.......................................................................................713 10QuadraticEquations 10.1 Objectives...............................................................................729 10.2 SolvingQuadraticEquations.............................................................730 10.3 SolvingQuadraticEquationsbyFactoring................................................735 10.4 SolvingQuadraticEquationsUsingtheMethodofExtractionofRoots...................742 10.5 SolvingQuadraticEquationsUsingtheMethodofCompletingtheSquare................749 10.6 SolvingQuadraticEquationsUsingtheQuadraticFormula...............................756 10.7 Applications.............................................................................764 10.8 GraphingQuadraticSolutions............................................................775 10.9 SummaryofKeyConcepts...............................................................788 10.10 ExerciseSupplement....................................................................789 10.11 ProciencyExam.......................................................................794 Solutions.......................................................................................798 11SystemsofLinearEquations 11.1 Objectives...............................................................................817 11.2 SolutionsbyGraphing...................................................................817 11.3 EliminationbySubstitution..............................................................828 11.4 EliminationbyAddition.................................................................836 11.5 Applications.............................................................................845 11.6 SummaryofKeyConcepts...............................................................851 11.7 ExerciseSupplement.....................................................................852 11.8 ProciencyExam........................................................................855 Solutions.......................................................................................857 12Appendix 12.1 TableofSymbols.........................................................................867 12.2 PropertiesofRealNumbers..............................................................868 12.3 ImportantandUsefulRules/Formulas....................................................869 12.4 The5-StepMethodofSolvingAppliedProblems.........................................869 Solutions........................................................................................ ??

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vi Glossary ............................................................................................870 Index ...............................................................................................871 Attributions ........................................................................................875

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Preface 1 ElementaryAlgebraisaworktextthatcoversthetraditionaltopicsstudiedinamodernelementaryalgebra course.Itisintendedforstudentswho: 1.Havenoexposuretoelementaryalgebra, 2.Havehadapreviouslyunpleasantexperiencewithelementaryalgebra,or 3.Needtoreviewalgebraicconceptsandtechniques. Useofthisbookwillhelpthestudentdeveloptheinsightandintuitionnecessarytomasteralgebraictechniquesandmanipulativeskills.Thetextiswrittentopromoteproblem-solvingabilitysothatthestudent hasthemaximumopportunitytoseethattheconceptsandtechniquesarelogicallybasedandtobecomfortableenoughwiththeseconceptstoknowwhenandhowtousetheminsubsequentsections,courses,and non-classroomsituations.Intuitionandunderstandingaresomeofthekeystocreativity;webelievethat thematerialpresentedwillhelpmakethesekeysavailabletothestudent. Thistextcanbeusedinstandardlectureorself-pacedclasses.Tohelpstudentsmeettheseobjectives andtomakethestudyofalgebraapleasantandrewardingexperience,ElementaryAlgebraisorganizedas follows. PedagogicalFeatures Theworktextformatgivesthestudentspacetopracticealgebraicskillswithreadyreferencetosample problems.Thechaptersaredividedintosections,andeachsectionisacompletetreatmentofaparticular topic,whichincludesthefollowingfeatures: SectionOverview SampleSets PracticeSets SectionExercises ExercisesforReview Thechaptersbeginwith Objectives andendwitha SummaryofKeyConcepts ,an ExerciseSupplement ,anda ProciencyExam Objectives Eachchapterbeginswithasetofobjectivesidentifyingthematerialtobecovered.Eachsectionbeginswith anoverviewthatrepeatstheobjectivesforthatparticularsection.Sectionsaredividedintosubsectionsthat correspondtothesectionobjectives,whichmakesforeasierreading. SampleSets ElementaryAlgebracontainsexamplesthataresetoinboxesforeasyreference.Theexamplesarereferred toasSampleSetsfortworeasons: 1 Thiscontentisavailableonlineat. 1

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2 1.Theyserveasarepresentationtobeimitated,whichwebelievewillfosterunderstandingofalgebra conceptsandprovideexperiencewithalgebraictechniques. 2.SampleSetsalsoserveasapreliminaryrepresentationofproblem-solvingtechniquesthatmaybe usedtosolvemoregeneralandmorecomplicatedproblems.Theexampleshavebeencarefullychosen toillustrateanddevelopconceptsandtechniquesinthemostinstructive,easilyrememberedway. Conceptsandtechniquesprecedingtheexamplesareintroducedatalevelbelowthatnormallyused insimilartextsandarethoroughlyexplained,assuminglittlepreviousknowledge. PracticeSet AparallelPracticeSetfollowseachSampleSet,whichreinforcestheconceptsjustlearned.Theanswers toallPracticeSetsaredisplayedwiththequestionwhenviewingthiscontentonline,orattheendofthe chapterintheprintversion. SectionExercises Theexercisesattheendofeachsectionaregradedintermsofdiculty,althoughtheyarenotgroupedinto categories.Thereareanamplenumberofproblems;afterworkingthroughtheexercises,thestudentwill becapableofsolvingavarietyofchallengingproblems. Theproblemsarepairedsothattheodd-numberedproblemsareequivalentinkindanddicultyto theeven-numberedproblems.Answerstotheodd-numberedproblemsareprovidedwiththeexercisewhen viewedonline,oratthebackofthechapterintheprintversion. ExercisesforReview Thissectionconsistsofproblemsthatformacumulativereviewofthematerialcoveredinthepreceding sectionsofthetextandisnotlimitedtomaterialinthatchapter.Theexercisesarekeyedbysectionfor easyreference. SummaryofKeyConcepts Asummaryoftheimportantideasandformulasusedthroughoutthechapterisincludedattheendof eachchapter.Morethanjustalistofterms,thesummaryisavaluabletoolthatreinforcesconceptsin preparationfortheProciencyExamattheendofthechapter,aswellasfutureexams.Thesummarykeys eachitemtothesectionofthetextwhereitisdiscussed. ExerciseSupplement Inadditiontonumeroussectionexercises,eachchapterincludesapproximately100supplementalproblems, whicharereferencedbysection.Answerstotheodd-numberedproblemsareincludedwiththeproblems whenviewedonlineandinthebackofthechapterintheprintversion. ProciencyExam EachchapterendswithaProciencyExamthatcanserveasachapterrevieworachapterevaluation.The prociencyExamiskeyedtosections,whichenablesthestudenttoreferbacktothetextforassistance. AnswerstoallProciencyExamproblemsareincludedwiththeexerciseswhenviewedonline,orintheback ofthechapterintheprintversion. Content Thewritingstyleisinformalandfriendly,oeringano-nonsense,straightforwardapproachtoalgebra.We havemadeadeliberateeortnottowriteanothertextthatminimizestheuseofwordsbecausewebelieve thatstudentscanbestudyalgebraicconceptsandunderstandalgebraictechniquesbyusingwords and symbolsratherthansymbolsalone.Ithasbeenourexperiencethatstudentsattheelementarylevelarenot experiencedenoughwithmathematicstounderstandsymbolicexplanationsalone;theyalsoneedtoread theexplanation. Wehavetakengreatcaretopresentconceptsandtechniquessotheyareunderstandableandeasily remembered.Afterconceptshavebeendeveloped,studentsarewarnedaboutcommonpitfalls.

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3 ArithmeticReview Thischaptercontainsmanyexamplesofarithmetictechniquesthatareuseddirectlyorindirectlyinalgebra.Sincethechapterisintendedasareview,theproblem-solvingtechniquesarepresentedwithoutbeing developed.Therefore,noworkspaceisprovided,nordoesthechaptercontainallofthepedagogicalfeatures ofthetext.Asareview,thischaptercanbeassignedatthediscretionoftheinstructorandcanalsobea valuablereferencetoolforthestudent. BasicPropertiesofRealNumbers Thesymbols,notations,andpropertiesofnumbersthatformthebasisofalgebra,aswellasexponentsand therulesofexponents,areintroducedinBasicPropertiesofRealNumbers.Eachpropertyofrealnumbers andtherulesofexponentsareexpressedbothsymbolicallyandliterally.Literalexplanationsareincluded becausesymbolicexplanationsalonemaybedicultforastudenttointerpret. BasicOperationswithRealNumbers Thebasicoperationswithrealnumbersarepresentedinthischapter.Theconceptofabsolutevalueis discussedbothgeometricallyandsymbolically.Thegeometricpresentationoersavisualunderstandingof themeaningof j x j .Thesymbolicpresentationincludesaliteralexplanationofhowtousethedenition. Negativeexponentsaredeveloped,usingreciprocalsandtherulesofexponentsthestudenthasalready learned.Scienticnotationisalsoincluded,usinguniqueandreal-lifeexamples. AlgebraicExpressionsandEquations OperationswithalgebraicexpressionsandnumericalevaluationsareintroducedinAlgebraicExpressions andEquations.Coecientsaredescribedratherthanmerelydened.Specialbinomialproductshaveboth literalsymbolicexplanationandsincetheyoccursofrequentlyinmathematics,wehavebeencarefultohelp thestudentrememberthem.Ineachexampleproblem,thestudentistalkedthroughthesymbolicform. SolvingLinearEquationsandInequalities Inthischapter,theemphasisisonthemechanicsofequationsolving,whichclearlyexplainshowtoisolate avariable.Thegoalistohelpthestudentfeelmorecomfortablewithsolvingappliedproblems.Ample opportunityisprovidedforthestudenttopracticetranslatingwordstosymbols,whichisanimportantpart oftheFive-StepMethodofsolvingappliedproblemsdiscussedinSection5.6andSection5.7. FactoringPolynomials Factoringisanessentialskillforsuccessinalgebraandhigherlevelmathematicscourses.Therefore,we havetakengreatcareindevelopingthestudent'sunderstandingofthefactorizationprocess.Thetechnique isconsistentlyillustratedbydisplayinganemptysetofparenthesesanddescribingthethoughtprocessused todiscoverthetermsthataretobeplacedinsidetheparentheses. Thefactoringschemeforspecialproductsispresentedwithbothverbalandsymbolicdescriptions,since notallstudentscaninterpretsymbolicdescriptionsalone.Twotechniques,thestandardtrialanderror method,andthecollectanddiscardmethodamethodsimilartotheacmethod,arepresentedfor factoringtrinomialswithleadingcoecientsdierentfrom1. GraphingLinearEquationsandInequalitiesinOneandTwoVariables Inthischapterthestudentisshownhowgraphsprovideinformationthatisnotalwaysevidentfromthe equationalone.Thechapterbeginsbyestablishingtherelationshipbetweenthevariablesinanequation, thenumberofcoordinateaxesnecessarytoconstructthegraph,andthespatialdimensionofboththe

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4 coordinatesystemandthegraph.Interpretationofgraphsisalsoemphasizedthroughoutthechapter, beginningwiththeplottingofpoints.Theslopeformulaisfullydeveloped,progressingfromverbalphrases tomathematicalexpressions.Theexpressionsarethenformedintoanequationbyexplicitlystatingthata ratioisacomparisonoftwoquantitiesofthesametypee.g.,distance,weight,ormoney.Thisapproach benetsstudentswhotakefuturecoursesthatusegraphstodisplayinformation. Thestudentisshownhowtographlinesusingtheinterceptmethod,thetablemethod,andtheslopeinterceptmethod,aswellashowtodistinguish,byinspection,obliqueandhorizontal/verticallines. RationalExpressions Adetailedstudyofarithmeticoperationswithrationalexpressionsispresentedinthischapter,beginning withthedenitionofarationalexpressionandthenproceedingimmediatelytoadiscussionofthedomain. Theprocessofreducingarationalexpressionandillustrationsofmultiplying,dividing,adding,andsubtractingrationalexpressionsarealsoincluded.Sincetheoperationsofadditionandsubtractioncancause themostdiculty,theyaregivenparticularattention.Wehavetriedtomakethewrittenexplanationof theexamplesclearerbyusingafreezeframeapproach. Theve-stepmethodofsolvingappliedproblemsisincludedinthischaptertoshowtheproblem-solving approachtonumberproblems,workproblems,andgeometryproblems.Thechapteralsoillustratessimplicationofcomplexrationalexpressions,usingthecombine-dividemethodandtheLCD-multiply-divide method. Roots,Radicals,andSquareRootEquations Thedistinctionbetweentheprincipalsquarerootofthenumber x p x ,andthesecondarysquarerootofthe number x p x ,ismadebyexplanationandbyexample.Thesimplicationofradicalexpressionsthatboth involveanddonotinvolvefractionsisshowninmanydetailedexamples;thisisfollowedbyanexplanation ofhowandwhyradicalsareeliminatedfromthedenominatorofaradicalexpression.Real-lifeapplications ofradicalequationshavebeenincluded,suchasproblemsinvolvingdailyoutput,dailysales,electronic resonancefrequency,andkineticenergy. QuadraticEquations Methodsofsolvingquadraticequationsaswellasthelogicunderlyingeachmethodarediscussed.Factoring, extractionofroots,completingthesquare,andthequadraticformulaarecarefullydeveloped.Thezerofactorpropertyofrealnumbersisreintroduced.Thechapteralsoincludesgraphsofquadraticequations basedonthestandardparabola, y = x 2 ,andappliedproblemsfromtheareasofmanufacturing,population, physics,geometry,mathematicsnumberandvolumes,andastronomy,whicharesolvedusingtheve-step method. SystemsofLinearEquations Beginningwiththegraphicalsolutionofsystems,thischapterincludesaninterpretationofindependent, inconsistent,anddependentsystemsandexamplestoillustratetheapplicationsforthesesystems.The substitutionmethodandtheadditionmethodofsolvingasystembyeliminationareexplained,notingwhen touseeachmethod.Theve-stepmethodisagainusedtoillustratethesolutionsofvalueandrateproblems coinandmixtureproblems,usingdrawingsthatcorrespondtotheactualsolution.

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Acknowledgments 2 Manyextraordinarilytalentedpeopleareresponsibleforhelpingtocreatethistext.Wewishtoacknowledge theeortsandskillsofthefollowingmathematicians.Theircontributionshavebeeninvaluable. JeraldT.Ball,ChabotCollege RonL.Bohuslov,CollegeofAlameda AnitaBuker,Miami-DadeCommunityCollege AnnBretscher,UniversityofGeorgia LorenGaither,PaulD.CampCommunityCollege JohnGordon,GeorgiaStateUniversity PatriciaHauss,ArapahoeCommunityCollege JeanHolton,TidewaterCommunityCollege KatherineHuppler,St.CloudStateUniversity BruceJacobs,LaneyCollege DonaldR.Johnson,ScottsdateCommunityCollege JohnLenhert,LongBeachCommunityCollege RolandE.Lentz,MankatoStateUniversity JeanMoran,DonnelleyCollege PatriciaMorgan,SanDiegoStateUniversity CharlesPeselnick,DevryInstituteofTechnology MazinaS.Porter,PaulD.CampCommunityCollege DavidPrice,TarrantCountyJuniorCollege HarveyReynolds,GoldenWestCollege J.DougRichey,NortheastTexasCommunityCollege JoyceL.Riseberg,MontgomeryCollege MarkSaks,CommunityCollegeofPhiladelphia NancyWadlingtonSpears,EverettCommunityCollege MollySumner,PikesPeakCommunityCollege IanWalton,MissionCollege ElizabethM.Wayt,TennesseeStateUniversity JohnWhitcomb,UniversityofNorthDakota Specialthankstothefollowingindividualsfortheircarefulaccuracyreviewsofmanuscript,galleys,and pageproofs:SteveBlasberg,WestValleyCollege;WadeEllisSr.,UniversityofMichigan;JohnR.Martin, TarrantCountyJuniorCollege;JaneEllis,AmyMiller,andGuySanders,BranhamHighSchoolfortheir help. OursincerethankstoDebbieWiedemannforherencouragement,suggestionsconcerningpsychobiological examples,proofreadingmuchofthemanuscript,andtypingmanyofthesectionexercises;SandiWiedermann forcollatingtheannotatedreviews,countingtheexamplesandexercises,andheruntiringuseof"white-out"; andJaneEllisforsolvingandtypingalltheexercisesolutions. 2 Thiscontentisavailableonlineat. 5

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6 Wethankthefollowingpeoplefortheirexcellentworkonthevariousancillaryitemsthataccompanied theoriginalreleaseofElementaryAlgebranotcurrentlyincludedwiththeConnexionsversion:JaneEllis Instructor'sManual;JohnR.Martin,TarrantCountyJuniorCollegeStudentSolutionsManualand StudyGuide;VirginiaHamilton,ShawneeStateUniversityComputerizedTestBank;PatriciaMorgan, SanDiegoStateUniversityPreparedTests;andGeorgeW.Bergeman,NorthernVirginiaCommunity CollegeMAXISInteractiveSoftware. WealsowishtothankthetalentedpeopleatSaundersCollegePublishingwhoseeortsmadethistextrun smoothlyandlesspainfullythanwehadimagined.OurparticularthankstoBobStern,MathematicsEditor; EllenNewman,DevelopmentalEditor;andJanetB.Nuciforo,ProjectEditor.Theirguidance,suggestions, openmindstooursuggestionsandconcerns,andencouragementhavebeenextraordinarilyhelpful.Although thereweretimeswethoughtwemightbepermanentlydamagedfromrereadingandrewriting,theireorts haveimprovedthistextimmensely.Itisapleasuretoworkwithsuchhigh-qualityprofessionals. DennyBurzynski WadeEllis,Jr. SanJose,California IwouldliketothankDougCampbell,EdLodi,andGuySandersforlisteningtomyfrustrationsand encouragingmeon.Thanksalsogotomycousin,DavidRaety,wholongagoinSequoiaNationalForest toldmewhatadierentialequationis. ParticularthanksgotoeachofmycolleaguesatWestValleyCollege.Oureverydayconversations regardingmathematicsinstructionhavebeenoftheutmostimportancetothedevelopmentofthistextand tomyteachingcareer. D.B. Sandi C'estpourtoi,l'toileaucentredemonunivers.

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Chapter1 ArithmeticReview 1.1Objectives 1 Thischaptercontainsmanyexamplesofarithmetictechniquesthatareuseddirectlyorindirectlyinalgebra. Sincethechapterisintendedasareview,theproblem-solvingtechniquesarepresentedwithoutbeingdeveloped.Ifyouwouldlikeaquickreviewofarithmeticbeforeattemptingthestudyofalgebra,thischapteris recommendedreading.Ifyoufeelyourarithmeticskillsareprettygood,thenmoveontoBasicProperties ofRealNumbersSection2.1.Howeveryoufeel,donothesitatetousethischapterasa quickreference ofarithmetictechniques TheotherchaptersincludePracticeSetspairedwithSampleSetswithsucientspaceforthestudent toworkouttheproblems.Inaddition,thesechaptersincludeaSummaryofKeyConcepts,Exercise Supplements,andProciencyExams. 1.2Factors,Products,andExponents 2 1.2.1Overview Factors ExponentialNotation 1.2.2Factors Let'sbeginourreviewofarithmeticbyrecallingthemeaningofmultiplicationforwholenumbersthe countingnumbersandzero. Multiplication Multiplication isadescriptionofrepeatedaddition. Intheaddition 7+7+7+7 thenumber7isrepeatedasan addend* 4 times. Therefore,wesaywehave fourtimesseven anddescribeitbywriting 4 7 1 Thiscontentisavailableonlineat. 2 Thiscontentisavailableonlineat. 7

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8 CHAPTER1.ARITHMETICREVIEW Theraiseddotbetweenthenumbers4and7indicatesmultiplication.Thedotdirectsustomultiplythetwonumbersthatitseparates.Inalgebra,thedotispreferredoverthesymbol todenote multiplicationbecausetheletter x isoftenusedtorepresentanumber.Thus, 4 7=7+7+7+7 FactorsandProducts Inamultiplication,thenumbersbeingmultipliedarecalled factors. Theresultofamultiplicationiscalled the product. Forexample,inthemultiplication 4 7=28 thenumbers4and7arefactors,andthenumber28istheproduct.Wesaythat4and7arefactorsof28.Theyarenottheonlyfactorsof28.Canyouthinkofothers? Nowweknowthat factor factor = product Thisindicatesthatarstnumberisafactorofasecondnumberiftherstnumberdividesintothe secondnumberwithnoremainder.Forexample,since 4 7=28 both4and7arefactorsof28sinceboth4and7divideinto28withnoremainder. 1.2.3ExponentialNotation Quiteoften,aparticularnumberwillberepeatedasafactorinamultiplication.Forexample,inthe multiplication 7 7 7 7 thenumber7isrepeatedasafactor4times.Wedescribethisbywriting 7 4 : Thus, 7 7 7 7=7 4 Therepeatedfactoristhelowernumberthebase,andthenumberrecordinghowmanytimesthe factorisrepeatedisthehighernumberthesuperscript.Thesuperscriptnumberiscalledan exponent. Exponent An exponent isanumberthatrecordshowmanytimesthenumbertowhichitisattachedoccursasa factorinamultiplication. 1.2.4SampleSetA ForExamples1,2,and3,expresseachproductusingexponents. Example1.1 3 3 3 3 3 3 : Since3occursasafactor6times, 3 3 3 3 3 3=3 6

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9 Example1.2 8 8 : Since8occursasafactor2times, 8 8=8 2 Example1.3 5 5 5 9 9 : Since5occursasafactor3times,wehave 5 3 : Since9occursasafactor2times, wehave 9 2 : Weshouldseethefollowingreplacements. 5 5 5 | {z } 5 3 9 9 | {z } 9 2 Thenwehave 5 5 5 9 9=5 3 9 2 Example1.4 Expand 3 5 : Thebaseis3soitistherepeatedfactor.Theexponentis5anditrecordsthenumber oftimesthebase3isrepeated.Thus,3istoberepeatedasafactor5times. 3 5 =3 3 3 3 3 Example1.5 Expand 6 2 10 4 : Thenotation 6 2 10 4 recordsthefollowingtwofacts:6istoberepeatedasa factor2timesand10istoberepeatedasafactor4times.Thus, 6 2 10 4 =6 6 10 10 10 10 1.2.5Exercises Forthefollowingproblems,expresseachproductusingexponents. Exercise1.1 Solutiononp.41. 8 8 8 Exercise1.2 12 12 12 12 12 Exercise1.3 Solutiononp.41. 5 5 5 5 5 5 5 Exercise1.4 1 1 Exercise1.5 Solutiononp.41. 3 3 3 3 3 4 4 Exercise1.6 8 8 8 15 15 15 15 Exercise1.7 Solutiononp.41. 2 2 2 9 9 9 9 9 9 9 9 Exercise1.8 3 3 10 10 10 Exercise1.9 Solutiononp.41. Supposethattheletters x and y areeachusedtorepresentnumbers.Useexponentstoexpress thefollowingproduct.

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10 CHAPTER1.ARITHMETICREVIEW x x x y y Exercise1.10 Supposethattheletters x and y areeachusedtorepresentnumbers.Useexponentstoexpress thefollowingproduct. x x x x x y y y Forthefollowingproblems,expandeachproductdonotcomputetheactualvalue. Exercise1.11 Solutiononp.41. 3 4 Exercise1.12 4 3 Exercise1.13 Solutiononp.41. 2 5 Exercise1.14 9 6 Exercise1.15 Solutiononp.41. 5 3 6 2 Exercise1.16 2 7 3 4 Exercise1.17 Solutiononp.41. x 4 y 4 Exercise1.18 x 6 y 2 Forthefollowingproblems,specifyallthewholenumberfactorsofeachnumber.Forexample,thecomplete setofwholenumberfactorsof6is1,2,3,6. Exercise1.19 Solutiononp.41. 20 Exercise1.20 14 Exercise1.21 Solutiononp.41. 12 Exercise1.22 30 Exercise1.23 Solutiononp.41. 21 Exercise1.24 45 Exercise1.25 Solutiononp.41. 11 Exercise1.26 17 Exercise1.27 Solutiononp.41. 19 Exercise1.28 2

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11 1.3PrimeFactorization 3 1.3.1Overview PrimeAndCompositeNumbers TheFundamentalPrincipleOfArithmetic ThePrimeFactorizationOfAWholeNumber 1.3.2PrimeAndCompositeNumbers Noticethattheonlyfactorsof7are1and7itself,andthattheonlyfactorsof23are1and23itself. PrimeNumber Awholenumbergreaterthan1whoseonlywholenumberfactorsareitselfand1PrimeNumberiscalleda primenumber. Therstsevenprimenumbersare 2,3,4,5,7,11,13,and17 Thenumber1isnotconsideredtobeaprimenumber,andthenumber2istherstandonlyeven primenumber. Manynumbershavefactorsotherthanthemselvesand1.Forexample,thefactorsof28are1,2,4,7, 14,and28sinceeachofthesewholenumbersandonlythesewholenumbersdivideinto28withouta remainder. CompositeNumbers Awholenumberthatiscomposedoffactorsotherthanitselfand1iscalleda compositenumber. Compositenumbersarenotprimenumbers. Somecompositenumbersare4,6,8,10,12,and15. 1.3.3TheFundamentalPrincipleOfArithmetic Primenumbersareveryimportantinthestudyofmathematics.Wewillusethemsooninourstudyof fractions.Wewillnow,however,beintroducedtoanimportantmathematicalprinciple. TheFundamentalPrincipleofArithmetic Exceptfortheorderofthefactors,everywholenumber,otherthan1,canbefactoredinoneandonlyone wayasaproductofprimenumbers. PrimeFactorization Whenanumberisfactoredsothatallitsfactorsareprimenumbers,thefactorizationiscalledthe prime factorization ofthenumber. 1.3.4SampleSetA Example1.6 Findtheprimefactorizationof10. 10=2 5 3 Thiscontentisavailableonlineat.

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12 CHAPTER1.ARITHMETICREVIEW Both2and5areprimenumbers.Thus, 2 5 istheprimefactorizationof10. Example1.7 Findtheprimefactorizationof60. 60=2 30 30isnotprime : 30=2 15 =2 2 1515 isnotprime : 15=3 5 =2 2 3 5 We'lluseexponents. 2 2=2 2 =2 2 3 5 Thenumbers2,3,and5areallprimes.Thus, 2 2 3 5 istheprimefactorizationof60. Example1.8 Findtheprimefactorizationof11. 11isaprimenumber.Primefactorizationappliesonlytocompositenumbers. 1.3.5ThePrimeFactorizationOfAWholeNumber Thefollowingmethodprovidesawayofndingtheprimefactorizationofawholenumber.Theexamples thatfollowwillusethemethodandmakeitmoreclear. 1.Dividethenumberrepeatedlybythesmallestprimenumberthatwilldivideintothenumberwithout aremainder. 2.Whentheprimenumberusedinstep1nolongerdividesintothegivennumberwithoutaremainder, repeattheprocesswiththenextlargestprimenumber. 3.Continuethisprocessuntilthequotientis1. 4.Theprimefactorizationofthegivennumberistheproductofalltheseprimedivisors. 1.3.6SampleSetB Example1.9 Findtheprimefactorizationof60. Since60isanevennumber,itisdivisibleby2.Wewillrepeatedlydivideby2untilweno longercanwhenwestartgettingaremainder.Weshalldivideinthefollowingway. 30isdivisibleby2again. 15isnotdivisibleby2,butisdivisibleby3,thenextlargestprime. 5isnotdivisibleby3,butisdivisibleby5,thenextlargestprime. Thequotientis1sowestopthedivisionprocess. Theprimefactorizationof60istheproductofallthesedivisors.

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13 60=2 2 3 5 Wewilluseexponentswhenpossible : 60=2 2 3 5 Example1.10 Findtheprimefactorizationof441. Since441isanoddnumber,itisnotdivisibleby2.We'lltry3,thenextlargestprime. 147isdivisibleby3 : 49isnotdivisibleby3norby5 ; butby7 : 7isdivisibleby7. Thequotientis1sowestopthedivisionprocess : Theprimefactorizationof441istheproductofallthedivisors. 441=3 3 7 7 Wewilluseexponentswhenpossible : 441=3 2 7 2 1.3.7Exercises Forthefollowingproblems,determinewhichwholenumbersareprimeandwhicharecomposite. Exercise1.29 Solutiononp.41. 23 Exercise1.30 25 Exercise1.31 Solutiononp.41. 27 Exercise1.32 2 Exercise1.33 Solutiononp.41. 3 Exercise1.34 5 Exercise1.35 Solutiononp.41. 7 Exercise1.36 9 Exercise1.37 Solutiononp.41. 11 Exercise1.38 34

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14 CHAPTER1.ARITHMETICREVIEW Exercise1.39 Solutiononp.41. 55 Exercise1.40 63 Exercise1.41 Solutiononp.41. 1044 Exercise1.42 339 Exercise1.43 Solutiononp.41. 209 Forthefollowingproblems,ndtheprimefactorizationofeachwholenumber.Useexponentsonrepeated factors. Exercise1.44 26 Exercise1.45 Solutiononp.41. 38 Exercise1.46 54 Exercise1.47 Solutiononp.41. 62 Exercise1.48 56 Exercise1.49 Solutiononp.41. 176 Exercise1.50 480 Exercise1.51 Solutiononp.41. 819 Exercise1.52 2025 Exercise1.53 Solutiononp.42. 148,225 1.4TheLeastCommonMultiple 4 1.4.1Overview Multiples CommonMultiples TheLeastCommonMultipleLCM FindingTheLeastCommonMultiple 4 Thiscontentisavailableonlineat.

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15 1.4.2Multiples Multiples Whenawholenumberismultipliedbyotherwholenumbers,withtheexceptionofMultipleszero,the resultingproductsarecalled multiples ofthegivenwholenumber. Multiplesof2 Multiplesof3 Multiplesof8 Multiplesof10 2 1=2 3 1=3 8 1=8 10 1=10 2 2=4 3 2=6 8 2=16 10 2=20 2 3=6 3 3=9 8 3=24 10 3=30 2 4=8 3 4=12 8 4=32 10 4=40 2 5=10 3 5=15 8 5=40 10 5=50 ::: ::: ::: ::: Table1.1 1.4.3CommonMultiples Therewillbetimeswhenwearegiventwoormorewholenumbersandwewillneedtoknowifthereareany multiplesthatarecommontoeachofthem.Ifthereare,wewillneedtoknowwhattheyare.Forexample, someofthemultiplesthatarecommonto2and3are6,12,and18. 1.4.4SampleSetA Example1.11 Wecanvisualizecommonmultiplesusingthenumberline. Noticethatthecommonmultiplescanbedividedbybothwholenumbers. 1.4.5TheLeastCommonMultipleLCM Noticethatinournumberlinevisualizationofcommonmultiplesabovetherstcommonmultipleisalso thesmallest,or leastcommonmultiple, abbreviatedby LCM. LeastCommonMultiple The leastcommonmultiple,LCM, oftwoormorewholenumbersisthesmallestwholenumberthat eachofthegivennumberswilldivideintowithoutaremainder.

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16 CHAPTER1.ARITHMETICREVIEW 1.4.6FindingTheLeastCommonMultiple FindingtheLCM TondtheLCMoftwoormorenumbers, 1.Writetheprimefactorizationofeachnumber,usingexponentsonrepeatedfactors. 2.Writeeachbasethatappearsineachoftheprimefactorizations. 3.Toeachbase,attachthelargestexponentthatappearsonitintheprimefactorizations. 4.TheLCMistheproductofthenumbersfoundinstep3. 1.4.7SampleSetB FindtheLCMofthefollowingnumber. Example1.12 9and12 1. 9=3 3=3 2 12=2 6=2 2 3=2 2 3 2.Thebasesthatappearintheprimefactorizationsare2and3. 3.Thelargestexponentsappearingon2and3intheprimefactorizationsare,respectively,2 and2or 2 2 from12,and 3 2 from9. 4.TheLCMistheproductofthesenumbers. LCM =2 2 3 2 =4 9=36 Thus,36isthesmallestnumberthatboth9and12divideintowithoutremainders. Example1.13 90and630 1. 90=2 45=2 3 15=2 3 3 5=2 3 2 5 630=2 315=2 3 105=2 3 3 35=2 3 3 5 7 =2 3 2 5 7 2.Thebasesthatappearintheprimefactorizationsare2,3,5,and7. 3.Thelargestexponentsthatappearon2,3,5,and7are,respectively,1,2,1,and1. 2 1 fromeither9 0 or63 0 3 2 fromeither9 0 or63 0 5 1 fromeither9 0 or63 0 7 1 from63 0 4.TheLCMistheproductofthesenumbers. LCM =2 3 2 5 7=2 9 5 7=630 Thus,630isthesmallestnumberthatboth90and630divideintowithnoremainders. Example1.14 33,110,and484

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17 1. 33=3 11 110=2 55=2 5 11 484=2 242=2 2 121=2 2 11 11=2 2 11 2 2.Thebasesthatappearintheprimefactorizationsare2,3,5,and11. 3.Thelargestexponentsthatappearon2,3,5,and11are,respectively,2,1,1,and2. 2 2 from 484 3 1 from 33 5 1 from 110 11 2 from 484 4.TheLCMistheproductofthesenumbers. LCM =2 2 3 5 11 2 =4 3 5 121 =7260 Thus,7260isthesmallestnumberthat33,110,and484divideintowithoutremainders. 1.4.8Exercises Forthefollowingproblems,ndtheleastcommonmultipleofgivennumbers. Exercise1.54 Solutiononp.42. 8,12 Exercise1.55 8,10 Exercise1.56 Solutiononp.42. 6,12 Exercise1.57 9,18 Exercise1.58 Solutiononp.42. 5,6 Exercise1.59 7,9 Exercise1.60 Solutiononp.42. 28,36 Exercise1.61 24,36 Exercise1.62 Solutiononp.42. 28,42 Exercise1.63 20,24 Exercise1.64 Solutiononp.42. 25,30

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18 CHAPTER1.ARITHMETICREVIEW Exercise1.65 24,54 Exercise1.66 Solutiononp.42. 16,24 Exercise1.67 36,48 Exercise1.68 Solutiononp.42. 15,21 Exercise1.69 7,11,33 Exercise1.70 Solutiononp.42. 8,10,15 Exercise1.71 4,5,21 Exercise1.72 Solutiononp.42. 45,63,98 Exercise1.73 15,25,40 Exercise1.74 Solutiononp.42. 12,16,20 Exercise1.75 12,16,24 Exercise1.76 Solutiononp.42. 12,16,24,36 Exercise1.77 6,9,12,18 Exercise1.78 Solutiononp.42. 8,14,28,32 1.5EquivalentFractions 5 1.5.1Overview EquivalentFractions ReducingFractionsToLowestTerms RaisingFractionsToHigherTerms 5 Thiscontentisavailableonlineat.

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19 1.5.2EquivalentFractions EquivalentFractions Fractionsthathavethesamevaluearecalled equivalentfractions. Forexample, 2 3 and 4 6 representthesamepartofawholequantityandarethereforeequivalent.Several morecollectionsofequivalentfractionsarelistedbelow. Example1.15 15 25 ; 12 20 ; 3 5 Example1.16 1 3 ; 2 6 ; 3 9 ; 4 12 Example1.17 7 6 ; 14 12 ; 21 18 ; 28 24 ; 35 30 1.5.3ReducingFractionsToLowestTerms ReducedtoLowestTerms Itisoftenusefultoconvertonefractiontoanequivalentfractionthathasreducedvaluesinthenumerator anddenominator.Whenafractionisconvertedtoanequivalentfractionthathasthesmallestnumerator anddenominatorinthecollectionofequivalentfractions,itissaidtobe reducedtolowestterms. The conversionprocessiscalled reducingafraction. Wecanreduceafractiontolowesttermsby 1.Expressingthenumeratoranddenominatorasaproductofprimenumbers.Findtheprimefactorizationofthenumeratoranddenominator.SeeSectionSection1.3forthistechnique. 2.Dividethenumeratoranddenominatorbyallcommonfactors.Thistechniqueiscommonlycalled cancelling. 1.5.4SampleSetA Reduceeachfractiontolowestterms. Example1.18 6 18 = 2 3 2 3 3 = 3 2and3arecommonfactors : = 1 3 Example1.19 16 20 = 2 2 2 2 2 2 5 = 2 2 5 2istheonlycommonfactor. = 4 5 Example1.20 56 70 = 2 4 7 2 5 7 = 4 5 2and7arecommonfactors : = 4 5

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20 CHAPTER1.ARITHMETICREVIEW Example1.21 8 15 = 2 2 2 3 5 Therearenocommonfactors : Thus ; 8 15 isreducedtolowestterms : 1.5.5RaisingaFractiontoHigherTerms Equallyimportantasreducingfractionsis raisingfractionstohigherterms. Raisingafractionto highertermsistheprocessofconstructinganequivalentfractionthathashighervaluesinthenumeratoranddenominator.Thehigher,equivalentfractionisconstructedbymultiplyingtheoriginalfractionby1. Noticethat 3 5 and 9 15 areequivalent,thatis 3 5 = 9 15 : Also, Thisobservationhelpsussuggestthefollowingmethodforraisingafractiontohigherterms. RaisingaFractiontoHigherTerms Afractioncanberaisedtohighertermsbymultiplyingboththenumeratoranddenominatorbythesame nonzeronumber. Forexample, 3 4 canberaisedto 24 32 bymultiplyingboththenumeratoranddenominatorby8,thatis, multiplyingby1intheform 8 8 : 3 4 = 3 8 4 8 = 24 32 Howdidweknowtochoose8astheproperfactor?Sincewewishtoconvert4to32bymultiplyingitbysomenumber,weknowthat4mustbeafactorof32.Thismeansthat4dividesinto32.Infact, 32 4=8 : Wedividedtheoriginaldenominatorintothenew,specieddenominatortoobtaintheproper factorforthemultiplication. 1.5.6SampleSetB Determinethemissingnumeratorordenominator. Example1.22 3 7 = ? 35 : Dividetheoriginaldenominator ; 7 ; intothenewdenominator ; 35 : 35 7=5 : Multiplytheoriginalnumeratorby5 : 3 7 = 3 5 7 5 = 15 35 Example1.23 5 6 = 45 ? : Dividetheoriginalnumerator ; 5 ; intothenewnumerator ; 45 : 45 5=9 : Multiplytheoriginaldenominatorby9 : 5 6 = 5 9 6 9 = 45 54

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21 1.5.7Exercises Forthefollowingproblems,reduce,ifpossible,eachfractionlowestterms. Exercise1.79 Solutiononp.42. 6 8 Exercise1.80 5 10 Exercise1.81 Solutiononp.42. 6 14 Exercise1.82 4 14 Exercise1.83 Solutiononp.42. 18 12 Exercise1.84 20 8 Exercise1.85 Solutiononp.42. 10 6 Exercise1.86 14 4 Exercise1.87 Solutiononp.42. 10 12 Exercise1.88 32 28 Exercise1.89 Solutiononp.42. 36 10 Exercise1.90 26 60 Exercise1.91 Solutiononp.42. 12 18 Exercise1.92 18 27 Exercise1.93 Solutiononp.42. 18 24 Exercise1.94 32 40 Exercise1.95 Solutiononp.42. 11 22 Exercise1.96 17 51 Exercise1.97 Solutiononp.42. 27 81 Exercise1.98 16 42

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22 CHAPTER1.ARITHMETICREVIEW Exercise1.99 Solutiononp.42. 39 13 Exercise1.100 44 11 Exercise1.101 Solutiononp.43. 121 132 Exercise1.102 30 105 Exercise1.103 Solutiononp.43. 108 76 Forthefollowingproblems,determinethemissingnumeratorordenominator. Exercise1.104 1 3 = ? 12 Exercise1.105 Solutiononp.43. 1 5 = ? 30 Exercise1.106 3 3 = ? 9 Exercise1.107 Solutiononp.43. 3 4 = ? 16 Exercise1.108 5 6 = ? 18 Exercise1.109 Solutiononp.43. 4 5 = ? 25 Exercise1.110 1 2 = 4 ? Exercise1.111 Solutiononp.43. 9 25 = 27 ? Exercise1.112 3 2 = 18 ? Exercise1.113 Solutiononp.43. 5 3 = 80 ? 1.6OperationswithFractions 6 1.6.1Overview MultiplicationofFractions DivisionofFractions AdditionandSubtractionofFractions 6 Thiscontentisavailableonlineat.

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23 1.6.2MultiplicationofFractions MultiplicationofFractions Tomultiplytwofractions,multiplythenumeratorstogetherandmultiplythedenominatorstogether.Reduce tolowesttermsifpossible. Example1.24 Forexample,multiply 3 4 1 6 : 3 4 1 6 = 3 1 4 6 = 3 24 Nowreduce : = 3 1 2 2 2 3 = 1 2 2 2 3istheonlycommonfactor : = 1 8 Noticethatwesincehadtoreduce,wenearlystartedoveragainwiththeoriginaltwofractions.If wefactorrst,thencancel,thenmultiply,wewillsavetimeandenergyandstillobtainthecorrect product. 1.6.3SampleSetA Performthefollowingmultiplications. Example1.25 1 4 8 9 = 1 2 2 2 2 2 3 3 = 1 2 3 3 2isacommonfactor. = 1 1 2 3 3 = 1 2 1 3 3 = 2 9 Example1.26 3 4 8 9 5 12 = 3 2 2 2 2 2 3 3 5 2 2 3 = 3 5 2 3 2and3arecommonfactors. = 1 1 5 3 2 3 = 5 18 1.6.4DivisionofFractions Reciprocals Twonumberswhoseproductis1are reciprocals ofeachother.Forexample,since 4 5 5 4 =1 ; 4 5 and 5 4 are reciprocalsofeachother.Someotherpairsofreciprocalsarelistedbelow.

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24 CHAPTER1.ARITHMETICREVIEW 2 7 ; 7 2 3 4 ; 4 3 6 1 ; 1 6 Reciprocalsareusedindivisionoffractions. DivisionofFractions Todividearstfractionbyasecondfraction,multiplytherstfractionbythereciprocalofthesecond fraction.Reduceifpossible. Thismethodissometimescalledtheinvertandmultiplymethod. 1.6.5SampleSetB Performthefollowingdivisions. Example1.27 1 3 3 4 : Thedivisoris 3 4 .Itsreciprocalis 4 3 : 1 3 3 4 = 1 3 4 3 = 1 4 3 3 = 4 9 Example1.28 3 8 5 4 : Thedivisoris 5 4 .Itsreciprocalis 4 5 : 3 8 5 4 = 3 8 4 5 = 3 2 2 2 2 2 5 = 3 2 5 2isacommonfactor. = 3 1 2 5 = 3 10 Example1.29 5 6 5 12 : Thedivisoris 5 12 .Itsreciprocalis 12 5 : 5 6 5 12 = 5 6 12 5 = 5 2 3 2 2 3 5 = 2 = 1 2 1 =2

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25 1.6.6AdditionandSubtractionofFractions FractionswithLikeDenominators Toaddorsubtracttwoormorefractionsthathavethesamedenominators,addorsubtractthenumerators andplacetheresultingsumoverthecommondenominator.Reduceifpossible. CAUTION Addorsubtractonlythenumerators.Do not addorsubtractthedenominators! 1.6.7SampleSetC Findthefollowingsums. Example1.30 3 7 + 2 7 : Thedenominatorsarethesame : Addthenumeratorsandplacethesumover7 : 3 7 + 2 7 = 3+2 7 = 5 7 Example1.31 7 9 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(4 9 : Thedenominatorsarethesame : Subtract4from7andplacethedierenceover9 : 7 9 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(4 9 = 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 9 = 3 9 = 1 3 1.6.8 Fractionscanonlybeaddedorsubtractedconvenientlyiftheyhavelikedenominators. FractionswithUnlikeDenominators Toaddorsubtractfractionshavingunlikedenominators,converteachfractiontoanequivalentfraction havingasthedenominatortheleastcommonmultipleoftheoriginaldenominators. Theleastcommonmultipleoftheoriginaldenominatorsiscommonlyreferredtoasthe leastcommon denominator LCD.SeeSectionSection1.4forthetechniqueofndingtheleastcommonmultipleof severalnumbers. 1.6.9SampleSetD Findeachsumordierence. Example1.32 1 6 + 3 4 : Thedenominatorsarenotalike : FindtheLCDof6and4 : f 6=2 3 4=2 2 TheLCDis 2 2 3=4 3=12 : Converteachoftheoriginalfractionstoequivalentfractionshavingthecommondenominator12. 1 6 = 1 2 6 2 = 2 12 3 4 = 3 3 4 3 = 9 12 Nowwecanproceedwiththeaddition : 1 6 + 3 4 = 2 12 + 9 12 = 2+9 12 = 11 12

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26 CHAPTER1.ARITHMETICREVIEW Example1.33 5 9 )]TJ/F7 6.9738 Tf 13.144 3.922 Td [(5 12 : Thedenominatorsarenotalike : FindtheLCDof9and12 : f 9=3 2 12=2 2 3 TheLCDis 2 2 3 2 =4 9=36 : Converteachoftheoriginalfractionstoequivalentfractionshavingthecommondenominator36. 5 9 = 5 4 9 4 = 20 36 5 12 = 5 3 12 3 = 15 36 Nowwecanproceedwiththesubtraction : 5 9 )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(5 12 = 20 36 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(15 36 = 20 )]TJ/F7 6.9738 Tf 6.226 0 Td [(15 36 = 5 36 1.6.10Exercises Forthefollowingproblems,performeachindicatedoperation. Exercise1.114 Solutiononp.43. 1 3 4 3 Exercise1.115 1 3 2 3 Exercise1.116 Solutiononp.43. 2 5 5 6 Exercise1.117 5 6 14 15 Exercise1.118 Solutiononp.43. 9 16 20 27 Exercise1.119 35 36 48 55 Exercise1.120 Solutiononp.43. 21 25 15 14 Exercise1.121 76 99 66 38 Exercise1.122 Solutiononp.43. 3 7 14 18 6 2 Exercise1.123 14 15 21 28 45 7 Exercise1.124 Solutiononp.43. 5 9 5 6 Exercise1.125 9 16 15 8 Exercise1.126 Solutiononp.43. 4 9 6 15 Exercise1.127 25 49 4 9

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27 Exercise1.128 Solutiononp.43. 15 4 27 8 Exercise1.129 24 75 8 15 Exercise1.130 Solutiononp.43. 57 8 7 8 Exercise1.131 7 10 10 7 Exercise1.132 Solutiononp.43. 3 8 + 2 8 Exercise1.133 3 11 + 4 11 Exercise1.134 Solutiononp.43. 5 12 + 7 12 Exercise1.135 11 16 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(2 16 Exercise1.136 Solutiononp.43. 15 23 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(2 23 Exercise1.137 3 11 + 1 11 + 5 11 Exercise1.138 Solutiononp.43. 16 20 + 1 20 + 2 20 Exercise1.139 3 8 + 2 8 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 8 Exercise1.140 Solutiononp.43. 11 16 + 9 16 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(5 16 Exercise1.141 1 2 + 1 6 Exercise1.142 Solutiononp.43. 1 8 + 1 2 Exercise1.143 3 4 + 1 3 Exercise1.144 Solutiononp.43. 5 8 + 2 3 Exercise1.145 6 7 )]TJ/F7 6.9738 Tf 11.159 3.922 Td [(1 4 Exercise1.146 Solutiononp.43. 8 15 )]TJ/F7 6.9738 Tf 13.143 3.922 Td [(3 10 Exercise1.147 1 15 + 5 12 Exercise1.148 Solutiononp.43. 25 36 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(7 10 Exercise1.149 9 28 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(4 45 Exercise1.150 Solutiononp.44. 8 15 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(3 10

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28 CHAPTER1.ARITHMETICREVIEW Exercise1.151 1 16 + 3 4 )]TJ/F7 6.9738 Tf 11.159 3.923 Td [(3 8 Exercise1.152 Solutiononp.44. 8 3 )]TJ/F7 6.9738 Tf 11.159 3.923 Td [(1 4 + 7 36 Exercise1.153 3 4 )]TJ/F7 6.9738 Tf 13.144 3.922 Td [(3 22 + 5 24 1.7DecimalFractions 7 1.7.1Overview DecimalFractions AddingandSubtractingDecimalFractions MultiplyingDecimalFractions DividingDecimalFractions ConvertingDecimalFractionstoFractions ConvertingFractionstoDecimalFractions 1.7.2DecimalFractions Fractionsareonewaywecanrepresentpartsofwholenumbers.Decimalfractionsareanotherwayof representingpartsofwholenumbers. DecimalFractions A decimalfraction isafractioninwhichthedenominatorisapowerof10. Adecimalfractionusesa decimalpoint toseparatewholepartsandfractionalparts.Wholepartsare writtentothe left ofthedecimalpointandfractionalpartsarewrittentothe right ofthedecimalpoint. Justaseachdigitinawholenumberhasaparticularvalue,sodothedigitsindecimalpositions. 1.7.3SampleSetA Thefollowingnumbersaredecimalfractions. Example1.34 57 : 9 The9isinthe tenths position : 57 : 9=57 9 10 : 7 Thiscontentisavailableonlineat.

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29 Example1.35 6 : 8014 The8isinthe tenths position. The0isinthe hundredths position. The1isinthe thousandths position. The4isintheten thousandths position. 6 : 8014=6 8014 10000 : 1.7.4AddingandSubtractingDecimalFractions Adding/SubtractingDecimalFractions Toaddorsubtractdecimalfractions, 1.Alignthenumbersverticallysothatthedecimalpointslineupundereachotherandcorresponding decimalpositionsareinthesamecolumn.Addzerosifnecessary. 2.Addorsubtractthenumbersasiftheywerewholenumbers. 3.Placeadecimalpointintheresultingsumordierencedirectlyundertheotherdecimalpoints. 1.7.5SampleSetB Findeachsumordierence. Example1.36 9 : 183+2 : 140 # Thedecimalpointsarealignedinthesamecolumn. 9.183 +2.140 11.323 Example1.37 841 : 0056+47 : 016+19 : 058 # Thedecimalpointsarealignedinthesamecolumn. 841 : 0056 47 : 016 Placea0intothethousandthsposition. +19 : 058 Placea0intothethousandthsposition. # Thedecimalpointsarealignedinthesamecolumn. 841 : 0056 47 : 0160 +19 : 0580 907 : 0796

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30 CHAPTER1.ARITHMETICREVIEW Example1.38 16 : 01 )]TJ/F8 9.9626 Tf 11.656 0 Td [(7 : 053 # Thedecimalpointsarealignedinthesamecolumn. 16 : 01 Placea0intothethousandthsposition. )]TJ/F8 9.9626 Tf 9.442 0 Td [(7 : 053 # Thedecimalpointsarealignedinthesamecolumn. 16 : 010 )]TJ/F8 9.9626 Tf 9.442 0 Td [(7 : 053 8 : 957 1.7.6MultiplyingDecimalFractions MultiplyingDecimalFractions Tomultiplydecimals, 1.Multiplytbenumbersasiftheywerewholenumbers. 2.Findthesumofthenumberofdecimalplacesinthefactors. 3.Thenumberofdecimalplacesintheproductisthesumfoundinstep2. 1.7.7SampleSetC Findthefollowingproducts. Example1.39 6 : 5 4 : 3 6 : 5 4 : 3=27 : 95 Example1.40 23 : 4 1 : 96

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31 23 : 4 1 : 96=45 : 864 1.7.8DividingDecimalFractions DividingDecimalFractions Todivideadecimalbyanonzerodecimal, 1.Convertthedivisortoawholenumberbymovingthedecimalpointtothepositionimmediatelyto therightofthedivisor'slastdigit. 2.Movethedecimalpointofthedividendtotherightthesamenumberofdigitsitwasmovedinthe divisor. 3.Setthedecimalpointinthequotientbyplacingadecimalpointdirectlyabovethedecimalpointin thedividend. 4.Divideasusual. 1.7.9SampleSetD Findthefollowingquotients. Example1.41 32 : 66 7 : 1 32 : 66 7 : 1=4 : 6 Check :32 : 66 7 : 1=4 : 6 if 4 : 6 7 : 1=32 : 66 4 : 6 7 : 1 4 : 6 322 32 : 66 True

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32 CHAPTER1.ARITHMETICREVIEW Example1.42 Checkbymultiplying 2 : 1 and 0 : 513 : Thiswillshowthatwehaveobtainedthecorrectresult. Example1.43 12 0 : 00032 1.7.10ConvertingDecimalFractionstoFractions Wecanconvertadecimalfractiontoafractionbyreadingitandthenwritingthephrasewehavejustread. Aswereadthedecimalfraction,wenotetheplacevaluefarthesttotheright.Wemayhavetoreducethe fraction. 1.7.11SampleSetE Converteachdecimalfractiontoafraction. Example1.44 0 : 6 0 : 6 tenthsposition Reading:sixtenths 6 10 Reduce: 0 : 6= 6 10 = 3 5

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33 Example1.45 21 : 903 21 : 903 thousandthsposition Reading:twenty-oneandninehundredthreethousandths 21 903 1000 1.7.12ConvertingFractionstoDecimalFractions 1.7.13SampleSetF Convertthefollowingfractionstodecimals.Ifthedivisionisnonterminating,roundto2decimalplaces. Example1.46 3 4 3 4 =0 : 75 Example1.47 1 5 1 5 =0 : 2 Example1.48 5 6 5 6 =0 : 833 ::: Wearetoroundto2decimalplaces : 5 6 =0 : 83 to2decimalplaces. Example1.49 5 1 8 Notethat 5 1 8 =5+ 1 8 :

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34 CHAPTER1.ARITHMETICREVIEW 1 8 = : 125 Thus, 5 1 8 =5+ 1 8 =5+ : 125=5 : 125 : Example1.50 0 : 16 1 4 Thisisacomplexdecimal.Theisinthehundredthsposition.Thenumber 0 : 16 1 4 is readassixteenandone-fourthhundredths. 0 : 16 1 4 = 16 1 4 100 = 16 4+1 4 100 = 65 4 100 1 = 13 4 1 20 = 13 1 4 20 = 13 80 Now,convert 13 80 toadecimal. 0 : 16 1 4 =0 : 1625 : 1.7.14Exercises Forthefollowingproblems,performeachindicatedoperation. Exercise1.154 Solutiononp.44. 1 : 84+7 : 11 Exercise1.155 15 : 015 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 : 527 Exercise1.156 Solutiononp.44. 4 : 904 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : 67 Exercise1.157 156 : 33 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 : 095 Exercise1.158 Solutiononp.44. : 0012+1 : 53+5 : 1

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35 Exercise1.159 44 : 98+22 : 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 : 76 Exercise1.160 Solutiononp.44. 5 : 0004 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 : 00004+1 : 6837 Exercise1.161 1 : 11+12 : 1212 )]TJ/F8 9.9626 Tf 9.963 0 Td [(13 : 131313 Exercise1.162 Solutiononp.44. 4 : 26 3 : 2 Exercise1.163 2 : 97 3 : 15 Exercise1.164 Solutiononp.44. 23 : 05 1 : 1 Exercise1.165 5 : 009 2 : 106 Exercise1.166 Solutiononp.44. 0 : 1 3 : 24 Exercise1.167 100 12 : 008 Exercise1.168 Solutiononp.44. 1000 12 : 008 Exercise1.169 10 ; 000 12 : 008 Exercise1.170 Solutiononp.44. 75 : 642 18 : 01 Exercise1.171 51 : 811 1 : 97 Exercise1.172 Solutiononp.44. 0 : 0000448 0 : 014 Exercise1.173 0 : 129516 1004 Forthefollowingproblems,converteachdecimalfractiontoafraction. Exercise1.174 Solutiononp.44. 0 : 06 Exercise1.175 0 : 115 Exercise1.176 Solutiononp.44. 3 : 7 Exercise1.177 48 : 1162 Exercise1.178 Solutiononp.44. 712 : 00004 Forthefollowingproblems,converteachfractiontoadecimalfraction.Ifthedecimalformisnonterminating,roundto3decimalplaces. Exercise1.179 5 8

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36 CHAPTER1.ARITHMETICREVIEW Exercise1.180 Solutiononp.44. 9 20 Exercise1.181 15 22 Exercise1.182 Solutiononp.44. 7 11 Exercise1.183 2 9 1.8Percent 8 1.8.1Overview TheMeaningofPercent ConvertingAFractionToAPercent ConvertingADecimalToAPercent ConvertingAPercentToADecimal 1.8.2TheMeaningofPercent Theword percent comesfromtheLatinwordpercentum,permeaningforeach,andcentummeaning hundred. Percent% Percent meansforeachhundredorforeveryhundred.Thesymbol%isusedtorepresenttheword percent. Thus, 1%= 1 100 or 1%=0 : 01 : 1.8.3ConvertingAFractionToAPercent Wecanseehowafractioncanbeconvertedtoapercentbyanalyzingthemethodthat 3 5 isconvertedtoa percent.Inordertoconvert 3 5 toapercent,weneedtointroduce 1 100 sincepercentmeansforeachhundred. Example1.51 3 5 = 3 5 100 100 Multiplythefractionby1 : = 3 5 100 1 100 Since 100 100 =100 1 100 : = 300 5 1 100 Divide 300 by5 : =60 1 100 Multiplythefractions : =60% Replace 1 100 withthe % symbol : FractiontoPercent Toconvertafractiontoapercent,multiplythefractionby1intheform 100 1 100 ,thenreplace 1 100 with the%symbol. 8 Thiscontentisavailableonlineat.

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37 1.8.4SampleSetA Converteachfractiontoapercent. Example1.52 1 4 = 1 4 100 1 100 = 100 4 1 100 =25 1 100 =25% Example1.53 8 5 = 8 5 100 1 100 = 800 5 1 100 =160% Example1.54 4 9 = 4 9 100 1 100 = 400 9 1 100 = : 4 ::: 1 100 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(44 : 4 1 100 =44 : 4% 1.8.5ConvertingADecimalToAPercent Wecanseehowadecimalisconvertedtoapercentbyanalyzingthemethodthat 0 : 75 isconvertedtoa percent.Weneedtointroduce 1 100 : 0 : 75=0 : 75 100 1 100 Multiplythedecimalby1. =75 1 100 =75% Replace 1 100 withthe%symbol : DecimaltoPercent Toconvertafractiontoapercent,multiplythedecimalby1intheform 100 1 100 ,thenreplace 1 100 with the%symbol.Thisamountstomovingthedecimalpoint2placestotheright. 1.8.6SampleSetB Converteachdecimaltoapercent. Example1.55 0 : 62=0 : 62 100 1 100 =62 1 100 =62%

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38 CHAPTER1.ARITHMETICREVIEW Noticethatthedecimalpointintheoriginalnumberhasbeenmovedtotheright2places. Example1.56 8 : 4=8 : 4 100 1 100 =840 1 100 =840% Noticethatthedecimalpointintheoriginalnumberhasbeenmovedtotheright2places. Example1.57 0 : 47623=0 : 47623 100 1 100 =0 : 47623 1 100 =47 : 623% Noticethatthedecimalpointintheoriginalnumberhasbeenmovedtotheright2places. 1.8.7ConvertingAPercentToADecimal Wecanseehowapercentisconvertedtoadecimalbyanalyzingthemethodthat12%isconvertedtoa decimal.Weneedtointroduce 1 100 : 12%=12 1 100 Replace % with 1 100 : = 12 100 Multiplythefractions : =0 : 12 Divide12by1 00 : PercenttoDecimal Toconvertapercenttoadecimal,replacethe%symbolwith 1 100 ; thendividethenumberby100.This amountstomovingthedecimalpoint2placestotheleft. 1.8.8SampleSetC Converteachpercenttoadecimal. Example1.58 48%=48 1 100 = 48 100 =0 : 48 Noticethatthedecimalpointintheoriginalnumberhasbeenmovedtotheleft2places.

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39 Example1.59 659%=659 1 100 = 659 100 =6 : 59 Noticethatthedecimalpointintheoriginalnumberhasbeenmovedtotheleft2places. Example1.60 0 : 4113%=0 : 4113 1 100 = 0 : 4113 100 =0 : 004113 Noticethatthedecimalpointintheoriginalnumberhasbeenmovedtotheleft2places. 1.8.9Exercises Forthefollowingproblems,converteachfractiontoapercent. Exercise1.184 Solutiononp.44. 2 5 Exercise1.185 7 8 Exercise1.186 Solutiononp.44. 1 8 Exercise1.187 5 16 Exercise1.188 Solutiononp.44. 15 22 Exercise1.189 2 11 Exercise1.190 Solutiononp.44. 2 9 Exercise1.191 16 45 Exercise1.192 Solutiononp.44. 27 55 Exercise1.193 7 27 Exercise1.194 Solutiononp.44. 15 Exercise1.195 8 Forthefollowingproblems,converteachdecimaltoapercent.

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40 CHAPTER1.ARITHMETICREVIEW Exercise1.196 Solutiononp.44. 0 : 36 Exercise1.197 0 : 42 Exercise1.198 Solutiononp.44. 0 : 446 Exercise1.199 0 : 1298 Exercise1.200 Solutiononp.45. 4 : 25 Exercise1.201 5 : 875 Exercise1.202 Solutiononp.45. 86 : 98 Exercise1.203 21 : 26 Exercise1.204 Solutiononp.45. 14 Exercise1.205 12 Forthefollowingproblems,converteachpercenttoadecimal. Exercise1.206 Solutiononp.45. 35% Exercise1.207 76% Exercise1.208 Solutiononp.45. 18 : 6% Exercise1.209 67 : 2% Exercise1.210 Solutiononp.45. 9 : 0145% Exercise1.211 3 : 00156% Exercise1.212 Solutiononp.45. 0 : 00005% Exercise1.213 0 : 00034%

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41 SolutionstoExercisesinChapter1 SolutiontoExercise1.1p.9 8 3 SolutiontoExercise1.3p.9 5 7 SolutiontoExercise1.5p.9 3 5 4 2 SolutiontoExercise1.7p.9 2 3 9 8 SolutiontoExercise1.9p.9 x 3 y 2 SolutiontoExercise1.11p.10 3 3 3 3 SolutiontoExercise1.13p.10 2 2 2 2 2 SolutiontoExercise1.15p.10 5 5 5 6 6 SolutiontoExercise1.17p.10 x x x x y y y y SolutiontoExercise1.19p.10 1 ; 2 ; 4 ; 5 ; 10 ; 20 SolutiontoExercise1.21p.10 1 ; 2 ; 3 ; 4 ; 6 ; 12 SolutiontoExercise1.23p.10 1 ; 3 ; 7 ; 21 SolutiontoExercise1.25p.10 1 ; 11 SolutiontoExercise1.27p.10 1 ; 19 SolutiontoExercise1.29p.13 prime SolutiontoExercise1.31p.13 composite SolutiontoExercise1.33p.13 prime SolutiontoExercise1.35p.13 prime SolutiontoExercise1.37p.13 prime SolutiontoExercise1.39p.14 composite SolutiontoExercise1.41p.14 composite SolutiontoExercise1.43p.14 composite SolutiontoExercise1.45p.14 2 19 SolutiontoExercise1.47p.14 2 31 SolutiontoExercise1.49p.14 2 4 11

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42 CHAPTER1.ARITHMETICREVIEW SolutiontoExercise1.51p.14 3 2 7 13 SolutiontoExercise1.53p.14 5 2 7 2 11 2 SolutiontoExercise1.54p.17 2 3 3 SolutiontoExercise1.56p.17 2 2 3 SolutiontoExercise1.58p.17 2 3 5 SolutiontoExercise1.60p.17 2 2 3 2 7 SolutiontoExercise1.62p.17 2 2 3 7 SolutiontoExercise1.64p.17 2 3 5 2 SolutiontoExercise1.66p.18 2 4 3 SolutiontoExercise1.68p.18 3 5 7 SolutiontoExercise1.70p.18 2 3 3 5 SolutiontoExercise1.72p.18 2 3 2 5 7 2 SolutiontoExercise1.74p.18 2 4 3 5 SolutiontoExercise1.76p.18 2 4 3 2 SolutiontoExercise1.78p.18 2 5 7 SolutiontoExercise1.79p.21 3 4 SolutiontoExercise1.81p.21 3 7 SolutiontoExercise1.83p.21 3 2 SolutiontoExercise1.85p.21 5 3 SolutiontoExercise1.87p.21 5 6 SolutiontoExercise1.89p.21 18 5 SolutiontoExercise1.91p.21 2 3 SolutiontoExercise1.93p.21 3 4 SolutiontoExercise1.95p.21 1 2 SolutiontoExercise1.97p.21 1 3

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43 SolutiontoExercise1.99p.21 3 SolutiontoExercise1.101p.22 11 12 SolutiontoExercise1.103p.22 27 19 SolutiontoExercise1.105p.22 6 SolutiontoExercise1.107p.22 12 SolutiontoExercise1.109p.22 20 SolutiontoExercise1.111p.22 75 SolutiontoExercise1.113p.22 48 SolutiontoExercise1.114p.26 4 9 SolutiontoExercise1.116p.26 1 3 SolutiontoExercise1.118p.26 5 12 SolutiontoExercise1.120p.26 9 10 SolutiontoExercise1.122p.26 1 SolutiontoExercise1.124p.26 2 3 SolutiontoExercise1.126p.26 10 9 SolutiontoExercise1.128p.26 10 9 SolutiontoExercise1.130p.27 57 7 SolutiontoExercise1.132p.27 5 8 SolutiontoExercise1.134p.27 1 SolutiontoExercise1.136p.27 13 23 SolutiontoExercise1.138p.27 19 20 SolutiontoExercise1.140p.27 15 16 SolutiontoExercise1.142p.27 5 8 SolutiontoExercise1.144p.27 31 24 SolutiontoExercise1.146p.27 5 6

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44 CHAPTER1.ARITHMETICREVIEW SolutiontoExercise1.148p.27 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 180 SolutiontoExercise1.150p.27 7 30 SolutiontoExercise1.152p.28 47 18 SolutiontoExercise1.154p.34 8 : 95 SolutiontoExercise1.156p.34 2 : 234 SolutiontoExercise1.158p.34 6 : 6312 SolutiontoExercise1.160p.35 3 : 68406 SolutiontoExercise1.162p.35 13 : 632 SolutiontoExercise1.164p.35 25 : 355 SolutiontoExercise1.166p.35 0 : 324 SolutiontoExercise1.168p.35 12 ; 008 SolutiontoExercise1.170p.35 4 : 2 SolutiontoExercise1.172p.35 0 : 0032 SolutiontoExercise1.174p.35 3 50 SolutiontoExercise1.176p.35 3 7 10 SolutiontoExercise1.178p.35 712 1 25000 SolutiontoExercise1.180p.35 0 : 45 SolutiontoExercise1.182p.36 0 : 636 SolutiontoExercise1.184p.39 40% SolutiontoExercise1.186p.39 12 : 5% SolutiontoExercise1.188p.39 68 : 18% SolutiontoExercise1.190p.39 22 : 22% SolutiontoExercise1.192p.39 49 : 09% SolutiontoExercise1.194p.39 1500% SolutiontoExercise1.196p.39 36%

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45 SolutiontoExercise1.198p.40 44 : 6% SolutiontoExercise1.200p.40 425% SolutiontoExercise1.202p.40 8698% SolutiontoExercise1.204p.40 1400% SolutiontoExercise1.206p.40 0 : 35 SolutiontoExercise1.208p.40 0 : 186 SolutiontoExercise1.210p.40 0 : 090145 SolutiontoExercise1.212p.40 0 : 0000005

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46 CHAPTER1.ARITHMETICREVIEW

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Chapter2 BasicPropertiesofRealNumbers 2.1Objectives 1 Aftercompletingthischapter,youshould SymbolsandNotationsSection2.2 understandthedierencebetweenvariablesandconstants befamiliarwiththesymbolsofoperation,equality,andinequality befamiliarwithgroupingsymbols beabletocorrectlyusetheorderofoperations TheRealNumberLineandtheRealNumbersSection2.3 befamiliarwiththerealnumberlineandtherealnumbers understandtheorderingoftherealnumbers PropertiesoftheRealNumbersSection2.4 understandtheclosure,commutative,associative,anddistributiveproperties understandtheidentityandinverseproperties ExponentsSection2.5 understandexponentialnotation beabletoreadexponentialnotation understandhowtouseexponentialnotationwiththeorderofoperations RulesofExponentsSection2.6 understandtheproductandquotientrulesforexponents understandthemeaningofzeroasanexponent ThePowerRulesforExponentsSection2.7 understandthepowerrulesforpowers,products,andquotients 1 Thiscontentisavailableonlineat. 47

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48 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.2SymbolsandNotations 2 2.2.1Overview VariablesandConstants SymbolsofOperation,Equality,andInequality GroupingSymbols TheOrderofOperations 2.2.2VariablesandConstants Abasiccharacteristicofalgebraistheuseofsymbolsusuallyletterstorepresentnumbers. Variable Aletterorsymbolthatrepresentsanymemberofacollectionoftwoormorenumbersiscalleda variable Constant Aletterorsymbolthatrepresentsaspecicnumber,knownorunknowniscalleda constant Inthefollowingexamples,theletter x isavariablesinceitcanbeanymemberofthecollectionof numbers f 35 ; 25 ; 10 g .Theletter h isaconstantsinceitcanassumeonlythevalue5890. Example2.1 Supposethatthestreetsonyourwayfromhometoschoolhavespeedlimitsof35mph,25mph, and10mph.Inalgebrawecanlettheletter x representourspeedaswetravelfromhometo school.Themaximumvalueof x dependsonwhatsectionofstreetweareon.Theletter x can assumeanyoneofthe various values35,25,10. Example2.2 SupposethatinwritingatermpaperforageographyclassweneedtospecifytheheightofMount Kilimanjaro.Ifwedonothappentoknowtheheightofthemountain,wecanrepresentitatleast temporarilyonourpaperwiththeletter h .Later,welookuptheheightinareferencebookand ndittobe5890meters.Theletter h canassumeonlytheonevalue,5890,andnoothers.The valueof h is constant 2.2.3SymbolsofOperation,Equality,andInequality BinaryOperation A binaryoperation onacollectionofnumbersisaprocessthatassignsanumbertotwogivennumbersin thecollection.Thebinaryoperationsusedinalgebraareaddition,subtraction,multiplication,anddivision. SymbolsofOperation Ifwelet x and y eachrepresentanumber,wehavethefollowingnotations: Addition x + y Subtraction x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y Multiplication x y x y x y xy Division x y x=yx yy p x 2.2.4SampleSetA Example2.3 a + b representsthe sum of a and b 2 Thiscontentisavailableonlineat.

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49 Example2.4 4+ y representsthe sum of4and y Example2.5 8 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x representsthe dierence of8and x Example2.6 6 x representsthe product of6and x Example2.7 ab representsthe product of a and b Example2.8 h 3 representsthe product of h and3. Example2.9 : 2 a representsthe product of 14 : 2 and a Example2.10 representsthe product of8and24. Example2.11 5 6 b representsthe product of5,6,and b Example2.12 6 x representsthe quotient of6and x 2.2.5PracticeSetA Exercise2.1 Solutiononp.106. Representtheproductof29and x vedierentways. Ifwelet a and b representtwonumbers,then a and b arerelatedinexactlyoneofthreeways: EqualityandInequalitySymbols a = ba and b areequal a>ba isstrictlygreaterthan b ab or a b : a isnotless b thancanbeexpressedaseither a
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50 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.2.6GroupingSymbols Groupingsymbolsareusedtoindicatethataparticularcollectionofnumbersandmeaningfuloperations aretobegroupedtogetherandconsideredasonenumber.Thegroupingsymbolscommonlyusedinalgebra are Parentheses: Brackets: hi Braces: fg Bar: Inacomputationinwhichmorethanoneoperationisinvolved,groupingsymbolshelptelluswhich operationstoperformrst.Ifpossible,weperformoperationsinsidegroupingsymbolsrst. 2.2.7SampleSetB Example2.13 +17 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=21 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=15 Example2.14 8+6=8=72 Example2.15 5[8+ )]TJ/F8 9.9626 Tf 9.963 0 Td [(4]=5[8+6]=5[14]=70 Example2.16 2 f 3[4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(11] g =2 f 3[4] g =2 f 3[24] g =2 f 72 g =144 Example2.17 9+1 24+3 Thefractionbarseparatesthetwogroupsofnumbers 9+1 and 24+3 .Performthe operationsinthenumeratoranddenominatorseparately. 9+1 24+3 = 9 24+3 = 54 24+3 = 54 27 =2 2.2.8PracticeSetB Usethegroupingsymbolstohelpperformthefollowingoperations. Exercise2.2 Solutiononp.106. 3+8 Exercise2.3 Solutiononp.106. 4[2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5] Exercise2.4 Solutiononp.106. 6 f 2[2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9] g Exercise2.5 Solutiononp.106. 1+19 2+3 Thefollowingexamplesshowhowtousealgebraicnotationtowriteeachexpression. Example2.18 9minus y becomes 9 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y Example2.19 46times x becomes 46x

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51 Example2.20 7times x + y becomes 7 x + y Example2.21 4dividedby3,times z becomes )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(4 3 z Example2.22 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b times b )]TJ/F11 9.9626 Tf 9.963 0 Td [(a dividedbytimes a becomes a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b b )]TJ/F10 6.9738 Tf 6.226 0 Td [(a 2 a Example2.23 Introduceavariable any letterwilldobutherewe'lllet x representthenumberanduseappropriatealgebraicsymbolstowritethestatement:Anumberplus4isstrictlygreaterthan6.The answeris x +4 > 6 2.2.9TheOrderofOperations Supposewewishtondthevalueof 16+4 9 .Wecould 1.add16and4,thenmultiplythissumby9. 16+4 9=20 9=180 2.multiply4and9,thenadd16tothisproduct. 16+4 9=16+36=52 Wenowhavetwovaluesforonenumber.Todeterminethecorrectvaluewemustusethestandard order ofoperations OrderofOperations 1.Performalloperationsinsidegroupingsymbols,beginningwiththeinnermostset. 2.Performallmultiplicationsanddivisions,asyoucometothem,movingleft-to-right. 3.Performalladditionsandsubtractions,asyoucometothem,movingleft-to-right. Asweproceedinourstudyofalgebra,wewillcomeuponanotheroperation,exponentiation,thatwillneed tobeinsertedbeforemultiplicationanddivision.SeeSectionSection2.5. 2.2.10SampleSetC USetheorderofoperationstondthevalueofeachnumber. Example2.24 16+4 9 Multiplyrst. =16+36 Nowadd. =52 Example2.25 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8+7+12 Combinewithinparentheses. =19+7 Multiply. =19+126 Nowadd. =145

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52 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Example2.26 8+2[4+3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] Beginwiththeinnermostsetofgroupingsymbols ; =8+2[4+3] Nowworkwithinthenextsetofgroupingsymbols ; hi =8+2[4+15] =8+2[19] =8+38 =46 Example2.27 6+4[2+3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(17] 18 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2[2+2] = 6+4[2+3] 18 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2[6+2] = 6+4[2+6] 18 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2[8] = 6+4[8] 18 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 = 6+32 2 = 38 2 =19 2.2.11PracticeSetC Usetheorderofoperationstondeachvalue. Exercise2.6 Solutiononp.106. 25+8 Exercise2.7 Solutiononp.106. 2+3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 2 Exercise2.8 Solutiononp.106. 4+3[2+3+8 4] Exercise2.9 Solutiononp.106. 19+2 f 5+2[18+6+1] g 5 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2.2.12Exercises Forthefollowingproblems,usetheorderofoperationstondeachvalue. Exercise2.10 Solutiononp.106. 2+3 Exercise2.11 18 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise2.12 Solutiononp.106. 8 4 16+5 Exercise2.13 +4 5 2 Exercise2.14 Solutiononp.106. 3+2 6+3 Exercise2.15 6+1 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15

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53 Exercise2.16 Solutiononp.106. 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+8+7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 Exercise2.17 +2+ Exercise2.18 Solutiononp.106. 61 )]TJ/F8 9.9626 Tf 9.962 0 Td [(22+4[3+11] Exercise2.19 +16 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 7 +5 Exercise2.20 Solutiononp.106. 8+20 8 + 3+16 22 Exercise2.21 18 2+55 Exercise2.22 Solutiononp.106. 21 7 3 Exercise2.23 85 5 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(85 Exercise2.24 Solutiononp.106. )]TJ/F8 9.9626 Tf 9.962 0 Td [(25 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise2.25 4 3+8 28 )]TJ/F8 9.9626 Tf 9.962 0 Td [(+17+11 Exercise2.26 Solutiononp.106. 2 f +7+6[4+2] g Exercise2.27 0+10+15[4+1] Exercise2.28 Solutiononp.106. 6 : 1 : 2+1 : 8 Exercise2.29 5 : 9 2 +0 : 6 Exercise2.30 Solutiononp.106. +7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise2.31 +5+5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise2.32 Solutiononp.106. )]TJ/F7 6.9738 Tf 7.748 -4.147 Td [(5 12 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 4 + )]TJ/F7 6.9738 Tf 5.761 -4.147 Td [(1 6 + 2 3 Exercise2.33 4 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(3 5 )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(8 15 +9 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 3 + 1 4 Exercise2.34 Solutiononp.106. 0 5 + 0 1 +0[2+4] Exercise2.35 0 9+4 0 7+0[2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2] Forthefollowingproblems,statewhetherthegivenstatementsarethesameordierent. Exercise2.36 Solutiononp.106. x y and x>y Exercise2.37

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54 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.38 Solutiononp.106. x = y and y = x Exercise2.39 Representtheproductof3and x vedierentways. Exercise2.40 Solutiononp.106. Representthesumof a and b twodierentways. Forthefollowingproblems,rewriteeachphraseusingalgebraicnotation. Exercise2.41 Tenminusthree Exercise2.42 Solutiononp.107. x plussixteen Exercise2.43 51dividedby a Exercise2.44 Solutiononp.107. 81times x Exercise2.45 3times x + y Exercise2.46 Solutiononp.107. x + b times x +7 Exercise2.47 3times x times y Exercise2.48 Solutiononp.107. x dividedbytimes b Exercise2.49 a + b dividedby a +4 Forthefollowingproblems,introduceavariableanyletterwilldoanduseappropriatealgebraicsymbols towritethegivenstatement. Exercise2.50 Solutiononp.107. Anumberminuseightequalsseventeen. Exercise2.51 Fivetimesanumber,minusone,equalszero. Exercise2.52 Solutiononp.107. Anumberdividedbysixisgreaterthanorequaltoforty-four. Exercise2.53 Sixteenminustwiceanumberequalsve. Determinewhetherthestatementsforthefollowingproblemsaretrueorfalse. Exercise2.54 Solutiononp.107. 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 10 Exercise2.55 5+2 10 110 Exercise2.56 Solutiononp.107. 8 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(48 0 Exercise2.57 20+4 : 3 16 < 5

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55 Exercise2.58 Solutiononp.107. 2[6+4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8] > 3+6 Exercise2.59 6[4+8+3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15] 3[7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4] Exercise2.60 Solutiononp.107. Thenumberofdierentways5peoplecanbearrangedinarowis 5 4 3 2 1 .Howmanyways isthis? Exercise2.61 Aboxcontains10computerchips.Threechipsaretobechosenatrandom.Thenumberofways thiscanbedoneis 10 9 8 7 6 5 4 3 2 1 3 2 1 7 6 5 4 3 2 1 Howmanywaysisthis? Exercise2.62 Solutiononp.107. Theprobabilityofobtainingfourofakindinave-cardpokerhandis 13 48 51 50 49 48 4 3 2 1 Whatisthisprobability? Exercise2.63 Threepeopleareonanelevatorinavestorybuilding.Ifeachpersonrandomlyselectsaoor onwhichtogeto,theprobabilitythatatleasttwopeoplegetoonthesameooris 1 )]TJ/F7 6.9738 Tf 11.159 3.923 Td [(5 4 3 5 5 5 Whatisthisprobability? 2.3TheRealNumberLineandtheRealNumbers 3 2.3.1Overview TheRealNumberLine TheRealNumbers OrderingtheRealNumbers 2.3.2TheRealNumberLine RealNumberLine Inourstudyofalgebra,wewilluseseveralcollectionsofnumbers.The realnumberline allowsusto visually displaythenumbersinwhichweareinterested. Alineiscomposedofinnitelymanypoints.Toeachpointwecanassociateauniquenumber,andwith eachnumberwecanassociateaparticularpoint. Coordinate Thenumberassociatedwithapointonthenumberlineiscalledthe coordinate ofthepoint. Graph Thepointonalinethatisassociatedwithaparticularnumberiscalledthe graph ofthatnumber. 3 Thiscontentisavailableonlineat.

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56 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Weconstructtherealnumberlineasfollows: ConstructionoftheRealNumberLine 1.Drawahorizontalline. 2.Chooseanypointonthelineandlabelit0.Thispointiscalledthe origin 3.Chooseaconvenientlength.Thislengthiscalled"1unit."Startingat0,markthislengthoinboth directions,beingcarefultohavethelengthslookliketheyareaboutthesame. Wenowdenearealnumber. RealNumber A realnumber isanynumberthatisthecoordinateofapointontherealnumberline. PositiveandNegativeRealNumbers Thecollectionoftheseinnitelymanynumbersiscalledthe collectionofrealnumbers .Therealnumbers whosegraphsaretotherightof0arecalledthe positiverealnumbers .Therealnumberswhosegraphs appeartotheleftof0arecalledthe negativerealnumbers Thenumber0isneitherpositivenornegative. 2.3.3TheRealNumbers Thecollectionofrealnumbershasmanysubcollections.Thesubcollectionsthatareofmostinteresttous arelistedbelowalongwiththeirnotationsandgraphs. NaturalNumbers The naturalnumbers N : f 1 ; 2 ; 3 ;::: g WholeNumbers The wholenumbers W : f 0 ; 1 ; 2 ; 3 ;::: g Noticethateverynaturalnumberisawholenumber. Integers The integers Z : f :::; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.442 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 0 ; 1 ; 2 ; 3 ;::: g Noticethateverywholenumberisaninteger.

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57 RationalNumbers The rationalnumbers Q : Rationalnumbersarerealnumbersthatcanbewrittenintheform a=b ,where a and b areintegers,and b 6 =0 Fractions Rationalnumbersarecommonlycalled fractions. Divisionby1 Since b canequal1,everyintegerisarationalnumber: a 1 = a Divisionby0 Recallthat 10 = 2=5 since 2 5=10 .However,if 10 = 0= x ,then 0 x =10 .But 0 x =0 ,not10.This suggeststhatnoquotientexists. Nowconsider 0 = 0= x .If 0 = 0= x ,then 0 x =0 .Butthismeansthat x couldbeanynumber,thatis, 0 = 0=4 since 0 4=0 ,or 0 = 0=28 since 0 28=0 .Thissuggeststhatthequotientisindeterminant. x= 0 IsUndenedorIndeterminant Divisionby0isundenedorindeterminant. Donotdivideby0. Rationalnumbershavedecimalrepresentationsthateitherterminateordonotterminatebutcontaina repeatingblockofdigits.Someexamplesare: 3 4 =0 : 75 | {z } Terminating 15 11 =1 : 36363636 ::: | {z } Nonterminating,butrepeating Somerationalnumbersaregraphedbelow. IrrationalNumbers The irrationalnumbers Ir : Irrationalnumbersarenumbersthatcannotbewrittenasthequotientof twointegers.Theyarenumberswhosedecimalrepresentationsarenonterminatingandnonrepeating.Some examplesare 4 : 01001000100001 ::: =3 : 1415927 ::: Noticethatthecollectionsofrationalnumbersandirrationalnumbershavenonumbersincommon. Whengraphedonthenumberline,therationalandirrationalnumbersaccountforeverypointonthe numberline.Thuseachpointonthenumberlinehasacoordinatethatiseitherarationaloranirrational number. Insummary,wehave 2.3.4SampleSetA Thesummaraychartillustratesthat

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58 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Example2.28 Everynaturalnumberisarealnumber. Example2.29 Everywholenumberisarealnumber. Example2.30 Nointegerisanirrationalnumber. 2.3.5PracticeSetA Exercise2.64 Solutiononp.107. Iseverynaturalnumberawholenumber? Exercise2.65 Solutiononp.107. Iseverywholenumberaninteger? Exercise2.66 Solutiononp.107. Iseveryintegerarationalnumber? Exercise2.67 Solutiononp.107. Iseveryrationalnumberarealnumber? Exercise2.68 Solutiononp.107. Iseveryintegeranaturalnumber? Exercise2.69 Solutiononp.107. Isthereanintegerthatisanaturalnumber? 2.3.6OrderingtheRealNumbers OrderingtheRealNumbers Arealnumber b issaidtobegreaterthanarealnumber a ,denoted b>a ,ifthegraphof b istotheright ofthegraphof a onthenumberline.

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59 2.3.7SampleSetB Aswewouldexpect, 5 > 2 since5istotherightof2onthenumberline.Also, )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 > )]TJ/F8 9.9626 Tf 10.207 0 Td [(5 since )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 isto therightof )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 onthenumberline. 2.3.8PracticeSetB Exercise2.70 Solutiononp.107. Areallpositivenumbersgreaterthan0? Exercise2.71 Solutiononp.107. Areallpositivenumbersgreaterthanallnegativenumbers? Exercise2.72 Solutiononp.107. Is0greaterthanallnegativenumbers? Exercise2.73 Solutiononp.107. Istherealargestpositivenumber?Isthereasmallestnegativenumber? Exercise2.74 Solutiononp.107. Howmanyrealnumbersarethere?Howmanyrealnumbersaretherebetween0and1? 2.3.9SampleSetC Example2.31 Whatintegerscanreplace x sothatthefollowingstatementistrue? )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x< 2 Thisstatementindicatesthatthenumberrepresentedby x isbetween )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 and2.Specically, )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 islessthanorequalto x ,andatthesametime, x isstrictlylessthan2.Thisstatementisan exampleofacompoundinequality. Theintegersare )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; 0 ; 1 Example2.32 Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 to7.Placepointsatallwholenumbersbetweenand including )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 and6. Example2.33 Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 to6andplacepointsatallrealnumbersgreaterthan orequalto3butstrictlylessthan5. Itiscustomarytousea closedcircle toindicatethatapointisincludedinthegraphandan opencircle toindicatethatapointisnotincluded.

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60 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.3.10PracticeSetC Exercise2.75 Solutiononp.107. Whatwholenumberscanreplace x sothatthefollowingstatementistrue? )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x< 3 Exercise2.76 Solutiononp.107. Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to3andplacepointsatallnumbersgreaterthanor equalto )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 butstrictlylessthan2. 2.3.11Exercises Forthefollowingproblems,nexttoeachrealnumber,noteallcollectionstowhichitbelongsbywriting N fornaturalnumbers, W forwholenumbers, Z forintegers, Q forrationalnumbers, Ir forirrationalnumbers, and R forrealnumbers.Somenumbersmayrequiremorethanoneletter. Exercise2.77 Solutiononp.107. 1 2 Exercise2.78 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise2.79 Solutiononp.108. 0 Exercise2.80 )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 7 8 Exercise2.81 Solutiononp.108. 86 : 3333 ::: Exercise2.82 49 : 125125125 ::: Exercise2.83 Solutiononp.108. )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 : 07 Forthefollowingproblems,drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 to3.Locateeachrealnumberonthe numberlinebyplacingapointclosedcircleatitsapproximatelocation. Exercise2.84 1 1 2 Exercise2.85 Solutiononp.108. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise2.86 )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(1 8

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61 Exercise2.87 Solutiononp.108. Is0apositivenumber,negativenumber,neither,orboth? Exercise2.88 Anintegerisanevenintegerifitcanbedividedby2withoutaremainder;otherwisethenumber isodd.Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to5andplacepointsatallnegativeevenintegers andatallpositiveoddintegers. Exercise2.89 Solutiononp.108. Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to5.Placepointsatallintegersstrictlygreaterthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 butstrictlylessthan4. Forthefollowingproblems,drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to5.Placepointsatallrealnumbers betweenandincludingeachpairofnumbers. Exercise2.90 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 and )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise2.91 Solutiononp.108. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 and4 Exercise2.92 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 and0 Exercise2.93 Solutiononp.108. Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to5.Isitpossibletolocateanynumbersthatare strictlygreaterthan3butalsostrictlylessthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ? Forthepairsofrealnumbersshowninthefollowingproblems,writetheappropriaterelationsymbol <;>; = inplaceofthe Exercise2.94 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 )]TJ/F8 9.9626 Tf 14.944 0 Td [(1 Exercise2.95 Solutiononp.108. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 0 Exercise2.96 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 7 Exercise2.97 Solutiononp.108. 6 )]TJ/F8 9.9626 Tf 14.944 0 Td [(1 Exercise2.98 )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 4 )]TJ/F7 6.9738 Tf 16.14 3.922 Td [(3 4 Exercise2.99 Solutiononp.108. Istherealargestrealnumber?Ifso,whatisit? Exercise2.100 Istherealargestinteger?Ifso,whatisit? Exercise2.101 Solutiononp.108. Istherealargesttwo-digitinteger?Ifso,whatisit? Exercise2.102 Isthereasmallestinteger?Ifso,whatisit? Exercise2.103 Solutiononp.108. Isthereasmallestwholenumber?Ifso,whatisit? Forthefollowingproblems,whatnumberscanreplace x sothatthefollowingstatementsaretrue? Exercise2.104 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 x 5 x aninteger

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62 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.105 Solutiononp.108. )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 n ? Exercise2.120 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a and )]TJ/F11 9.9626 Tf 7.749 0 Td [(b )]TJ/F11 9.9626 Tf 7.749 0 Td [(b> )]TJ/F11 9.9626 Tf 9.963 0 Td [(a ? 2.3.12ExercisesforReview Exercise2.121 Solutiononp.108. Section2.2 Findthevalueof 6+3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise2.122 Section2.2 Findthevalueof 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6+3+2 3 Exercise2.123 Solutiononp.108. Section2.2 Arethestatements y< 4 and y 4 thesameordierent?

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63 Exercise2.124 Section2.2 Usealgebraicnotationtowritethestatement"sixtimesanumberislessthanor equaltoeleven." Exercise2.125 Solutiononp.109. Section2.2 Isthestatement 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 7 3 trueorfalse? 2.4PropertiesoftheRealNumbers 4 2.4.1Overview TheClosureProperties TheCommutativeProperties TheAssociativeProperties TheDistributiveProperties TheIdentityProperties TheInverseProperties Property A property ofacollectionofobjectsisacharacteristicthatdescribesthecollection.Weshallnowexamine someofthepropertiesofthecollectionofrealnumbers.Thepropertieswewillexamineareexpressedin termsofadditionandmultiplication. 2.4.2TheClosureProperties TheClosureProperties If a and b arerealnumbers,then a + b isauniquerealnumber,and a b isauniquerealnumber. Forexample,3and11arerealnumbers; 3+11=14 and 3 11=33 ; andboth14and33arereal numbers.Althoughthispropertyseemsobvious,somecollectionsarenotclosedundercertainoperations. Forexample, Example2.34 Therealnumbersarenotclosedunderdivisionsince,although5and0arerealnumbers, 5 = 0 and 0 = 0 arenotrealnumbers. Example2.35 Thenaturalnumbersarenotclosedundersubtractionsince,although8isanaturalnumber, 8 )]TJ/F8 9.9626 Tf 9.278 0 Td [(8 isnot. 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8=0 and0isnotanaturalnumber. 2.4.3TheCommutativeProperties Let a and b representrealnumbers. TheCommutativeProperties COMMUTATIVEPROPERTY OFADDITION COMMUTATIVEPROPERTY OFMULTIPLICATION a + b = b + aa b = b a Thecommutativepropertiestellusthattwonumberscanbeaddedormultipliedinanyorderwithout aectingtheresult. 4 Thiscontentisavailableonlineat.

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64 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.4.4SampleSetA Thefollowingareexamplesofthecommutativeproperties. Example2.36 3+4=4+3 Bothequal 7 : Example2.37 5+ x = x +5 Bothrepresentthesamesum. Example2.38 4 8=8 4 Bothequal32. Example2.39 y 7=7 y Bothrepresentthesameproduct. Example2.40 5 a +1= a +15 Bothrepresentthesameproduct. Example2.41 x +4 y +2= y +2 x +4 Bothrepresentthesameproduct. 2.4.5PracticeSetA Fillinthe withthepropernumberorlettersoastomakethestatementtrue.Usethecommutative properties. Exercise2.126 Solutiononp.109. 6+5= +6 Exercise2.127 Solutiononp.109. m +12=12+ Exercise2.128 Solutiononp.109. 9 7= 9 Exercise2.129 Solutiononp.109. 6 a = a Exercise2.130 Solutiononp.109. 4 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(5= 4 Exercise2.131 Solutiononp.109. a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = b +7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1

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65 2.4.6TheAssociativeProperties Let a;b; and c representrealnumbers. TheAssociativeProperties ASSOCIATIVEPROPERTY OFADDITION ASSOCIATIVEPROPERTY OFMULTIPLICATION a + b + c = a + b + c ab c = a bc Theassociativepropertiestellusthatwemaygrouptogetherthequantitiesaswepleasewithoutaecting theresult. 2.4.7SampleSetB Thefollowingexamplesshowhowtheassociativepropertiescanbeused. Example2.42 +6+1=2++1 8+1=2+7 9=9 Bothequal9. Example2.43 + x +17=3+ x +17 Bothrepresentthesamesum. Example2.44 3 5=2 5 6 5=2 15 30=30 Bothequal 30 : Example2.45 y 4=9 y 4 Bothrepresentthesameproduct. 2.4.8PracticeSetB Fillinthe tomakeeachstatementtrue.Usetheassociativeproperties. Exercise2.132 Solutiononp.109. +2+5=9+ Exercise2.133 Solutiononp.109. x ++ y = + y Exercise2.134 Solutiononp.109. a 6=11 Exercise2.135 Solutiononp.109. [ m )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 m +3] m +4= m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 hi

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66 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.4.9SampleSetC Example2.46 Simplifyrearrangeintoasimplerform: 5 x 6 b 8 ac 4 Accordingtothecommutativepropertyofmultiplication,wecanmakeaseriesofconsecutive switchesandgetallthenumberstogetherandalltheletterstogether. 5 6 8 4 x b a c 960 xbac Multiplythenumbers. 960 abcx Byconvention,wewill,whenpossible,writealllettersinalphabeticalorder. 2.4.10PracticeSetC Simplifyeachofthefollowingquantities. Exercise2.136 Solutiononp.109. 3 a 7 y 9 d Exercise2.137 Solutiononp.109. 6 b 8 acz 4 5 Exercise2.138 Solutiononp.109. 4 p 6 qr 3 a + b 2.4.11TheDistributiveProperties Whenwewererstintroducedtomultiplicationwesawthatitwasdevelopedasadescriptionforrepeated addition. 4+4+4=3 4 Noticethattherearethree4's,thatis,4appears3 times .Hence,3times4. Weknowthatalgebraisgeneralizedarithmetic.Wecannowmakeanimportantgeneralization. Whenanumber a isaddedrepeatedly n times,wehave a + a + a + + a | {z } a appears n times Then,usingmultiplicationasadescriptionforrepeatedaddition,wecanreplace a + a + a + + a | {z } n times with na Forexample: Example2.47 x + x + x + x canbewrittenas 4 x since x isrepeatedlyadded4times. x + x + x + x =4 x Example2.48 r + r canbewrittenas 2 r since r isrepeatedlyadded2times. r + r =2 r Thedistributivepropertyinvolvesbothmultiplicationandaddition.Let'srewrite 4 a + b : Weproceedby reading 4 a + b asamultiplication:4timesthequantity a + b .Thisdirectsustowrite 4 a + b = a + b + a + b + a + b + a + b = a + b + a + b + a + b + a + b Nowweusethecommutativepropertyofadditiontocollectallthe a s togetherandallthe b s together.

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67 4 a + b = a + a + a + a | {z } 4 a s + b + b + b + b | {z } 4 b s Now,usingmultiplicationasadescriptionforrepeatedaddition,wehave 4 a + b =4 a +4 b Wehave distributed the4overthesumtoboth a and b TheDistributiveProperty a b + c = a b + a c b + c a = a b + a c Thedistributivepropertyisusefulwhenwecannotordonotwishtoperformoperationsinsideparentheses. 2.4.12SampleSetD Usethedistributivepropertytorewriteeachofthefollowingquantities. Example2.49 Example2.50 Example2.51 2.4.13PracticeSetD Exercise2.139 Solutiononp.109. Whatpropertyofrealnumbersjusties a b + c = b + c a ? Usethedistributivepropertytorewriteeachofthefollowingquantities. Exercise2.140 Solutiononp.109. 3+1 Exercise2.141 Solutiononp.109. x +67 Exercise2.142 Solutiononp.109. 4 a + y Exercise2.143 Solutiononp.109. +2 a Exercise2.144 Solutiononp.109. a x +5 Exercise2.145 Solutiononp.109. 1 x + y

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68 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.4.14TheIdentityProperties AdditiveIdentity Thenumber0iscalledthe additiveidentity sincewhenitisaddedtoanyrealnumber,itpreservesthe identityofthatnumber.Zeroistheonlyadditiveidentity. Forexample, 6+0=6 MultiplicativeIdentity Thenumber1iscalledthe multiplicativeidentity sincewhenitmultipliesanyrealnumber,itpreserves theidentityofthatnumber.Oneistheonlymultiplicativeidentity. Forexample 6 1=6 Wesummarizetheidentitypropertiesasfollows. ADDITIVEIDENTITY PROPERTY MULTIPLICATIVEIDENTITY PROPERTY If a isarealnumber,thenIf a isarealnumber,then a +0= a and 0+ a = aa 1= a and 1 a = a 2.4.15TheInverseProperties AdditiveInverses Whentwonumbersareaddedtogetherandtheresultistheadditiveidentity,0,thenumbersarecalled additiveinverses ofeachother.Forexample,when3isaddedto )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 theresultis0,thatis, 3+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3=0 Thenumbers3and )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 areadditiveinversesofeachother. MultiplicativeInverses Whentwonumbersaremultipliedtogetherandtheresultisthemultiplicativeidentity,1,thenumbersare called multiplicativeinverses ofeachother.Forexample,when6and 1 6 aremultipliedtogether,theresult is1,thatis, 6 1 6 =1 .Thenumbers6and 1 6 aremultiplicativeinversesofeachother. Wesummarizetheinversepropertiesasfollows. TheInverseProperties 1.If a isanyrealnumber,thenthereisauniquerealnumber )]TJ/F11 9.9626 Tf 7.749 0 Td [(a ,suchthat a + )]TJ/F11 9.9626 Tf 7.749 0 Td [(a =0 and )]TJ/F11 9.9626 Tf 7.749 0 Td [(a + a =0 Thenumbers a and )]TJ/F11 9.9626 Tf 7.749 0 Td [(a arecalled additiveinverses ofeachother. 2.If a isanynonzerorealnumber,thenthereisauniquerealnumber 1 a suchthat a 1 a =1 and 1 a a =1 Thenumbers a and 1 a arecalled multiplicativeinverses ofeachother. ExpandingQuantities Whenweperformoperationssuchas 6 a +3=6 a +18 ,wesayweare expanding thequantity 6 a +3 2.4.16Exercises Usethecommutativepropertyofadditionandmultiplicationtowriteexpressionsforanequalnumberfor thefollowingproblems.Youneednotperformanycalculations. Exercise2.146 Solutiononp.109. x +3 Exercise2.147 5+ y Exercise2.148 Solutiononp.109. 10 x

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69 Exercise2.149 18 z Exercise2.150 Solutiononp.109. r 6 Exercise2.151 ax Exercise2.152 Solutiononp.109. xc Exercise2.153 7+ b Exercise2.154 Solutiononp.110. 6 s +1 Exercise2.155 + a x +6 Exercise2.156 Solutiononp.110. x +16 a +7 Exercise2.157 x + y x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y Exercise2.158 Solutiononp.110. 0 : 06 m Exercise2.159 Exercise2.160 Solutiononp.110. 5 h +1 Exercise2.161 m a +2 b Exercise2.162 Solutiononp.110. k a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b Exercise2.163 c : 008 Exercise2.164 Solutiononp.110. )]TJ/F8 9.9626 Tf 7.748 0 Td [(16 Exercise2.165 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise2.166 Solutiononp.110. [U+25CB] Exercise2.167 Simplifyusingthecommutativepropertyofmultiplicationforthefollowingproblems.Youneednotusethe distributiveproperty. Exercise2.168 Solutiononp.110. 9 x 2 y Exercise2.169 5 a 6 b Exercise2.170 Solutiononp.110. 2 a 3 b 4 c

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70 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.171 5 x 10 y 5 z Exercise2.172 Solutiononp.110. 1 u 3 r 2 z 5 m 1 n Exercise2.173 6 d 4 e 1 f 2 g +2 h Exercise2.174 Solutiononp.110. )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 d )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 4 e )]TJ/F7 6.9738 Tf 5.761 -4.147 Td [(1 2 a Exercise2.175 3 a +62 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(96 b Exercise2.176 Solutiononp.110. 1 x +2 y + z 9 x +5 y Forthefollowingproblems,usethedistributivepropertytoexpandthequantities. Exercise2.177 2 y +9 Exercise2.178 Solutiononp.110. b r +5 Exercise2.179 m u + a Exercise2.180 Solutiononp.110. k j +1 Exercise2.181 x y +5 Exercise2.182 Solutiononp.110. z x +9 w Exercise2.183 + d e Exercise2.184 Solutiononp.110. +2 f g Exercise2.185 c a +10 b Exercise2.186 Solutiononp.110. 15 x y +3 z Exercise2.187 8 y a + b Exercise2.188 Solutiononp.110. z x + y + m Exercise2.189 a +6 x + y Exercise2.190 Solutiononp.110. x +10 a + b + c Exercise2.191 1 x + y Exercise2.192 Solutiononp.110. 1 a +16

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71 Exercise2.193 0 : 48 : 34 a +0 : 61 Exercise2.194 Solutiononp.110. 21 : 5 : 2 a +3 : 8 b +0 : 7 c Exercise2.195 Exercise2.196 Solutiononp.110. 2 z t L m +8 k 2.4.17ExercisesforReview Exercise2.197 Section2.2 Findthevalueof 4 2+5 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 5 Exercise2.198 Solutiononp.110. Section2.2 Isthestatement 3 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 5+6 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 4 < 0 trueorfalse? Exercise2.199 Section2.3 Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 to2andplacepointsatallintegers betweenandincluding )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 and3. Exercise2.200 Solutiononp.110. Section2.3 Replacethe withtheappropriaterelationsymbol <;> : )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 )]TJ/F8 9.9626 Tf 14.944 0 Td [(3 Exercise2.201 Section2.3 Whatwholenumberscanreplace x sothatthestatement )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x< 2 istrue? 2.5Exponents 5 2.5.1Overview ExponentialNotation ReadingExponentialNotation TheOrderofOperations 2.5.2ExponentialNotation InSectionSection2.4wewereremindedthatmultiplicationisadescriptionforrepeatedaddition.Anatural questionisIsthereadescriptionfor repeated multiplication?Theanswerisyes.Thenotationthat describesrepeatedmultiplicationis exponentialnotation Factors Inmultiplication,thenumbersbeingmultipliedtogetherarecalled factors .Inrepeatedmultiplication,all thefactorsarethesame.Innonrepeatedmultiplication,noneofthefactorsarethesame.Forexample, 5 Thiscontentisavailableonlineat.

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72 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Example2.52 18 18 18 18 Repeatedmultiplicationof 18 : Allfourfactors ; 18 ; arethesame. x x x x x Repeatedmultiplicationof x: Allvefactors ;x; arethesame. 3 7 a Nonrepeatedmultiplication.Noneofthefactorsarethesame. Exponentialnotationisusedtoshowrepeatedmultiplicationofthesamefactor.Thenotationconsistsof usinga superscriptonthefactorthatisrepeated .Thesuperscriptiscalledan exponent ExponentialNotation If x isanyrealnumberand n isanaturalnumber,then x n = x x x ::: x | {z } n factorsof x Anexponentrecordsthenumberofidenticalfactorsinamultiplication. Notethatthedenitionforexponentialnotationonlyhasmeaningfornaturalnumberexponents.We willextendthisnotationtoincludeothernumbersasexponentslater. 2.5.3SampleSetA Example2.53 7 7 7 7 7 7=7 6 Therepeatedfactoris7.Theexponent6recordsthefactthat7appears6timesinthe multiplication. Example2.54 x x x x = x 4 Therepeatedfactoris x .Theexponent4recordsthefactthat x appears4timesinthe multiplication. Example2.55 y y y = y 3 Therepeatedfactoris 2 y .Theexponent3recordsthefactthatthefactor 2 y appears3times inthemultiplication. Example2.56 2 yyy =2 y 3 Therepeatedfactoris y .Theexponent3recordsthefactthatthefactor y appears3timesin themultiplication. Example2.57 a + b a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b = a + b 2 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 3 Therepeatedfactorsare a + b and a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b a + b appearing2timesand a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b appearing3 times. 2.5.4PracticeSetA Writeeachofthefollowingusingexponents. Exercise2.202 Solutiononp.110. a a a a Exercise2.203 Solutiononp.111. b b c c c c

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73 Exercise2.204 Solutiononp.111. 2 2 7 7 7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise2.205 Solutiononp.111. 8 xxxyzzzzz CAUTION Itisextremelyimportanttorealizeandrememberthatanexponentappliesonlytothefactortowhichitis directlyconnected. 2.5.5SampleSetB Example2.58 8 x 3 means 8 xxx and not 8 x 8 x 8 x .Theexponent3appliesonlytothefactor x sinceitisonlyto thefactor x thatthe3isconnected. Example2.59 x 3 means x x x sincetheparenthesesindicatethattheexponent3isdirectlyconnected tothefactor 8 x .Rememberthatthegroupingsymbols indicatethatthequantitiesinsideare tobeconsideredasonesinglenumber. Example2.60 34 a +1 2 means 34 a +1 a +1 sincetheexponent2appliesonlytothefactor a +1 2.5.6PracticeSetB Writeeachofthefollowingwithoutexponents. Exercise2.206 Solutiononp.111. 4 a 3 Exercise2.207 Solutiononp.111. a 3 2.5.7SampleSetC Example2.61 Selectanumbertoshowthat x 2 isnotalwaysequalto 2 x 2 Supposewechoose x tobe5.Considerboth x 2 and 2 x 2 x 2 2 x 2 5 2 2 5 2 2 2 25 100 6 =50 .1 Noticethat x 2 =2 x 2 onlywhen x =0 .

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74 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.5.8PracticeSetC Exercise2.208 Solutiononp.111. Selectanumbertoshowthat x 2 isnotalwaysequalto 5 x 2 2.5.9ReadingExponentialNotation In x n Base x isthe base Exponent n isthe exponent Power Thenumberrepresentedby x n iscalleda power x tothe n thPower Theterm x n isreadas" x tothe n thpower,"ormoresimplyas" x tothe n th." x Squaredand x Cubed Thesymbol x 2 isoftenreadas" x squared,"and x 3 isoftenreadas" x cubed."Anaturalquestionis"Why aregeometrictermsappearingintheexponentexpression?"Theanswerfor x 3 isthis: x 3 means x x x Ingeometry,thevolumeofarectangularboxisfoundbymultiplyingthelengthbythewidthbythedepth. Acubehasthesamelengthoneachside.Ifwerepresentthislengthbytheletter x thenthevolumeofthe cubeis x x x ,which,ofcourse,isdescribedby x 3 .Canyouthinkofwhy x 2 isreadas x squared? Cubewith length = x width = x depth = x Volume = xxx = x 3 2.5.10TheOrderofOperations InSectionSection2.2wewereintroducedtotheorderofoperations.Itwasnotedthatwewouldinsert anotheroperationbeforemultiplicationanddivision.Wecandothatnow. TheOrderofOperations 1.Performalloperationsinsidegroupingsymbolsbeginningwiththeinnermostset. 2.Performallexponential operationsasyoucometo them,movingleft-to-right. 3.Performallmultiplicationsanddivisionsasyoucometothem,movingleft-to-right. 4.Performalladditionsandsubtractionsasyoucometothem,movingleft-to-right.

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75 2.5.11SampleSetD Usetheorderofoperationstosimplifyeachofthefollowing. Example2.62 2 2 +5=4+5=9 Example2.63 5 2 +3 2 +10=25+9+10=44 Example2.64 2 2 + )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=4+ )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =4+40 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =43 Example2.65 7 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 +1 5 =7 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16+1 =42 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16+1 =27 Example2.66 +3 3 +7 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+1 2 = 3 +7 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 =125+49 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 =125+49 )]TJ/F8 9.9626 Tf 9.963 0 Td [(75 =99 Example2.67 h 4+2 3 i 2 = h 4 3 i 2 =[4] 2 =[2048] 2 =4 ; 194 ; 304 Example2.68 6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 2 +2 2 +4 2 =6+4+4 2 =6+4 2 =6+16 =78+16 =94 Example2.69 6 2 +2 2 4 2 +6 2 2 + 1 3 +8 2 10 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [( = 36+4 16+6 4 + 1+64 100 )]TJ/F7 6.9738 Tf 6.227 0 Td [(95 = 36+4 16+24 + 1+64 100 )]TJ/F7 6.9738 Tf 6.227 0 Td [(95 = 40 40 + 65 5 =1+13 =14

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76 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.5.12PracticeSetD Usetheorderofoperationstosimplifythefollowing. Exercise2.209 Solutiononp.111. 3 2 +4 5 Exercise2.210 Solutiononp.111. 2 3 +3 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 4 Exercise2.211 Solutiononp.111. 1 4 + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 2 +4 2 2 3 Exercise2.212 Solutiononp.111. 6 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(10 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 3 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 2 Exercise2.213 Solutiononp.111. 5 2 +6 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 1+4 2 + 0 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(0 5 7 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 2 3 2.5.13Exercises Forthefollowingproblems,writeeachofthequantitiesusingexponentialnotation. Exercise2.214 Solutiononp.111. b tothefourth Exercise2.215 a squared Exercise2.216 Solutiononp.111. x totheeighth Exercise2.217 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 cubed Exercise2.218 Solutiononp.111. 5times s squared Exercise2.219 3squaredtimes y tothefth Exercise2.220 Solutiononp.111. a cubedminus b +7 squared Exercise2.221 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x cubedplus x +5 totheseventh Exercise2.222 Solutiononp.111. xxxxx Exercise2.223 xxxx Exercise2.224 Solutiononp.111. 2 3 3 3 3 xxyyyyy Exercise2.225 2 2 5 6 6 6 xyyzzzwwww Exercise2.226 Solutiononp.111. 7 xx a +8 a +8 Exercise2.227 10 xyy c +5 c +5 c +5

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77 Exercise2.228 Solutiononp.111. 4 x 4 x 4 x 4 x 4 x Exercise2.229 a a a a Exercise2.230 Solutiononp.111. )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 aabbba )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 baab Exercise2.231 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a +10 Exercise2.232 Solutiononp.111. z + w z + w z + w z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w Exercise2.233 y y 2 y 2 y Exercise2.234 Solutiononp.111. 3 xyxxy )]TJ/F8 9.9626 Tf 9.963 0 Td [( x +1 x +1 x +1 Forthefollowingproblems,expandthequantitiessothatnoexponentsappear. Exercise2.235 4 3 Exercise2.236 Solutiononp.111. 6 2 Exercise2.237 7 3 y 2 Exercise2.238 Solutiononp.111. 8 x 3 y 2 Exercise2.239 )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(18 x 2 y 4 2 Exercise2.240 Solutiononp.111. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(9 a 3 b 2 3 Exercise2.241 5 x 2 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 y 3 3 Exercise2.242 Solutiononp.112. 10 a 3 b 2 c 2 Exercise2.243 a +10 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 +10 2 Exercise2.244 Solutiononp.112. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 2 )]TJ/F11 9.9626 Tf 10.792 -8.07 Td [(x 2 + y 2 Forthefollowingproblems,selectanumberornumberstoshowthat Exercise2.245 x 2 isnotgenerallyequalto 5 x 2 Exercise2.246 Solutiononp.112. x 2 isnotgenerallyequalto 7 x 2 Exercise2.247 a + b 2 isnotgenerallyequalto a 2 + b 2 Exercise2.248 Solutiononp.112. Forwhatrealnumberis a 2 equalto 6 a 2 ?

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78 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.249 Forwhatrealnumbers, a and b ,is a + b 2 equalto a 2 + b 2 ? Usetheorderofoperationstosimplifythequantitiesforthefollowingproblems. Exercise2.250 Solutiononp.112. 3 2 +7 Exercise2.251 4 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 Exercise2.252 Solutiononp.112. 5 2 +2 Exercise2.253 8 2 +3+5+7 Exercise2.254 Solutiononp.112. 2 5 +3+1 Exercise2.255 3 4 +2 4 +5 3 Exercise2.256 Solutiononp.112. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 5 Exercise2.257 2 2 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(10 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 3 Exercise2.258 Solutiononp.112. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 3 17 Exercise2.259 +3 2 +1 5 Exercise2.260 Solutiononp.112. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 4 +2 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 3 5 2 4 2 Exercise2.261 1 6 +0 8 +5 2 +8 3 Exercise2.262 Solutiononp.112. )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 2 +4 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1 1 +3 2 Exercise2.263 2 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 5 2 Exercise2.264 Solutiononp.112. +6 2 +2 19 Exercise2.265 6 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 5 + 4 3 + 10 Exercise2.266 Solutiononp.112. 5 [ 8 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 ] 2 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 + 7 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 2 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise2.267 +1 3 +2 3 +1 3 6 2 )]TJ/F7 6.9738 Tf 11.158 4.832 Td [(15 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [([2] 2 5 5 2 Exercise2.268 Solutiononp.112. 6 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 10 2 2 2 + 18 2 3 +7 2 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 3

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79 2.5.14ExercisesforReview Exercise2.269 Section2.2 Usealgebraicnotationtowritethestatement"anumberdividedbyeight,plus ve,isequaltoten." Exercise2.270 Solutiononp.112. Section2.3 Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to5andplacepointsatallrealnumbers thatarestrictlygreaterthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 butlessthanorequalto2. Exercise2.271 Section2.3 Iseveryintegerawholenumber? Exercise2.272 Solutiononp.112. Section2.4 Usethecommutativepropertyofmultiplicationtowriteanumberequaltothe number yx Exercise2.273 Section2.4 Usethedistributivepropertytoexpand 3 x +6 2.6RulesofExponents 6 2.6.1Overview TheProductRuleforExponents TheQuotientRuleforExponents ZeroasanExponent Wewillbeginourstudyoftherulesofexponentsbyrecallingthedenitionofexponents. DenitionofExponents If x isanyrealnumberand n isanaturalnumber,then x n = x x x ::: x | {z } n factorsof x Anexponentrecordsthenumberofidenticalfactorsinamultiplication. BaseExponentPower In x n x isthe base n isthe exponent Thenumberrepresentedby x n iscalleda power Theterm x n isreadas" x tothe n th." 2.6.2TheProductRuleforExponents Therstrulewewishtodevelopistheruleformultiplyingtwoexponentialquantitieshavingthe same base andnaturalnumberexponents.Thefollowingexamplessuggestthisrule: 6 Thiscontentisavailableonlineat.

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80 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Example2.70 x 2 x 4 = xx |{z} xxxx | {z } = xxxxxx | {z } = x 6 2+4=6 factors factors Example2.71 a a 2 = a |{z} aa |{z} = aaa |{z} = a 3 1+2=3 factorsfactors PRODUCTRULEFOREXPONENTS If x isarealnumberand n and m arenaturalnumbers, x n x m = x n + m Tomultiplytwoexponentialquantitieshavingthesamebase,addtheexponents.Keepinmind thattheexponentialquantitiesbeingmultiplied must havethe samebase forthisruletoapply. 2.6.3SampleSetA Findthefollowingproducts.Allexponentsarenaturalnumbers. Example2.72 x 3 x 5 = x 3+5 = x 8 Example2.73 a 6 a 14 = a 6+14 = a 20 Example2.74 y 5 y = y 5 y 1 = y 5+1 = y 6 Example2.75 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 5 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 8+5 = x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y 13 Example2.76 x 3 y 4 6 = xy 3+4 Sincethebasesarenotthesame,the productruledoesnotapply. 2.6.4PracticeSetA Findeachproduct. Exercise2.274 Solutiononp.112. x 2 x 5 Exercise2.275 Solutiononp.112. x 9 x 4 Exercise2.276 Solutiononp.112. y 6 y 4

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81 Exercise2.277 Solutiononp.112. c 12 c 8 Exercise2.278 Solutiononp.112. x +2 3 x +2 5 2.6.5SampleSetB Wecanusetherstruleofexponentsandtheothersthatwewilldevelopalongwiththepropertiesofreal numbers. Example2.77 2 x 3 7 x 5 = 2 7 x 3+5 =14 x 8 Weusedthecommutativeandassociativepropertiesofmultiplication.Inpractice,weusethese propertiesmentallyassigniedbythedrawingofthebox.Wedon'tactuallywritethesecond step. Example2.78 4 y 3 6 y 2 = 4 6 y 3+2 =24 y 5 Example2.79 9 a 2 b 6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 ab 4 2 b 3 = 9 8 2 a 2+1 b 6+4+3 =144 a 3 b 13 Example2.80 5 a +6 2 3 a +6 8 = 5 3 a +6 2+8 =15 a +6 10 Example2.81 4 x 3 12 y 2 =48 x 3 y 2 Example2.82 Thebasesarethesame,soweaddtheexponents.Althoughwedon'tknowexactlywhatnumber is,thenotation indicatestheaddition. 2.6.6PracticeSetB Performeachmultiplicationinonestep. Exercise2.279 Solutiononp.112. 3 x 5 2 x 2 Exercise2.280 Solutiononp.112. 6 y 3 3 y 4 Exercise2.281 Solutiononp.112. 4 a 3 b 2 9 a 2 b Exercise2.282 Solutiononp.112. x 4 4 y 2 2 x 2 7 y 6 Exercise2.283 Solutiononp.113. x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 3 4 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 2

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82 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.284 Solutiononp.113. 8 x 4 y 2 xx 3 y 5 Exercise2.285 Solutiononp.113. 2 aaa 3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(ab 2 a 3 b 6 ab 2 Exercise2.286 Solutiononp.113. a n a m a r 2.6.7TheQuotientRuleforExponents Thesecondrulewewishtodevelopistherulefordividingtwoexponentialquantitieshavingthesamebase andnaturalnumberexponents. Thefollowingexamplessuggestarulefordividingtwoexponentialquantitieshavingthesamebaseand naturalnumberexponents. Example2.83 x 5 x 2 = xxxxx xx = xx xxx xx = xxx = x 3 : Noticethat 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=3 : Example2.84 a 8 a 3 = aaaaaaaa aaa = aaa aaaaa aaa = aaaaa = a 5 : Noticethat 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=5 : QUOTIENTRULEFOREXPONENTS If x isarealnumberand n and m arenaturalnumbers, x n x m = x n )]TJ/F10 6.9738 Tf 6.226 0 Td [(m ;x 6 =0 Todividetwoexponentialquantitieshavingthesamenonzerobase,subtracttheexponentofthedenominatorfromtheexponentofthenumerator.Keepinmindthattheexponentialquantitiesbeingdivided must havethe samebase forthisruletoapply. 2.6.8SampleSetC Findthefollowingquotients.Allexponentsarenaturalnumbers. Example2.85 x 5 x 2 = x 5-2 = x 3 Thepartintheboxisusallydonementally. Example2.86 27 a 3 b 6 c 2 3 a 2 bc = 9 a 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 c 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =9 ab 5 c Example2.87 15 x 3 x 4 =5 x Thebasesarethesame,sowesubtracttheexponents.Althoughwedon'tknowexactlywhat )-222(4 is,thenotation )-222(4 indicatesthesubtraction.

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83 2.6.9PracticeSetC Findeachquotient Exercise2.287 Solutiononp.113. y 9 y 5 Exercise2.288 Solutiononp.113. a 7 a Exercise2.289 Solutiononp.113. x +6 5 x +6 3 Exercise2.290 Solutiononp.113. 26 x 4 y 6 z 2 13 x 2 y 2 z Whenwemakethesubtraction, n )]TJ/F11 9.9626 Tf 9.878 0 Td [(m ,inthedivision x n x m ,therearethreepossibilitiesforthevaluesofthe exponents: 1.Theexponentofthenumeratorisgreaterthantheexponentofthedenominator,thatis, n>m Thus,theexponent, n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m ,isanaturalnumber. 2.Theexponentsarethesame,thatis, n = m .Thus,theexponent, n )]TJ/F11 9.9626 Tf 9.963 0 Td [(m ,iszero,awholenumber. 3.Theexponentofthedenominatorisgreaterthantheexponentofthenumerator,thatis, n
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84 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.6.11SampleSetD Findeachvalue.Assumethebaseisnotzero. Example2.88 6 0 =1 Example2.89 247 0 =1 Example2.90 a +5 0 =1 Example2.91 4 y 0 =4 1=4 Example2.92 y 6 y 6 = y 0 =1 Example2.93 2 x 2 x 2 =2 x 0 =2 1=2 Example2.94 5 x +4 8 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 5 5 x +4 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 5 = x +4 8 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = x +4 5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 0 = x +4 5 2.6.12PracticeSetD Findeachvalue.Assumethebaseisnotzero. Exercise2.291 Solutiononp.113. y 7 y 3 Exercise2.292 Solutiononp.113. 6 x 4 2 x 3 Exercise2.293 Solutiononp.113. 14 a 7 7 a 2 Exercise2.294 Solutiononp.113. 26 x 2 y 5 4 xy 2 Exercise2.295 Solutiononp.113. 36 a 4 b 3 c 8 8 ab 3 c 6 Exercise2.296 Solutiononp.113. 51 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 17 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4

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85 Exercise2.297 Solutiononp.113. 52 a 7 b 3 a + b 8 26 a 2 b a + b 8 Exercise2.298 Solutiononp.113. a n a 3 Exercise2.299 Solutiononp.113. 14 x r y p z q 2 x r y h z 5 Wewillstudythecasewheretheexponentofthedenominatorisgreaterthantheexponentofthenumerator inSectionSection3.7. 2.6.13Exercises Usetheproductruleandquotientruleofexponentstosimplifythefollowingproblems.Assumethatall basesarenonzeroandthatallexponentsarewholenumbers. Exercise2.300 Solutiononp.113. 3 2 3 3 Exercise2.301 5 2 5 4 Exercise2.302 Solutiononp.113. 9 0 9 2 Exercise2.303 7 3 7 0 Exercise2.304 Solutiononp.113. 2 4 2 5 Exercise2.305 x 5 x 4 Exercise2.306 Solutiononp.113. x 2 x 3 Exercise2.307 a 9 a 7 Exercise2.308 Solutiononp.113. y 5 y 7 Exercise2.309 m 10 m 2 Exercise2.310 Solutiononp.113. k 8 k 3 Exercise2.311 y 3 y 4 y 6 Exercise2.312 Solutiononp.113. 3 x 2 2 x 5 Exercise2.313 a 2 a 3 a 8 Exercise2.314 Solutiononp.113. 4 y 4 5 y 6 Exercise2.315 2 a 3 b 2 3 ab

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86 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.316 Solutiononp.114. 12 xy 3 z 2 4 x 2 y 2 z 3 x Exercise2.317 ab )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a 2 b Exercise2.318 Solutiononp.114. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(4 x 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(8 xy 3 Exercise2.319 xy y )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(4 x 2 y 5 Exercise2.320 Solutiononp.114. )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 4 a 2 b 4 )]TJ/F7 6.9738 Tf 15.342 -4.147 Td [(1 2 b 4 Exercise2.321 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(3 8 )]TJ/F7 6.9738 Tf 15.342 -4.147 Td [(16 21 x 2 y 3 )]TJ/F11 9.9626 Tf 14.147 -8.07 Td [(x 3 y 2 Exercise2.322 Solutiononp.114. 8 5 8 3 Exercise2.323 6 4 6 3 Exercise2.324 Solutiononp.114. 2 9 2 4 Exercise2.325 4 16 4 13 Exercise2.326 Solutiononp.114. x 5 x 3 Exercise2.327 y 4 y 3 Exercise2.328 Solutiononp.114. y 9 y 4 Exercise2.329 k 16 k 13 Exercise2.330 Solutiononp.114. x 4 x 2 Exercise2.331 y 5 y 2 Exercise2.332 Solutiononp.114. m 16 m 9 Exercise2.333 a 9 b 6 a 5 b 2 Exercise2.334 Solutiononp.114. y 3 w 10 yw 5 Exercise2.335 m 17 n 12 m 16 n 10 Exercise2.336 Solutiononp.114. x 5 y 7 x 3 y 4 Exercise2.337 15 x 20 y 24 z 4 5 x 19 yz

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87 Exercise2.338 Solutiononp.114. e 11 e 11 Exercise2.339 6 r 4 6 r 4 Exercise2.340 Solutiononp.114. x 0 x 0 Exercise2.341 a 0 b 0 c 0 Exercise2.342 Solutiononp.114. 8 a 4 b 0 4 a 3 Exercise2.343 24 x 4 y 4 z 0 w 8 9 xyw 7 Exercise2.344 Solutiononp.114. t 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 4 Exercise2.345 x 3 x 6 x 2 Exercise2.346 Solutiononp.114. a 4 b 6 a 10 b 16 a 5 b 7 Exercise2.347 3 a 2 b 3 14 a 2 b 5 2 b Exercise2.348 Solutiononp.114. x +3 y 11 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 x +3 y 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise2.349 40 x 5 z 10 z )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 4 12 x + z 2 10 z 7 z )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 4 5 Exercise2.350 Solutiononp.114. x n x r Exercise2.351 a x b y c 5 z Exercise2.352 Solutiononp.114. x n x n +3 Exercise2.353 x n +3 x n Exercise2.354 Solutiononp.114. x n +2 x 3 x 4 x n Exercise2.355 Exercise2.356 Solutiononp.114. Exercise2.357 y y r Exercise2.358 Solutiononp.114. a a r b b

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88 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.6.14ExercisesforReview Exercise2.359 Section2.3 Whatnaturalnumberscanreplace x sothatthestatement )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 .

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89 2.7.3SampleSetA Simplifyeachexpressionusingthepowerruleforpowers.Allexponentsarenaturalnumbers. Example2.97 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 4 = x 3 4 x 12 Theboxrepresentsastepdonementally. Example2.98 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 5 3 = y 5 3 = y 15 Example2.99 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(d 20 6 = d 20 6 = d 120 Example2.100 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 4 = x 4 Althoughwedon'tknowexactlywhatnumber 4 is,thenotation 4 indicatesthemultiplication. 2.7.4PracticeSetA Simplifyeachexpressionusingthepowerruleforpowers. Exercise2.364 Solutiononp.114. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 5 4 Exercise2.365 Solutiononp.115. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 7 7 2.7.5ThePowerRuleforProducts Thefollowingexamplessuggestaruleforraisingaproducttoapower: Example2.101 ab 3 = ab ab ab Usethecommutativepropertyofmultiplication : = aaabbb = a 3 b 3 Example2.102 xy 5 = xy xy xy xy xy = xxxxx yyyyy = x 5 y 5 Example2.103 xyz 2 =4 xyz 4 xyz =4 4 xx yy zz =16 x 2 y 2 z 2 POWERRULEFORPRODUCTS If x and y arerealnumbersand n isanaturalnumber, xy n = x n y n Toraiseaproducttoapower,applytheexponenttoeachandeveryfactor.

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90 CHAPTER2.BASICPROPERTIESOFREALNUMBERS 2.7.6SampleSetB Makeuseofeitherorboththepowerruleforproductsandpowerruleforpowerstosimplifyeachexpression. Example2.104 ab 7 = a 7 b 7 Example2.105 axy 4 = a 4 x 4 y 4 Example2.106 ab 2 =3 2 a 2 b 2 =9 a 2 b 2 Don'tforgettoapplytheexponenttothe3! Example2.107 st 5 =2 5 s 5 t 5 =32 s 5 t 5 Example2.108 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(ab 3 2 = a 2 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(b 3 2 = a 2 b 6 Weusedtworuleshere.First,thepowerrulefor products.Second,thepowerruleforpowers. Example2.109 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 a 4 b 2 c 8 2 =7 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 4 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b 2 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c 8 2 =49 a 8 b 4 c 16 Example2.110 If 6 a 3 c 7 6 =0 ; then )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6 a 3 c 7 0 =1 Recallthat x 0 =1 for x 6 =0 : Example2.111 h 2 x +1 4 i 6 =2 6 x +1 24 =64 x +1 24 2.7.7PracticeSetB Makeuseofeitherorboththepowerruleforproductsandthepowerruleforpowerstosimplifyeach expression. Exercise2.366 Solutiononp.115. ax 4 Exercise2.367 Solutiononp.115. bxy 2 Exercise2.368 Solutiononp.115. [4 t s )]TJ/F8 9.9626 Tf 9.963 0 Td [(5] 3 Exercise2.369 Solutiononp.115. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(9 x 3 y 5 2 Exercise2.370 Solutiononp.115. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1 a 5 b 8 c 3 d 6 Exercise2.371 Solutiononp.115. [ a +8 a +5] 4 Exercise2.372 Solutiononp.115. 12 c 4 u 3 w )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 i 5

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91 Exercise2.373 Solutiononp.115. h 10 t 4 y 7 j 3 d 2 v 6 n 4 g 8 )]TJ/F11 9.9626 Tf 9.962 0 Td [(k 17 i 4 Exercise2.374 Solutiononp.115. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 3 x 5 y 2 y 6 9 Exercise2.375 Solutiononp.115. )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(10 6 10 12 10 5 10 2.7.8ThePowerRuleforQuotients Thefollowingexamplesuggestsaruleforraisingaquotienttoapower. Example2.112 )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [(a b 3 = a b a b a b = a a a b b b = a 3 b 3 POWERRULEFORQUOTIENTS If x and y arerealnumbersand n isanaturalnumber, x y n = x n y n ;y 6 =0 Toraiseaquotienttoapower,distributetheexponenttoboththenumeratoranddenominator. 2.7.9SampleSetC Makeuseofthepowerruleforquotients,thepowerruleforproducts,thepowerruleforpowers,ora combinationoftheserulestosimplifyeachexpression.Allexponentsarenaturalnumbers. Example2.113 x y 6 = x 6 y 6 Example2.114 )]TJ/F10 6.9738 Tf 5.762 -4.147 Td [(a c 2 = a 2 c 2 Example2.115 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(2 x b 4 = x 4 b 4 = 2 4 x 4 b 4 = 16 x 4 b 4 Example2.116 a 3 b 5 7 = a 3 7 b 5 7 = a 21 b 35 Example2.117 3 c 4 r 2 2 3 g 5 3 = 3 3 c 12 r 6 2 9 g 15 = 27 c 12 r 6 2 9 g 15 or 27 c 12 r 6 512 g 15 Example2.118 h a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a +7 i 4 = a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 4 a +7 4 Example2.119 h 6 x )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 4 2 a y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 6 i 2 = 6 2 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 8 2 2 a 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 12 = 36 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 8 4 a 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 12 = 9 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 8 a 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 12

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92 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Example2.120 a 3 b 5 a 2 b 3 = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 Wecansimplifywithintheparentheses.We havearulethattellsustoproceedthisway. = )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(ab 4 3 = a 3 b 12 a 3 b 5 a 2 b 3 = a 9 b 15 a 6 b 3 = a 9 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 b 15 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = a 3 b 12 Wecouldhaveactuallyusedthepowerrulefor quotientsrst.Distributetheexponent,then simplifyusingtheotherrules. Itisprobablybetter,forthesakeofconsistency, toworkinsidetheparenthesesrst. Example2.121 )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [(a r b s c t w = a rw b sw c tw 2.7.10PracticeSetC Makeuseofthepowerruleforquotients,thepowerruleforproducts,thepowerruleforpowers,ora combinationoftheserulestosimplifyeachexpression. Exercise2.376 Solutiononp.115. )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [(a c 5 Exercise2.377 Solutiononp.115. 2 x 3 y 3 Exercise2.378 Solutiononp.115. x 2 y 4 z 7 a 5 b 9 Exercise2.379 Solutiononp.115. h 2 a 4 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 b 3 c +6 i 4 Exercise2.380 Solutiononp.115. 8 a 3 b 2 c 6 4 a 2 b 3 Exercise2.381 Solutiononp.115. h + w 2 + w 5 i 10 Exercise2.382 Solutiononp.115. h 5 x 4 y +1 5 x 4 y +1 i 6 Exercise2.383 Solutiononp.115. 16 x 3 v 4 c 7 12 x 2 vc 6 0

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93 2.7.11Exercises Usethepowerrulesforexponentstosimplifythefollowingproblems.Assumethatallbasesarenonzeroand thatallvariableexponentsarenaturalnumbers. Exercise2.384 Solutiononp.115. ac 5 Exercise2.385 nm 7 Exercise2.386 Solutiononp.115. a 3 Exercise2.387 a 5 Exercise2.388 Solutiononp.115. xy 4 Exercise2.389 xy 5 Exercise2.390 Solutiononp.115. ab 4 Exercise2.391 mn 2 Exercise2.392 Solutiononp.115. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 y 3 2 Exercise2.393 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 m 3 4 Exercise2.394 Solutiononp.116. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 x 6 3 Exercise2.395 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 x 2 3 Exercise2.396 Solutiononp.116. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(10 a 2 b 2 Exercise2.397 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 x 2 y 3 2 Exercise2.398 Solutiononp.116. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 y 3 z 5 4 Exercise2.399 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a 5 b 11 0 Exercise2.400 Solutiononp.116. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 y 2 z 4 5 Exercise2.401 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m 6 n 2 p 5 5 Exercise2.402 Solutiononp.116. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 4 b 7 c 6 d 8 8 Exercise2.403 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 y 3 z 9 w 7 3 Exercise2.404 Solutiononp.116. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(9 xy 3 0

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94 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.405 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 f 2 r 6 s 5 4 Exercise2.406 Solutiononp.116. )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 8 c 10 d 8 e 4 f 9 2 Exercise2.407 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(3 5 a 3 b 5 c 10 3 Exercise2.408 Solutiononp.116. xy 4 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 y 4 Exercise2.409 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 a 2 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 a 5 2 Exercise2.410 Solutiononp.116. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 b 3 3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 3 b 3 4 Exercise2.411 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(h 3 k 5 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(h 2 k 4 3 Exercise2.412 Solutiononp.116. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 4 y 3 z 4 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 5 yz 2 2 Exercise2.413 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(ab 3 c 2 5 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 b 2 c 2 Exercise2.414 Solutiononp.116. 6 a 2 b 8 2 ab 5 2 Exercise2.415 a 3 b 4 5 a 4 b 4 3 Exercise2.416 Solutiononp.116. x 6 y 5 3 x 2 y 3 5 Exercise2.417 a 8 b 10 3 a 7 b 5 3 Exercise2.418 Solutiononp.116. m 5 n 6 p 4 4 m 4 n 5 p 4 Exercise2.419 x 8 y 3 z 2 5 x 6 yz 6 Exercise2.420 Solutiononp.116. 10 x 4 y 5 z 11 3 xy 2 4 Exercise2.421 9 a 4 b 5 2 b 2 c a 3 b bc Exercise2.422 Solutiononp.116. 2 x 3 y 3 4 5 x 6 y 8 2 x 5 y 3 2 Exercise2.423 3 x 5 y 2 Exercise2.424 Solutiononp.116. 3 ab 4 xy 3

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95 Exercise2.425 x 2 y 2 2 z 3 5 Exercise2.426 Solutiononp.116. 3 a 2 b 3 c 4 3 Exercise2.427 4 2 a 3 b 7 b 5 c 4 2 Exercise2.428 Solutiononp.116. h x 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 x +6 i 4 Exercise2.429 )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x n t 2 m 4 Exercise2.430 Solutiononp.116. x n +2 3 x 2 n Exercise2.431 xy 4 Exercise2.432 Solutiononp.116. Exercise2.433 Exercise2.434 Solutiononp.116. Exercise2.435 4 3 a a 4 a r Exercise2.436 Solutiononp.116. 4 x 2 y r Exercise2.437 2.7.12ExercisesforReview Exercise2.438 Solutiononp.116. Section2.3 Isthereasmallestinteger?Ifso,whatisit? Exercise2.439 Section2.4 Usethedistributivepropertytoexpand 5 a x +8 Exercise2.440 Solutiononp.117. Section2.5 Findthevalueof )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 ++4 3 +2 4 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise2.441 Section2.6 Assumingthebasesarenotzero,ndthevalueof )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(4 a 2 b 3 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(5 ab 4 .

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96 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.442 Solutiononp.117. Section2.6 Assumingthebasesarenotzero,ndthevalueof 36 x 10 y 8 z 3 w 0 9 x 5 y 2 z 2.8SummaryofKeyConcepts 8 2.8.1SummaryofKeyConcepts VariablesandConstantsSection2.2 A variable isaletterorsymbolthatrepresentsanymemberofacollectionoftwoormorenumbers.A constant isaletterorsymbolthatrepresentsaspecicnumber. BinaryOperationSection2.2 A binaryoperation isaprocessthatassignstwonumberstoasinglenumber. + ; )]TJ/F11 9.9626 Tf 7.749 0 Td [(; ; arebinary operations. GroupingSymbolsSection2.2 Groupingsymbolsareusedtoindicatethataparticularcollectionofnumbersandmeaningfuloperations istobeconsideredasasinglenumber 5 0 isnotmeaningful.Groupingsymbolscanalsodirectusin operationswhenmorethantwooperationsaretobeperformed.Commonalgebraicgroupingsymbolsare Parentheses : Brackets : hi Braces : fg Bar : OrderofOperationsSection2.2,Section2.5 Whentwoormoreoperationsaretobeperformedonacollectionofnumbers,thecorrectvaluecanbe obtainedonlybyusingthecorrectorderofoperations. TheRealNumberLineSection2.3 The realnumberline allowsustovisuallydisplaysomeofthenumbersinwhichweareinterested. CoordinateandGraphSection2.3 Thenumberassociatedwithapointonthenumberlineiscalledthe coordinate ofthepoint.Thepoint associatedwithanumberiscalledthe graph ofthenumber. RealNumberSection2.3 A realnumber isanynumberthatisthecoordinateofapointontherealnumberline. TypesofRealNumbersSection2.3 Thecollectionofrealnumbershasmanysubcollections.Theonesofmostinteresttousare thenaturalnumbers: f 1 ; 2 ; 3 ;::: g thewholenumbers: f 0 ; 1 ; 2 ; 3 ;::: g theintegers: f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 0 ; 1 ; 2 ; 3 ;::: g therationalnumbers: {allnumbersthatcanbeexpressedasthequotientoftwointegers} theirrationalnumbers: {allnumbersthathavenonendingandnonrepeatingdecimalrepresentations} PropertiesofRealNumbersSection2.4 Closure: If a and b arerealnumbers,then a + b and a b areuniquerealnumbers. Commutative: a + b = b + a and a b = b a Associative: a + b + c = a + b + c and a b c = a b c 8 Thiscontentisavailableonlineat.

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97 Distributive: a b + c = a b + a c Additiveidentity: 0istheadditiveidentity. a +0= a and 0+ a = a Multiplicativeidentity: 1isthemultiplicativeidentity. a 1= a and 1 a = a Additiveinverse: Foreachrealnumber a thereisexactlyonenumber )]TJ/F8 9.9626 Tf 7.749 0 Td [(a suchthat a + )]TJ/F8 9.9626 Tf 7.749 0 Td [(a=0 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(a+ a =0 Multiplicativeinverse: Foreachnonzerorealnumber a thereisexactlyonenonzerorealnumber 1 a such that a 1 a =1 and 1 a a =1 ExponentsSection2.5 Exponentsrecordthenumberofidenticalfactorsthatappearinamultiplication. x x x ::: x | {z } n factorsof x = x n RulesofExponentsSection2.6,Section2.7 If x isarealnumberand n and m arenaturalnumbers,then : x n x m = x n + m : x n x m = x n )]TJ/F10 6.9738 Tf 6.227 0 Td [(m x 6 =0 : x 0 =1 x 6 =0 : x n m = x n m : x y n = x n y n y 6 =0 2.9ExerciseSupplement 9 2.9.1ExerciseSupplement 2.9.1.1SymbolsandNotationsSection2.2 Forthefollowingproblems,simplifytheexpressions. Exercise2.443 Solutiononp.117. 12+7+3 Exercise2.444 9 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+6+2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+4 Exercise2.445 Solutiononp.117. 6[1+8+2] Exercise2.446 26 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 Exercise2.447 Solutiononp.117. +17+1+4 14 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise2.448 51 3 7 Exercise2.449 Solutiononp.117. +5+6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(+7 Exercise2.450 8 12 13+2 5 11 )]TJ/F8 9.9626 Tf 9.963 0 Td [([1+4+2] Exercise2.451 Solutiononp.117. 3 4 + 1 12 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(3 4 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 9 Thiscontentisavailableonlineat.

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98 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.452 48 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 1+17 6 Exercise2.453 Solutiononp.117. 29+11 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise2.454 88 11 + 99 9 +1 54 9 )]TJ/F6 4.9813 Tf 7.423 2.678 Td [(22 11 Exercise2.455 Solutiononp.117. 8 6 2 + 9 9 3 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(10 4 5 Forthefollowingproblems,writetheappropriaterelationsymbol = ;<;> inplaceofthe Exercise2.456 22 6 Exercise2.457 Solutiononp.117. 9[4+3] 6[1+8] Exercise2.458 3 : 06+2 : 11 4 : 01 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 : 06 Exercise2.459 Solutiononp.117. 2 0 Forthefollowingproblems,statewhetherthelettersorsymbolsarethesameordierent. Exercise2.460 Exercise2.461 Solutiononp.117. > and Exercise2.462 a = b and b = a Exercise2.463 Solutiononp.117. Representthesumof c and d twodierentways. Forthefollowingproblems,usealgebraicnotataion. Exercise2.464 8plus9 Exercise2.465 Solutiononp.117. 62dividedby f Exercise2.466 8times x +4 Exercise2.467 Solutiononp.117. 6times x ,minus2 Exercise2.468 x +1 dividedby x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise2.469 Solutiononp.117. y +11 dividedby y +10 ,minus12 Exercise2.470 zerominus a times b

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99 2.9.1.2TheRealNumberLineandtheRealNumbersSection2.3 Exercise2.471 Solutiononp.117. Iseverynaturalnumberawholenumber? Exercise2.472 Iseveryrationalnumberarealnumber? Forthefollowingproblems,locatethenumbersonanumberlinebyplacingapointattheirapproximate position. Exercise2.473 Solutiononp.117. 2 Exercise2.474 3 : 6 Exercise2.475 Solutiononp.117. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 3 8 Exercise2.476 0 Exercise2.477 Solutiononp.117. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 1 2 Exercise2.478 Drawanumberlinethatextendsfrom10to20.Placeapointatalloddintegers. Exercise2.479 Solutiononp.117. Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 to 10 .Placeapointatallnegativeoddintegersand atallevenpositiveintegers. Exercise2.480 Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 to 10 .Placeapointatallintegersthataregreaterthen orequalto )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 butstrictlylessthan5. Exercise2.481 Solutiononp.117. Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 to 10 .Placeapointatallrealnumbersthatarestrictly greaterthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 butlessthanorequalto7. Exercise2.482 Drawanumberlinethatextendsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 to 10 .Placeapointatallrealnumbersbetweenand including )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 and4. Forthefollowingproblems,writetheappropriaterelationsymbol = ;<;> : Exercise2.483 Solutiononp.118. )]TJ/F8 9.9626 Tf 7.748 0 Td [(30 Exercise2.484 )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 Exercise2.485 Solutiononp.118. )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise2.486 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 1 2 Exercise2.487 Solutiononp.118. Isthereasmallesttwodigitinteger?Ifso,whatisit? Exercise2.488 Isthereasmallesttwodigitrealnumber?Ifso,whatisit? Forthefollowingproblems,whatintegerscanreplace x sothatthestatementsaretrue?

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100 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.489 Solutiononp.118. 4 x 7 Exercise2.490 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x< 1 Exercise2.491 Solutiononp.118. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3
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101 Exercise2.507 Solutiononp.118. 16 ab 2 c Exercise2.508 4 axyc 4 d 4 e Exercise2.509 Solutiononp.118. 3 x +25 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 x +6 Exercise2.510 8 b a )]TJ/F8 9.9626 Tf 9.963 0 Td [(69 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Forthefollowingproblems,usethedistributivepropertytoexpandtheexpressions. Exercise2.511 Solutiononp.118. 3 a +4 Exercise2.512 a b +3 c Exercise2.513 Solutiononp.118. 2 g h +2 k Exercise2.514 m +5 n 6 p Exercise2.515 Solutiononp.118. 3 y x +4 z +5 w Exercise2.516 a +2 b +2 c Exercise2.517 Solutiononp.118. x + y a +3 b Exercise2.518 10 a z b z + c 2.9.1.4ExponentsSection2.5 Forthefollowingproblems,writetheexpressionsusingexponentialnotation. Exercise2.519 Solutiononp.118. x tothefth. Exercise2.520 y +2 cubed. Exercise2.521 Solutiononp.118. a +2 b squaredminus a +3 b tothefourth. Exercise2.522 x cubedplus2times y )]TJ/F11 9.9626 Tf 9.963 0 Td [(x totheseventh. Exercise2.523 Solutiononp.118. aaaaaaa Exercise2.524 2 2 2 2 Exercise2.525 Solutiononp.118. )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 xxxyyyyy Exercise2.526 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9+ x +1 x +1 x +1

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102 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.527 Solutiononp.118. 2 zzyzyyy +7 zzyz a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Forthefollowingproblems,expandthetermssothatnoexponentsappear. Exercise2.528 x 3 Exercise2.529 Solutiononp.118. 3 x 3 Exercise2.530 7 3 x 2 Exercise2.531 Solutiononp.118. b 2 Exercise2.532 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 a 2 3 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 Exercise2.533 Solutiononp.119. )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x 3 +7 2 )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 3 z +10 Exercise2.534 Choosevaluesfor a and b toshowthat a. a + b 2 isnotalwaysequalto a 2 + b 2 b. a + b 2 maybeequalto a 2 + b 2 Exercise2.535 Solutiononp.119. Choosevaluefor x toshowthat a. x 2 isnotalwaysequalto 4 x 2 b. x 2 maybeequalto 4 x 2 2.9.1.5RulesofExponentsSection2.6-ThePowerRulesforExponentsSection2.7 Simplifythefollowingproblems. Exercise2.536 4 2 +8 Exercise2.537 Solutiononp.119. 6 3 +5 Exercise2.538 1 8 +0 10 +3 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 2 +2 3 Exercise2.539 Solutiononp.119. 12 2 +0 : 3 2 Exercise2.540 3 4 +1 2 2 +4 2 +3 2 Exercise2.541 Solutiononp.119. 6 2 +3 2 2 2 +1 + +4 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 2 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 Exercise2.542 a 4 a 3 Exercise2.543 Solutiononp.119. 2 b 5 2 b 3

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103 Exercise2.544 4 a 3 b 2 c 8 3 ab 2 c 0 Exercise2.545 Solutiononp.119. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 x 4 y 10 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(xy 3 Exercise2.546 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 xyz 2 )]TJ/F8 9.9626 Tf 10.792 -8.07 Td [(2 x 2 y 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 x 2 y 2 z 4 Exercise2.547 Solutiononp.119. a 4 Exercise2.548 xy 2 Exercise2.549 Solutiononp.119. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 y 4 6 Exercise2.550 )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(a 4 b 7 c 7 z 12 9 Exercise2.551 Solutiononp.119. )]TJ/F7 6.9738 Tf 5.762 -4.148 Td [(3 4 x 8 y 6 z 0 a 10 b 15 2 Exercise2.552 x 8 x 5 Exercise2.553 Solutiononp.119. 14 a 4 b 6 c 7 2 ab 3 c 2 Exercise2.554 11 x 4 11 x 4 Exercise2.555 Solutiononp.119. x 4 x 10 x 3 Exercise2.556 a 3 b 7 a 9 b 6 a 5 b 10 Exercise2.557 Solutiononp.119. x 4 y 6 z 10 4 xy 5 z 7 3 Exercise2.558 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 13 x +5 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 10 x +5 Exercise2.559 Solutiononp.119. 3 x 2 4 y 3 2 Exercise2.560 x + y 9 x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 4 x + y 3 Exercise2.561 Solutiononp.119. x n x m Exercise2.562 a n +2 a n +4 Exercise2.563 Solutiononp.119. 6 b 2 n +7 8 b 5 n +2 Exercise2.564 18 x 4 n +9 2 x 2 n +1 Exercise2.565 Solutiononp.119. )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x 5 t y 4 r 7

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104 CHAPTER2.BASICPROPERTIESOFREALNUMBERS Exercise2.566 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 n b 3 m c 4 p 6 r Exercise2.567 Solutiononp.119. u w u k 2.10ProciencyExam 10 2.10.1ProciencyExam Forthefollowingproblems,simplifyeachoftheexpressions. Exercise2.568 Solutiononp.119. Section2.2 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 4+3 4 3 Exercise2.569 Solutiononp.119. Section2.2 f 2+7 2 g 0 Exercise2.570 Solutiononp.119. Section2.2 1 8 +4 0 +3 3 +4 2 2 +15 Exercise2.571 Solutiononp.119. Section2.2 2 3 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 2 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 + 5 2 2 +3 2 11 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise2.572 Solutiononp.119. Section2.2 Writetheappropriaterelationsymbol >;< inplaceofthe 5+11 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Forthefollowingproblems,usealgebraicnotation. Exercise2.573 Solutiononp.119. Section2.2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 times x plus 2 Exercise2.574 Solutiononp.120. Section2.2 Anumberdividedbytwelveislessthanorequaltothesamenumberplusfour. Exercise2.575 Solutiononp.120. Section2.3 Locatetheapproximatepositionof )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 6 onthenumberline. Exercise2.576 Solutiononp.120. Section2.3 Is0apositivenumber,anegativenumber,neither,orboth? Exercise2.577 Solutiononp.120. Section2.3 Drawaportionofthenumberlineandplacepointsatallevenintegersstrictly between14and20. Exercise2.578 Solutiononp.120. Section2.3 Drawaportionofthenumberlineandplacepointsatallrealnumbersstrictly greaterthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 butlessthanorequalto4. Exercise2.579 Solutiononp.120. Section2.3 Whatwholenumberscanreplace x sothatthefollowingstatementistrue? )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x 5 Exercise2.580 Solutiononp.120. Section2.3 Istherealargestrealnumberbetweenandincluding6and10?Ifso,whatisit? 10 Thiscontentisavailableonlineat.

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105 Exercise2.581 Solutiononp.120. Section2.4 Usethecommutativepropertyofmultiplicationtowrite m a +3 inanequivalent form. Exercise2.582 Solutiononp.120. Section2.4 Usethecommutativepropertiestosimplify 3 a 4 b 8 cd Exercise2.583 Solutiononp.120. Section2.4 Usethecommutativepropertiestosimplify 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(92 y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(93 y Exercise2.584 Solutiononp.120. Section2.5 Simplify4squaredtimes x cubedtimes y tothefth. Exercise2.585 Solutiononp.120. Section2.5 Simplify aabbbbabba a Forthefollowingproblems,usetherulesofexponentstosimplifyeachoftheexpressions. Exercise2.586 Solutiononp.120. Section2.6,Section2.7 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 ab 2 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a 3 b 3 Exercise2.587 Solutiononp.120. Section2.6,Section2.7 x 10 y 12 x 2 y 5 Exercise2.588 Solutiononp.120. Section2.6,Section2.7 52 x 7 y 10 y )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 4 12 y + x 5 4 y 6 y )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 4 10 y + x Exercise2.589 Solutiononp.120. Section2.6,Section2.7 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x n y 3 m z 2 p 4 Exercise2.590 Solutiononp.120. x +4 0 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0 Exercise2.591 Solutiononp.120. x r x y x y r Exercise2.592 Solutiononp.120. Section2.6,Section2.7 Whatwordisusedtodescribetheletterorsymbolthatrepresents anunspeciedmemberofaparticularcollectionoftwoormorenumbersthatareclearlydened?

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106 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutionstoExercisesinChapter2 SolutiontoExercise2.1p.49 29 x; 29 x; x ; 29 x ; x SolutiontoExercise2.2p.50 27 SolutiontoExercise2.3p.50 48 SolutiontoExercise2.4p.50 24 SolutiontoExercise2.5p.50 4 SolutiontoExercise2.6p.52 49 SolutiontoExercise2.7p.52 26 SolutiontoExercise2.8p.52 37 SolutiontoExercise2.9p.52 17 SolutiontoExercise2.10p.52 20 SolutiontoExercise2.12p.52 7 SolutiontoExercise2.14p.52 8 SolutiontoExercise2.16p.53 78 SolutiontoExercise2.18p.53 203 SolutiontoExercise2.20p.53 29 SolutiontoExercise2.22p.53 1 SolutiontoExercise2.24p.53 91 2 3 SolutiontoExercise2.26p.53 508 SolutiontoExercise2.28p.53 24 : 4 SolutiontoExercise2.30p.53 55 SolutiontoExercise2.32p.53 1 SolutiontoExercise2.34p.53 0 SolutiontoExercise2.36p.53 dierent SolutiontoExercise2.38p.54 same

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107 SolutiontoExercise2.40p.54 a + b;b + a SolutiontoExercise2.42p.54 x +16 SolutiontoExercise2.44p.54 81 x SolutiontoExercise2.46p.54 x + b x +7 SolutiontoExercise2.48p.54 x 7 b SolutiontoExercise2.50p.54 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8=17 SolutiontoExercise2.52p.54 x 6 44 SolutiontoExercise2.54p.54 true SolutiontoExercise2.56p.54 true SolutiontoExercise2.58p.55 false SolutiontoExercise2.60p.55 120 SolutiontoExercise2.62p.55 0 : 00024 ; or 1 4165 SolutiontoExercise2.64p.58 yes SolutiontoExercise2.65p.58 yes SolutiontoExercise2.66p.58 yes SolutiontoExercise2.67p.58 yes SolutiontoExercise2.68p.58 no SolutiontoExercise2.69p.58 yes SolutiontoExercise2.70p.59 yes SolutiontoExercise2.71p.59 yes SolutiontoExercise2.72p.59 yes SolutiontoExercise2.73p.59 no,no SolutiontoExercise2.74p.59 innitelymany,innitelymany SolutiontoExercise2.75p.60 0,1,2 SolutiontoExercise2.76p.60

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108 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.77p.60 Q;R SolutiontoExercise2.79p.60 W;Z;Q;R SolutiontoExercise2.81p.60 Q;R SolutiontoExercise2.83p.60 Q;R SolutiontoExercise2.85p.60 SolutiontoExercise2.87p.60 neither SolutiontoExercise2.89p.61 SolutiontoExercise2.91p.61 SolutiontoExercise2.93p.61 ;no SolutiontoExercise2.95p.61 < SolutiontoExercise2.97p.61 > SolutiontoExercise2.99p.61 no SolutiontoExercise2.101p.61 99 SolutiontoExercise2.103p.61 yes,0 SolutiontoExercise2.105p.61 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise2.107p.62 Therearenonaturalnumbersbetween )]TJ/F15 9.9626 Tf 7.748 0 Td [(15and )]TJ/F15 9.9626 Tf 7.749 0 Td [(1. SolutiontoExercise2.109p.62 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(95 1 SolutiontoExercise2.111p.62 Yes,everyintegerisarationalnumber. SolutiontoExercise2.113p.62 Yes. 1 2 + 1 2 =1 or 1+1=2 SolutiontoExercise2.115p.62 5units SolutiontoExercise2.117p.62 8units SolutiontoExercise2.119p.62 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(n units SolutiontoExercise2.121p.62 23

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109 SolutiontoExercise2.123p.62 dierent SolutiontoExercise2.125p.63 true SolutiontoExercise2.126p.64 5 SolutiontoExercise2.127p.64 m SolutiontoExercise2.128p.64 7 SolutiontoExercise2.129p.64 6 SolutiontoExercise2.130p.64 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise2.131p.64 b +7 SolutiontoExercise2.132p.65 2+5 SolutiontoExercise2.133p.65 x +5 SolutiontoExercise2.134p.65 a 6 SolutiontoExercise2.135p.65 m +3 m +4 SolutiontoExercise2.136p.66 189 ady SolutiontoExercise2.137p.66 960 abcz SolutiontoExercise2.138p.66 72 pqr a + b SolutiontoExercise2.139p.67 thecommutativepropertyofmultiplication SolutiontoExercise2.140p.67 6+3 SolutiontoExercise2.141p.67 7 x +42 SolutiontoExercise2.142p.67 4 a +4 y SolutiontoExercise2.143p.67 9 a +2 a SolutiontoExercise2.144p.67 ax +5 a SolutiontoExercise2.145p.67 x + y SolutiontoExercise2.146p.68 3+ x SolutiontoExercise2.148p.68 10 x SolutiontoExercise2.150p.69 6 r

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110 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.152p.69 cx SolutiontoExercise2.154p.69 s +16 SolutiontoExercise2.156p.69 a +7 x +16 SolutiontoExercise2.158p.69 m : 06 SolutiontoExercise2.160p.69 h +15 SolutiontoExercise2.162p.69 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b k SolutiontoExercise2.164p.69 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 SolutiontoExercise2.166p.69 [U+25CB] SolutiontoExercise2.168p.69 18 xy SolutiontoExercise2.170p.69 24 abc SolutiontoExercise2.172p.70 30 mnruz SolutiontoExercise2.174p.70 1 16 ade SolutiontoExercise2.176p.70 9 x +2 y + z x +5 y SolutiontoExercise2.178p.70 br +5 b SolutiontoExercise2.180p.70 jk + k SolutiontoExercise2.182p.70 xz +9 wz SolutiontoExercise2.184p.70 8 g +2 fg SolutiontoExercise2.186p.70 30 xy +45 xz SolutiontoExercise2.188p.70 xz + yz + mz SolutiontoExercise2.190p.70 ax + bx + cx +10 a +10 b +10 c SolutiontoExercise2.192p.70 a +16 SolutiontoExercise2.194p.71 348 : 3 a +81 : 7 b +15 : 05 c SolutiontoExercise2.196p.71 2 L m z t +16 kz t SolutiontoExercise2.198p.71 false SolutiontoExercise2.200p.71 <

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111 SolutiontoExercise2.202p.72 a 4 SolutiontoExercise2.203p.72 b 2 c 4 SolutiontoExercise2.204p.72 2 2 7 3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 SolutiontoExercise2.205p.73 8 x 3 yz 5 SolutiontoExercise2.206p.73 4 aaa SolutiontoExercise2.207p.73 a a a SolutiontoExercise2.208p.74 Select x =3 .Then 3 2 = 2 =225 ,but 5 3 2 =5 9=45 225 6 =45 SolutiontoExercise2.209p.76 29 SolutiontoExercise2.210p.76 3 SolutiontoExercise2.211p.76 9 SolutiontoExercise2.212p.76 8 SolutiontoExercise2.213p.76 3 SolutiontoExercise2.214p.76 b 4 SolutiontoExercise2.216p.76 x 8 SolutiontoExercise2.218p.76 5 s 2 SolutiontoExercise2.220p.76 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [( b +7 2 SolutiontoExercise2.222p.76 x 5 SolutiontoExercise2.224p.76 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 4 x 2 y 5 SolutiontoExercise2.226p.76 7 x 2 a +8 2 SolutiontoExercise2.228p.76 x 5 or 4 5 x 5 SolutiontoExercise2.230p.77 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 4 a 5 b 5 SolutiontoExercise2.232p.77 z + w 3 z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w 2 SolutiontoExercise2.234p.77 3 x 3 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( x +1 3 SolutiontoExercise2.236p.77 6 6 SolutiontoExercise2.238p.77 8 x x x y y

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112 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.240p.77 aaabb aaabb aaabb or 9 9 9 aaaaaaaaabbbbbb SolutiontoExercise2.242p.77 10 aaabb c c or 10 3 3 aaabbcc SolutiontoExercise2.244p.77 xx )]TJ/F11 9.9626 Tf 9.963 0 Td [(yy xx + yy SolutiontoExercise2.246p.77 Select x =2 : Then, 196 6 =28 : SolutiontoExercise2.248p.77 zero SolutiontoExercise2.250p.78 16 SolutiontoExercise2.252p.78 105 SolutiontoExercise2.254p.78 59 SolutiontoExercise2.256p.78 4 SolutiontoExercise2.258p.78 1 SolutiontoExercise2.260p.78 4 SolutiontoExercise2.262p.78 71 SolutiontoExercise2.264p.78 51 19 SolutiontoExercise2.266p.78 5 SolutiontoExercise2.268p.78 1070 11 or 97 : 27 SolutiontoExercise2.270p.79 SolutiontoExercise2.272p.79 xy SolutiontoExercise2.274p.80 x 2+5 = x 7 SolutiontoExercise2.275p.80 x 9+4 = x 13 SolutiontoExercise2.276p.80 y 6+4 = y 10 SolutiontoExercise2.277p.80 c 12+8 = c 20 SolutiontoExercise2.278p.81 x +2 3+5 = x +2 8 SolutiontoExercise2.279p.81 6 x 7 SolutiontoExercise2.280p.81 18 y 7 SolutiontoExercise2.281p.81 36 a 5 b 3

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113 SolutiontoExercise2.282p.81 56 x 6 y 8 SolutiontoExercise2.283p.81 4 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 5 SolutiontoExercise2.284p.81 8 x 8 y 7 SolutiontoExercise2.285p.82 12 a 10 b 5 SolutiontoExercise2.286p.82 a n + m + r SolutiontoExercise2.287p.83 y 4 SolutiontoExercise2.288p.83 a 6 SolutiontoExercise2.289p.83 x +6 2 SolutiontoExercise2.290p.83 2 x 2 y 4 z SolutiontoExercise2.291p.84 y 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = y 4 SolutiontoExercise2.292p.84 3 x 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 =3 x SolutiontoExercise2.293p.84 2 a 7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 =2 a 5 SolutiontoExercise2.294p.84 13 2 xy 3 SolutiontoExercise2.295p.84 9 2 a 3 c 2 SolutiontoExercise2.296p.84 3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 SolutiontoExercise2.297p.84 2 a 5 b 2 SolutiontoExercise2.298p.85 a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise2.299p.85 7 y p )]TJ/F10 6.9738 Tf 6.227 0 Td [(h z q )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 SolutiontoExercise2.300p.85 3 5 =243 SolutiontoExercise2.302p.85 9 2 =81 SolutiontoExercise2.304p.85 2 9 =512 SolutiontoExercise2.306p.85 x 5 SolutiontoExercise2.308p.85 y 12 SolutiontoExercise2.310p.85 k 11 SolutiontoExercise2.312p.85 6 x 7

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114 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.314p.85 20 y 10 SolutiontoExercise2.316p.85 144 x 4 y 5 z 3 SolutiontoExercise2.318p.86 32 x 3 y 3 SolutiontoExercise2.320p.86 1 8 a 2 b 8 SolutiontoExercise2.322p.86 8 2 =64 SolutiontoExercise2.324p.86 2 5 =32 SolutiontoExercise2.326p.86 x 2 SolutiontoExercise2.328p.86 y 5 SolutiontoExercise2.330p.86 x 2 SolutiontoExercise2.332p.86 m 7 SolutiontoExercise2.334p.86 y 2 w 5 SolutiontoExercise2.336p.86 x 2 y 3 SolutiontoExercise2.338p.86 e 0 =1 SolutiontoExercise2.340p.87 x 0 =1 SolutiontoExercise2.342p.87 2 a SolutiontoExercise2.344p.87 t 2 y 4 SolutiontoExercise2.346p.87 a 9 b 15 SolutiontoExercise2.348p.87 x +3 y 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 3 SolutiontoExercise2.350p.87 x n + r SolutiontoExercise2.352p.87 x 2 n +3 SolutiontoExercise2.354p.87 x SolutiontoExercise2.356p.87 SolutiontoExercise2.358p.87 a + r b + SolutiontoExercise2.360p.88 8 ax +12 bx SolutiontoExercise2.362p.88 8

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115 SolutiontoExercise2.364p.89 x 20 SolutiontoExercise2.365p.89 y 49 SolutiontoExercise2.366p.90 a 4 x 4 SolutiontoExercise2.367p.90 9 b 2 x 2 y 2 SolutiontoExercise2.368p.90 64 t 3 s )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 3 SolutiontoExercise2.369p.90 81 x 6 y 10 SolutiontoExercise2.370p.90 a 30 b 48 c 18 d 6 SolutiontoExercise2.371p.90 a +8 4 a +5 4 SolutiontoExercise2.372p.90 12 5 c 20 u 15 w )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 10 SolutiontoExercise2.373p.91 10 4 t 16 y 28 j 12 d 8 v 24 n 16 g 32 )]TJ/F11 9.9626 Tf 9.963 0 Td [(k 68 SolutiontoExercise2.374p.91 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 8 y 8 9 = x 72 y 72 SolutiontoExercise2.375p.91 10 230 SolutiontoExercise2.376p.92 a 5 c 5 SolutiontoExercise2.377p.92 8 x 3 27 y 3 SolutiontoExercise2.378p.92 x 18 y 36 z 63 a 45 b 9 SolutiontoExercise2.379p.92 16 a 16 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 81 b 12 c +6 4 SolutiontoExercise2.380p.92 8 a 3 b 3 c 18 SolutiontoExercise2.381p.92 + w 20 + w 50 SolutiontoExercise2.382p.92 1 ; if x 4 y +1 6 =0 SolutiontoExercise2.383p.92 1 ; if x 2 vc 6 6 =0 SolutiontoExercise2.384p.93 a 5 c 5 SolutiontoExercise2.386p.93 8 a 3 SolutiontoExercise2.388p.93 81 x 4 y 4 SolutiontoExercise2.390p.93 81 a 4 b 4

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116 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.392p.93 49 y 6 SolutiontoExercise2.394p.93 125 x 18 SolutiontoExercise2.396p.93 100 a 4 b 2 SolutiontoExercise2.398p.93 x 8 y 12 z 20 SolutiontoExercise2.400p.93 x 15 y 10 z 20 SolutiontoExercise2.402p.93 a 32 b 56 c 48 d 64 SolutiontoExercise2.404p.93 1 SolutiontoExercise2.406p.94 1 64 c 20 d 16 e 8 f 18 SolutiontoExercise2.408p.94 x 6 y 8 SolutiontoExercise2.410p.94 a 18 b 21 SolutiontoExercise2.412p.94 x 26 y 14 z 8 SolutiontoExercise2.414p.94 4 a 2 b 6 SolutiontoExercise2.416p.94 x 8 SolutiontoExercise2.418p.94 m 4 n 4 p 12 SolutiontoExercise2.420p.94 1000 x 8 y 7 z 33 SolutiontoExercise2.422p.94 25 x 14 y 22 SolutiontoExercise2.424p.94 27 a 3 b 3 64 x 3 y 3 SolutiontoExercise2.426p.95 27 a 6 b 9 c 12 SolutiontoExercise2.428p.95 x 8 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 12 x +6 4 SolutiontoExercise2.430p.95 x n +6 SolutiontoExercise2.432p.95 SolutiontoExercise2.434p.95 SolutiontoExercise2.436p.95

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117 SolutiontoExercise2.438p.95 no SolutiontoExercise2.440p.95 147 SolutiontoExercise2.442p.96 4 x 5 y 6 z 2 SolutiontoExercise2.443p.97 61 SolutiontoExercise2.445p.97 438 SolutiontoExercise2.447p.97 2 SolutiontoExercise2.449p.97 79 SolutiontoExercise2.451p.97 37 48 SolutiontoExercise2.453p.98 8 SolutiontoExercise2.455p.98 43 SolutiontoExercise2.457p.98 252 > 246 SolutiontoExercise2.459p.98 2 > 0 SolutiontoExercise2.461p.98 dierent SolutiontoExercise2.463p.98 c + d ; d + c SolutiontoExercise2.465p.98 62 f or 62 f SolutiontoExercise2.467p.98 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise2.469p.98 y +11 y +10 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 or y +11 y +10 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 SolutiontoExercise2.471p.99 yes SolutiontoExercise2.473p.99 SolutiontoExercise2.475p.99 SolutiontoExercise2.477p.99 SolutiontoExercise2.479p.99

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118 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.481p.99 SolutiontoExercise2.483p.99 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 < 0 SolutiontoExercise2.485p.99 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise2.487p.99 yes, )]TJ/F8 9.9626 Tf 9.963 0 Td [(99 SolutiontoExercise2.489p.100 4 ; 5 ; 6 ; or 7 SolutiontoExercise2.491p.100 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; 0 ; 1 ; or 2 SolutiontoExercise2.493p.100 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise2.495p.100 4 SolutiontoExercise2.497p.100 commutative,multiplication SolutiontoExercise2.499p.100 4 b + a SolutiontoExercise2.501p.100 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 SolutiontoExercise2.503p.100 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 or )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 or )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 or )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 SolutiontoExercise2.505p.100 4 SolutiontoExercise2.507p.100 32 abc SolutiontoExercise2.509p.101 0 SolutiontoExercise2.511p.101 3 a +12 SolutiontoExercise2.513p.101 8 gh +4 gk SolutiontoExercise2.515p.101 6 xy +12 yz +15 wy SolutiontoExercise2.517p.101 4 ax +3 bx +4 ay +3 by SolutiontoExercise2.519p.101 x 5 SolutiontoExercise2.521p.101 a +2 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( a +3 b 4 SolutiontoExercise2.523p.101 a 7 SolutiontoExercise2.525p.101 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 4 x 3 y 5 SolutiontoExercise2.527p.101 2 y 4 z 3 +7 yz 3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 3 SolutiontoExercise2.529p.102 3 xxx

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119 SolutiontoExercise2.531p.102 4 b 4 b SolutiontoExercise2.533p.102 xxx +7 xxx +7 yy )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 yy )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 yy )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 z +10 SolutiontoExercise2.535p.102 a anyvalueexceptzero b onlyzero SolutiontoExercise2.537p.102 366 SolutiontoExercise2.539p.102 180 : 3 SolutiontoExercise2.541p.102 10 SolutiontoExercise2.543p.102 4 b 8 SolutiontoExercise2.545p.103 6 x 5 y 13 SolutiontoExercise2.547p.103 81 a 4 SolutiontoExercise2.549p.103 x 12 y 24 SolutiontoExercise2.551p.103 9 16 x 16 y 12 a 20 b 30 SolutiontoExercise2.553p.103 7 a 3 b 3 c 5 SolutiontoExercise2.555p.103 x 11 SolutiontoExercise2.557p.103 x 13 y 9 z 19 SolutiontoExercise2.559p.103 9 x 4 16 y 6 SolutiontoExercise2.561p.103 x n + m SolutiontoExercise2.563p.103 48 b 7 n +9 SolutiontoExercise2.565p.103 x 35 t y 28 r SolutiontoExercise2.567p.104 u w )]TJ/F10 6.9738 Tf 6.227 0 Td [(k SolutiontoExercise2.568p.104 40 SolutiontoExercise2.569p.104 1 SolutiontoExercise2.570p.104 137 68 SolutiontoExercise2.571p.104 75 SolutiontoExercise2.572p.104 >

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120 CHAPTER2.BASICPROPERTIESOFREALNUMBERS SolutiontoExercise2.573p.104 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x +2 SolutiontoExercise2.574p.104 x 12 x +4 SolutiontoExercise2.575p.104 SolutiontoExercise2.576p.104 Zeroisneitherpositivenornegative. SolutiontoExercise2.577p.104 SolutiontoExercise2.578p.104 SolutiontoExercise2.579p.104 0 ; 1 ; 2 ; 3 ; 4 ; 5 SolutiontoExercise2.580p.104 yes;10 SolutiontoExercise2.581p.105 a +3 m SolutiontoExercise2.582p.105 96 abcd SolutiontoExercise2.583p.105 24 y 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 2 SolutiontoExercise2.584p.105 16 x 3 y 5 SolutiontoExercise2.585p.105 81 a 5 b 6 SolutiontoExercise2.586p.105 72 a 11 b 7 SolutiontoExercise2.587p.105 x 8 y 7 SolutiontoExercise2.588p.105 13 x 7 y 4 )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(y )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 4 2 y + x 4 SolutiontoExercise2.589p.105 x 4 n y 12 m z 8 p SolutiontoExercise2.590p.105 1 SolutiontoExercise2.591p.105 SolutiontoExercise2.592p.105 avariable

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Chapter3 BasicOperationswithRealNumbers 3.1Objectives 1 Aftercompletingthischapter,youshould SignedNumbersSection3.2 befamiliarwithpositiveandnegativenumbersandwiththeconceptofopposites AbsoluteValueSection3.3 understandthegeometricandalgebraicdenitionsofabsolutevalue AdditionofSignedNumbersSection3.4 beabletoaddnumberswithlikesignsandunlikesigns understandadditionwithzero SubtractionofSignedNumbersSection3.5 understandthedenitionofsubtraction beabletosubtractsignednumbers MultiplicationandDivisionofSignedNumbersSection3.6 beabletomultiplyanddividesignednumbers NegativeExponentsSection3.7 understandtheconceptsofreciprocalsandnegativeexponents beabletoworkwithnegativeexponents ScienticNotationSection3.8 beabletoconvertanumberfromstandardformtoscienticformandfromscienticformtostandard form beabletoworkwithnumbersinscienticnotation 1 Thiscontentisavailableonlineat. 121

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122 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.2SignedNumbers 2 3.2.1Overview PositiveandNegativeNumbers Opposites 3.2.2PositiveandNegativeNumbers WhenwestudiedthenumberlineinSectionSection2.3wenotedthat Eachpointonthenumberlinecorrespondstoarealnumber,andeachrealnumberislocatedata uniquepointonthenumberline. PositiveandNegativeNumbers Eachrealnumberhasa sign inherentlyassociatedwithit.Arealnumberissaidtobea positivenumber ifitislocatedtotherightof0onthenumberline.Itisa negative numberifitislocatedtotheleftof0 onthenumberline. THENOTATIONOFSIGNEDNUMBERS Anumberisdenotedas positive ifitisdirectlyprecededbya "+" sign or nosignatall. Anumberisdenotedas negative ifitisdirectlyprecededbya )]TJ/F8 9.9626 Tf 9.962 0 Td [(" sign. The "+" and )]TJ/F8 9.9626 Tf 9.963 0 Td [(" signsnowhavetwomeanings: + candenotetheoperationofadditionorapositivenumber. )]TJ/F15 9.9626 Tf 11.069 0 Td [(candenotetheoperationofsubtractionoranegativenumber. Readthe )]TJ/F8 9.9626 Tf 9.963 0 Td [(" Signas"Negative" Toavoidanyconfusionbetween"sign"and"operation,"itispreferabletoreadthesignofanumberas "positive"or"negative." 3.2.3SampleSetA Example3.1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 shouldbereadas"negativeeight"ratherthan"minuseight." Example3.2 4+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 shouldbereadas"fourplusnegativetwo"ratherthan"fourplusminustwo." Example3.3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 shouldbereadas"negativesixplusnegativethree"ratherthan"minussixplusminus three." Example3.4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 )]TJ/F8 9.9626 Tf 8.7 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 shouldbereadas"negativefteenminusnegativesix"ratherthan"minusfteenminus minussix." Example3.5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5+7 shouldbereadas"negativeveplusseven"ratherthan"minusveplusseven." Example3.6 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 shouldbereadas"zerominustwo." 2 Thiscontentisavailableonlineat.

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123 3.2.4PracticeSetA Writeeachexpressioninwords. Exercise3.1 Solutiononp.179. 4+10 Exercise3.2 Solutiononp.179. 7+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Exercise3.3 Solutiononp.179. )]TJ/F8 9.9626 Tf 7.749 0 Td [(9+2 Exercise3.4 Solutiononp.179. )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 )]TJ/F8 9.9626 Tf 9.962 0 Td [(+8 Exercise3.5 Solutiononp.179. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise3.6 Solutiononp.179. 0+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 3.2.5Opposites Opposites Onthenumberline,eachrealnumberhasanimageontheoppositesideof0.Forthisreasonwesaythat eachrealnumberhasanopposite. Opposites arethesamedistancefromzerobuthaveoppositesigns. Theoppositeofarealnumberisdenotedbyplacinganegativesigndirectlyinfrontofthenumber.Thus, if a isanyrealnumber,then )]TJ/F11 9.9626 Tf 7.749 0 Td [(a isitsopposite. Notice thattheletter a isavariable.Thus, a neednot bepositive,and )]TJ/F11 9.9626 Tf 9.962 0 Td [(a neednotbenegative. If a isarealnumber, )]TJ/F11 9.9626 Tf 7.749 0 Td [(a isopposite a onthenumberlineand a isopposite )]TJ/F11 9.9626 Tf 7.749 0 Td [(a onthenumberline. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F11 9.9626 Tf 7.749 0 Td [(a isopposite )]TJ/F11 9.9626 Tf 7.749 0 Td [(a onthenumberline.Thisimpliesthat )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F11 9.9626 Tf 7.749 0 Td [(a = a Thispropertyofoppositessuggeststhedouble-negativepropertyforrealnumbers. THEDOUBLE-NEGATIVEPROPERTY If a isarealnumber,then )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F11 9.9626 Tf 7.748 0 Td [(a = a 3.2.6SampleSetB Example3.7 If a =3 ,then )]TJ/F11 9.9626 Tf 7.749 0 Td [(a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 and )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F11 9.9626 Tf 7.749 0 Td [(a = )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(3=3

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124 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Example3.8 If a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ,then )]TJ/F11 9.9626 Tf 7.749 0 Td [(a = )]TJ/F8 9.9626 Tf 9.41 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(4=4 and )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F11 9.9626 Tf 7.749 0 Td [(a = a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 3.2.7PracticeSetB Findtheoppositeofeachrealnumber. Exercise3.7 Solutiononp.179. 8 Exercise3.8 Solutiononp.179. 17 Exercise3.9 Solutiononp.179. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.10 Solutiononp.179. )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 Exercise3.11 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise3.12 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(7] Exercise3.13 Solutiononp.179. Supposethat a isapositivenumber.Whattypeofnumberis )]TJ/F11 9.9626 Tf 7.749 0 Td [(a ? Exercise3.14 Solutiononp.179. Supposethat a isanegativenumber.Whattypeofnumberis )]TJ/F11 9.9626 Tf 7.748 0 Td [(a ? Exercise3.15 Solutiononp.179. Supposewedonotknowthesignofthenumber m .Canwesaythat )]TJ/F11 9.9626 Tf 7.748 0 Td [(m ispositive,negative,or thatwedonotknow? 3.2.8Exercises Exercise3.16 Solutiononp.179. Anumberisdenotedaspositiveifitisdirectlyprecededby____________________. Exercise3.17 Anumberisdenotedasnegativeifitisdirectlyprecededby____________________. Forthefollowingproblems,howshouldtherealnumbersberead?Writeinwords. Exercise3.18 Solutiononp.179. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise3.19 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise3.20 Solutiononp.179. 12

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125 Exercise3.21 10 Exercise3.22 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.41 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.23 )]TJ/F8 9.9626 Tf 9.41 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Forthefollowingproblems,writetheexpressionsinwords. Exercise3.24 Solutiononp.179. 5+7 Exercise3.25 2+6 Exercise3.26 Solutiononp.179. 11+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.27 1+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 Exercise3.28 Solutiononp.179. 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.29 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(15 Rewritethefollowingproblemsinasimplerform. Exercise3.30 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.31 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise3.32 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.33 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise3.34 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise3.35 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.36 Solutiononp.179. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3] Exercise3.37 )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(10] Exercise3.38 Solutiononp.180. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(6] Exercise3.39 )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(15] Exercise3.40 Solutiononp.180. f)]TJ/F8 9.9626 Tf 22.139 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(26] g Exercise3.41 f)]TJ/F8 9.9626 Tf 22.139 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(11] g Exercise3.42 Solutiononp.180. f)]TJ/F8 9.9626 Tf 22.139 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(31] g

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126 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.43 f)]TJ/F8 9.9626 Tf 22.14 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(14] g Exercise3.44 Solutiononp.180. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [(] Exercise3.45 )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [(] Exercise3.46 Solutiononp.180. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [(] Exercise3.47 )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [(] Exercise3.48 Solutiononp.180. 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.49 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 Exercise3.50 Solutiononp.180. 10 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise3.51 18 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 Exercise3.52 Solutiononp.180. 31 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Exercise3.53 54 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 Exercise3.54 Solutiononp.180. 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Exercise3.55 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.56 Solutiononp.180. 15 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise3.57 24 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 3.2.9ExercisesforReview Exercise3.58 Solutiononp.180. Section2.5 Thereisonlyonerealnumberforwhich a 2 =5 a 2 .Whatisthenumber? Exercise3.59 Section2.6 Simplify xy )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x 2 y 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 x 2 y 4 Exercise3.60 Solutiononp.180. Section2.6 Simplify x n +3 x 5 Exercise3.61 Section2.7 Simplify )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 3 b 2 c 4 4 Exercise3.62 Solutiononp.180. Section2.7 Simplify 4 a 2 b 3 xy 3 2 .

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127 3.3AbsoluteValue 3 3.3.1Overview GeometricDenitionofAbsoluteValue AlgebraicDenitionofAbsoluteValue 3.3.2GeometricDenitionofAbsoluteValue AbsoluteValueGeometricApproach The absolutevalue ofanumber a ,denoted j a j ,isthedistancefrom a to0onthenumberline. Absolutevaluespeakstothequestionof"howfar,"andnot"whichway."Thephrasehowfarimplies length,andlengthisalwaysanonnegativezeroorpositivequantity.Thus,theabsolutevalueofanumber isanonnegativenumber.Thisisshowninthefollowingexamples: Example3.9 j 4 j =4 Example3.10 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(4 j =4 Example3.11 j 0 j =0 Example3.12 j 5 j = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 Thequantityontheleftsideoftheequalsignisreadas"negativetheabsolutevalueof5."The absolutevalueof5is5.Hence,negativetheabsolutevalueof5is )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Example3.13 j)]TJ/F8 9.9626 Tf 22.693 0 Td [(3 j = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Thequantityontheleftsideoftheequalsignisreadas"negativetheabsolutevalueof )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ."The absolutevalueof )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 is3.Hence,negativetheabsolutevalueof )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 is )]TJ/F8 9.9626 Tf 9.409 0 Td [(= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 3.3.3AlgebraicDenitionofAbsoluteValue Theproblemsintherstexamplemayhelptosuggestthefollowingalgebraicdenitionofabsolutevalue. Thedenitionisinterpretedbelow.Examplesfollowtheinterpretation. AbsoluteValueAlgebraicApproach The absolutevalue ofanumber a is j a j = f a if a 0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a if a< 0 Thealgebraicdenitiontakesintoaccountthefactthatthenumber a couldbeeitherpositiveorzero 0 ornegative < 0 3 Thiscontentisavailableonlineat.

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128 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 1.Ifthenumber a ispositiveorzero 0 ,therstpartofthedenitionapplies.Therstpartof thedenitiontellsusthatifthenumberenclosedintheabsolutebarsisanonnegativenumber,the absolutevalueofthenumberisthenumberitself. 2.Ifthenumber a isnegative < 0 ,thesecondpartofthedenitionapplies.Thesecondpartofthe denitiontellsusthatifthenumberenclosedwithintheabsolutevaluebarsisanegativenumber,the absolutevalueofthenumberistheoppositeofthenumber.Theoppositeofanegativenumberisa positivenumber. 3.3.4SampleSetA Usethealgebraicdenitionofabsolutevaluetondthefollowingvalues. Example3.14 j 8 j Thenumberenclosedwithintheabsolutevaluebarsisanonnegativenumbersotherstpartof thedenitionapplies.Thispartsaysthattheabsolutevalueof8is8itself. j 8 j =8 Example3.15 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(3 j Thenumberenclosedwithinabsolutevaluebarsisanegativenumbersothesecondpartofthe denitionapplies.Thispartsaysthattheabsolutevalueof )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 istheoppositeof )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ,whichis )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 .Bythedouble-negativeproperty, )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3=3 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(3 j =3 3.3.5PracticeSetA Usethealgebraicdenitionofabsolutevaluetondthefollowingvalues. Exercise3.63 Solutiononp.180. j 7 j Exercise3.64 Solutiononp.180. j 9 j Exercise3.65 Solutiononp.180. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(12 j Exercise3.66 Solutiononp.180. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(5 j Exercise3.67 Solutiononp.180. j 8 j Exercise3.68 Solutiononp.180. j 1 j Exercise3.69 Solutiononp.180. j)]TJ/F8 9.9626 Tf 22.693 0 Td [(52 j Exercise3.70 Solutiononp.180. j)]TJ/F8 9.9626 Tf 22.693 0 Td [(31 j

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129 3.3.6Exercises Forthefollowingproblems,determineeachofthevalues. Exercise3.71 Solutiononp.180. j 5 j Exercise3.72 j 3 j Exercise3.73 Solutiononp.180. j 6 j Exercise3.74 j 14 j Exercise3.75 Solutiononp.180. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(8 j Exercise3.76 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(10 j Exercise3.77 Solutiononp.180. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(16 j Exercise3.78 j 8 j Exercise3.79 Solutiononp.181. j 12 j Exercise3.80 j 47 j Exercise3.81 Solutiononp.181. j 9 j Exercise3.82 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(9 j Exercise3.83 Solutiononp.181. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(1 j Exercise3.84 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(4 j Exercise3.85 Solutiononp.181. j 3 j Exercise3.86 j 7 j Exercise3.87 Solutiononp.181. j)]TJ/F8 9.9626 Tf 22.693 0 Td [(14 j Exercise3.88 j)]TJ/F8 9.9626 Tf 22.693 0 Td [(19 j Exercise3.89 Solutiononp.181. j)]TJ/F8 9.9626 Tf 22.693 0 Td [(28 j Exercise3.90 j)]TJ/F8 9.9626 Tf 22.693 0 Td [(31 j Exercise3.91 Solutiononp.181. j)]TJ/F8 9.9626 Tf 22.693 0 Td [(68 j

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130 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.92 j 0 j Exercise3.93 Solutiononp.181. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(26 j Exercise3.94 j)]TJ/F8 9.9626 Tf 22.693 0 Td [(26 j Exercise3.95 Solutiononp.181. j)]TJ/F8 9.9626 Tf 22.693 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 j Exercise3.96 j)]TJ/F8 9.9626 Tf 22.693 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 j Exercise3.97 Solutiononp.181. j)]TJ/F8 9.9626 Tf 22.693 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 j Exercise3.98 j)]TJ/F8 9.9626 Tf 22.693 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 j Exercise3.99 Solutiononp.181. )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 4 j Exercise3.100 )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 2 j Exercise3.101 Solutiononp.181. )]TJ/F8 9.9626 Tf 9.409 0 Td [( j)]TJ/F8 9.9626 Tf 22.692 0 Td [(6 j Exercise3.102 )]TJ/F8 9.9626 Tf 9.409 0 Td [( j)]TJ/F8 9.9626 Tf 22.692 0 Td [(42 j Exercise3.103 Solutiononp.181. j)-222(j)]TJ/F8 9.9626 Tf 29.888 0 Td [(3 jj Exercise3.104 j)-222(j)]TJ/F8 9.9626 Tf 29.888 0 Td [(15 jj Exercise3.105 Solutiononp.181. j)-222(j)]TJ/F8 9.9626 Tf 29.888 0 Td [(12 jj Exercise3.106 j)-222(j)]TJ/F8 9.9626 Tf 29.888 0 Td [(29 jj Exercise3.107 Solutiononp.181. j 6 )-222(j)]TJ/F8 9.9626 Tf 24.906 0 Td [(2 jj Exercise3.108 j 18 )-222(j)]TJ/F8 9.9626 Tf 24.907 0 Td [(11 jj Exercise3.109 Solutiononp.181. j 5 )-222(j)]TJ/F8 9.9626 Tf 24.906 0 Td [(1 jj Exercise3.110 j 10 )-222(j)]TJ/F8 9.9626 Tf 24.907 0 Td [(3 jj Exercise3.111 Solutiononp.181. j)]TJ/F8 9.9626 Tf 14.944 0 Td [( )-222(j)]TJ/F8 9.9626 Tf 24.906 0 Td [(12 j j Exercise3.112 j)]TJ/F8 9.9626 Tf 14.944 0 Td [( )-222(j)]TJ/F8 9.9626 Tf 24.906 0 Td [(24 j j Exercise3.113 Solutiononp.181. j 5 j)-222(j)]TJ/F8 9.9626 Tf 29.888 0 Td [(2 j Exercise3.114 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(2 j 3

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131 Exercise3.115 Solutiononp.181. j)]TJ/F8 9.9626 Tf 14.944 0 Td [( 3 j Exercise3.116 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(2 j + j)]TJ/F8 9.9626 Tf 14.944 0 Td [(9 j Exercise3.117 Solutiononp.181. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(6 j + j 4 j 2 Exercise3.118 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(1 j)-222(j 1 j 3 Exercise3.119 Solutiononp.181. j 4 j + j)]TJ/F8 9.9626 Tf 14.944 0 Td [(6 j 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( j)]TJ/F8 9.9626 Tf 14.943 0 Td [(2 j 3 Exercise3.120 )]TJ/F8 9.9626 Tf 7.749 0 Td [([ j)]TJ/F8 9.9626 Tf 14.944 0 Td [(10 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(6] 2 Exercise3.121 Solutiononp.181. )]TJ/F1 9.9626 Tf 7.749 11.058 Td [(h )]TJ/F8 9.9626 Tf 7.748 0 Td [( j)]TJ/F8 9.9626 Tf 22.692 0 Td [(4 j + j)]TJ/F8 9.9626 Tf 14.944 0 Td [(3 j 3 i 2 Exercise3.122 AMissionControlOceratCapeCanaveralmakesthestatement"lift-o, T minus50seconds." Howlongbeforelift-o? Exercise3.123 Solutiononp.181. Duetoaslowdownintheindustry,aSiliconValleycomputercompanyndsitselfindebt $2 ; 400 ; 000 .Useabsolutevaluenotationtodescribethiscompany'sdebt. Exercise3.124 Aparticularmachineissetcorrectlyifuponactionitsmeterreads0units.Oneparticularmachine hasameterreadingof )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 6 uponaction.Howfaristhismachineoitscorrectsetting? 3.3.7ExercisesforReview Exercise3.125 Solutiononp.181. Section2.2 Writethefollowingphraseusingalgebraicnotation:"fourtimes a + b ." Exercise3.126 Section2.3 Isthereasmallestnaturalnumber?Ifso,whatisit? Exercise3.127 Solutiononp.181. Section2.4 Namethepropertyofrealnumbersthatmakes 5+ a = a +5 atruestatement. Exercise3.128 Section2.6 Findthequotientof x 6 y 8 x 4 y 3 Exercise3.129 Solutiononp.182. Section3.2 Simplify )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 .

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132 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.4AdditionofSignedNumbers 4 3.4.1Overview AdditionofNumberswithLikeSigns AdditionwithZero AdditionofNumberswithUnlikeSigns 3.4.2AdditionofNumberswithLikeSigns Letusaddthetwopositivenumbers2and3.Weperformthisadditiononthenumberlineasfollows. Webeginat0,theorigin. Since2ispositive,wemove2unitstotheright. Since3ispositive,wemove3moreunitstotheright. Wearenowlocatedat5. Thus, 2+3=5 Summarizing,wehave positiveunits + positiveunits = positiveunits Nowletusaddthetwonegativenumbers )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 .Weperformthisadditiononthenumberlineas follows. Webeginat0,theorigin. Since )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 isnegative,wemove2unitstotheleft. Since )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 isnegative,wemove3moreunitstotheleft. Wearenowlocatedat )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Thus, )]TJ/F8 9.9626 Tf 7.748 0 Td [(2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3= )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Summarizing,wehave negativeunits + negativeunits = negativeunits Thesetwoexamplessuggestthat positivenumber + positivenumber = positivenumber negativenumber + negativenumber = negativenumber AddingNumberswiththeSameSign Toaddtworealnumbersthathavethesamesign,addtheabsolutevaluesofthenumbersandassociatethe commonsignwiththesum. 3.4.3SampleSetA Findthesums. Example3.16 3+7 4 Thiscontentisavailableonlineat.

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133 Addtheseabsolutevalues. j 3 j =3 j 7 j =7 g 3+7=10 Thecommonsignis"+." 3+7=+10 or 3+7=10 Example3.17 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Addtheseabsolutevalues. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(4 j =4 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(9 j =9 g 4+9=13 Thecommonsignis" )]TJ/F15 9.9626 Tf 9.963 0 Td [(." )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(9= )]TJ/F8 9.9626 Tf 7.748 0 Td [(13 3.4.4PracticeSetA Findthesums. Exercise3.130 Solutiononp.182. 8+6 Exercise3.131 Solutiononp.182. 41+11 Exercise3.132 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.133 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.748 0 Td [(36+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise3.134 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(14+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 Exercise3.135 Solutiononp.182. )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(2 3 + )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F7 6.9738 Tf 8.944 3.922 Td [(5 3 Exercise3.136 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 : 8+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 : 6 3.4.5AdditionwithZero Noticethat Additionwith0 0 + apositivenumber = thatsamepositivenumber 0 + anegativenumber = thatsamenegativenumber TheAdditiveIdentityIs0 Sinceadding0toarealnumberleavesthatnumberunchanged,0iscalledthe additiveidentity .

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134 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.4.6AdditionofNumberswithUnlikeSigns Nowletusperformtheaddition 2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 .Thesetwonumbershaveunlikesigns.Thistypeofadditioncan alsobeillustratedusingthenumberline. Webeginat0,theorigin. Since2ispositive,wemove2unitstotheright. Since )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 isnegative,wemove,fromthe2,6unitstotheleft. Wearenowlocatedat )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Aruleforaddingtwonumbersthathaveunlikesignsissuggestedbynotingthatifthesignsaredisregarded,4canbeobtainedfrom2and6by subtracting 2from6.But2and6arepreciselytheabsolute valuesof2and )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 .Also,noticethatthesignofthenumberwiththelargerabsolutevalueisnegativeand thatthesignoftheresultingsumisnegative. AddingNumberswithUnlikeSigns Toaddtworealnumbersthathaveunlikesigns,subtractthesmallerabsolutevaluefromthelargerabsolute valueandassociatethesignofthenumberwiththelargerabsolutevaluewiththisdierence. 3.4.7SampleSetB Findthefollowingsums. Example3.18 7+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 j 7 j =7 | {z } Largerabsolutevalue. Signis "+" : j)]TJ/F8 9.9626 Tf 14.944 0 Td [(2 j =2 | {z } Smallerabsolutevalue. Subtractabsolutevalues: 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2=5 : Attachthepropersign: "+" : 7+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2=+5 or 7+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2=5 Example3.19 3+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 j 3 j =3 | {z } Smallerabsolutevalue. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(11 j =11 | {z } Largerabsolutevalue. Signis )]TJ/F8 9.9626 Tf 9.963 0 Td [(" : Subtractabsolutevalues: 11 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=8 : Attachthepropersign: )]TJ/F8 9.9626 Tf 9.962 0 Td [(" : 3+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(11= )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Example3.20 Themorningtemperatureonawinter'sdayinLakeTahoewas )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 degrees.Theafternoon temperaturewas25degreeswarmer.Whatwastheafternoontemperature? Weneedtond )]TJ/F8 9.9626 Tf 7.749 0 Td [(12+25 .

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135 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(12 j =12 | {z } Smallerabsolutevalue. j 25 j =25 | {z } Largerabsolutevalue. Signis"+". Subtractabsolutevalues: 25 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12=13 : Attachthepropersign: "+" : )]TJ/F8 9.9626 Tf 7.749 0 Td [(12+25=13 Thus,theafternoontemperatureis13degrees. Example3.21 Add )]TJ/F8 9.9626 Tf 7.748 0 Td [(147+84 DisplayReads Type 147147 Press + = )]TJETq1 0 0 1 185.877 500.312 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQq1 0 0 1 154.838 500.113 cm[]0 d 0 J 0.398 w 0 0 m 31.238 0 l SQBT/F14 9.9626 Tf 201.639 506.19 Td [()]TJ/F15 9.9626 Tf 9.963 0 Td [(147 Press + )]TJ/F15 9.9626 Tf 9.963 0 Td [(147 Type 8484 Press = )]TJ/F8 9.9626 Tf 9.963 0 Td [(63 3.4.8PracticeSetB Findthesums. Exercise3.137 Solutiononp.182. 4+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise3.138 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+5 Exercise3.139 Solutiononp.182. 15+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 Exercise3.140 Solutiononp.182. 0+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise3.141 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(26+12 Exercise3.142 Solutiononp.182. 35+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(78 Exercise3.143 Solutiononp.182. 15+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise3.144 Solutiononp.182. 1 : 5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.145 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8+0 Exercise3.146 Solutiononp.182. 0+ : 57 Exercise3.147 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(879+454 Exercise3.148 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1345 : 6+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6648 : 1

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136 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.4.9Exercises Findthesumsforthethefollowingproblems. Exercise3.149 Solutiononp.182. 4+12 Exercise3.150 8+6 Exercise3.151 Solutiononp.182. 6+2 Exercise3.152 7+9 Exercise3.153 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.748 0 Td [(3+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise3.154 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 Exercise3.155 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.156 )]TJ/F8 9.9626 Tf 7.748 0 Td [(11+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.157 Solutiononp.182. )]TJ/F8 9.9626 Tf 7.748 0 Td [(16+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.158 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 Exercise3.159 Solutiononp.183. 14+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise3.160 21+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Exercise3.161 Solutiononp.183. 14+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise3.162 18+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise3.163 Solutiononp.183. 10+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.164 40+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(31 Exercise3.165 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 Exercise3.166 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(20 Exercise3.167 Solutiononp.183. 10+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise3.168 8+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(15

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137 Exercise3.169 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.170 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(14 Exercise3.171 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(9+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.172 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise3.173 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(16+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 Exercise3.174 )]TJ/F8 9.9626 Tf 7.749 0 Td [(22+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Exercise3.175 Solutiononp.183. 0+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise3.176 0+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.177 Solutiononp.183. 0+ Exercise3.178 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+1+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 Exercise3.179 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(12+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.180 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5+5 Exercise3.181 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7+7 Exercise3.182 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14+14 Exercise3.183 Solutiononp.183. 4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.184 9+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise3.185 Solutiononp.183. 84+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(61 Exercise3.186 13+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(56 Exercise3.187 Solutiononp.183. 452+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(124 Exercise3.188 636+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(989 Exercise3.189 Solutiononp.183. 1811+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(935 Exercise3.190 )]TJ/F8 9.9626 Tf 7.749 0 Td [(373+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(14 Exercise3.191 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1221+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(44

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138 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.192 )]TJ/F8 9.9626 Tf 7.749 0 Td [(47 : 03+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(22 : 71 Exercise3.193 Solutiononp.183. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 998+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 : 086 Exercise3.194 [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4]+[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1] Exercise3.195 Solutiononp.183. [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8]+[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(7] Exercise3.196 [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8]+[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(12] Exercise3.197 Solutiononp.183. [ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6]+[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1] Exercise3.198 [4+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(12]+[12+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3] Exercise3.199 Solutiononp.183. [5+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(16]+[4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(11] Exercise3.200 [2+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(4]+[17+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(19] Exercise3.201 Solutiononp.183. [10+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6]+[12+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2] Exercise3.202 9+[ )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+7] Exercise3.203 Solutiononp.183. 14+[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+5] Exercise3.204 [2+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(7]+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Exercise3.205 Solutiononp.183. [14+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8]+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise3.206 Inorderforasmallbusinesstobreakevenonaproject,itmusthavesalesof $21 ; 000 .Ifthe amountofsaleswas $15 ; 000 ,howmuchmoneydidthiscompanyfallshort? Exercise3.207 Solutiononp.183. Supposeapersonhas $56 : 00 inhischeckingaccount.Hedeposits $100 : 00 intohischeckingaccount byusingtheautomatictellermachine.Hethenwritesacheckfor $84 : 50 .Ifanerrorcausesthe depositnottobelistedintothisperson'saccount,whatisthisperson'scheckingbalance? Exercise3.208 Apersonborrows $7 : 00 onMondayandthen $12 : 00 onTuesday.Howmuchhasthisperson borrowed? Exercise3.209 Solutiononp.184. Apersonborrows $11 : 00 onMondayandthenpaysback $8 : 00 onTuesday.Howmuchdoesthis personowe?

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139 3.4.10ExercisesforReview Exercise3.210 Section2.5 Simplify 4 7 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 2 3 2 2 Exercise3.211 Solutiononp.184. Section2.6 Simplify 35 a 6 b 2 c 5 7 b 2 c 4 Exercise3.212 Section2.7 Simplify 12 a 8 b 5 4 a 5 b 2 3 Exercise3.213 Solutiononp.184. Section3.3 Determinethevalueof j)]TJ/F8 9.9626 Tf 14.944 0 Td [(8 j Exercise3.214 Section3.3 Determinethevalueof j 2 j + j 4 j 2 + j)]TJ/F8 9.9626 Tf 14.944 0 Td [(5 j 2 3.5SubtractionofSignedNumbers 5 3.5.1Overview DenitionofSubtraction SubtractionofSignedNumbers 3.5.2DenitionofSubtraction Weknowfromourexperiencewitharithmeticthatthesubtraction 5 )]TJ/F8 9.9626 Tf 10.48 0 Td [(2 produces3,thatis, 5 )]TJ/F8 9.9626 Tf 10.48 0 Td [(2=3 Illustratingthisprocessonthenumberlinesuggestsaruleforsubtractingsignednumbers. Webeginat0,theorigin. Since5ispositive,wemove5unitstotheright. Then,wemove 2unitstotheleft togetto3.Thisremindsusofadditionwithanegativenumber. Thisillustrationsuggeststhat 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 isthesameas 5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Thisleadsusdirectlytothedenitionofsubtraction. DenitionofSubtraction If a and b arerealnumbers, a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b isthesameas a + )]TJ/F11 9.9626 Tf 7.749 0 Td [(b ,where )]TJ/F11 9.9626 Tf 7.748 0 Td [(b istheoppositeof b 3.5.3SubtractionofSignedNumbers Theprecedingdenitionsuggeststheruleforsubtractingsignednumbers. SubtractionofSignedNumbers Toperformthesubtraction a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b ,addtheoppositeof b to a ,thatis,changethesignof b andadd. 5 Thiscontentisavailableonlineat.

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140 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.5.4SampleSetA Performthesubtractions. Example3.22 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3=2 Example3.23 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9=4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(9= )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Example3.24 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6= )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Example3.25 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(12= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+12=9 Example3.26 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(15=0+15=15 Example3.27 ThehightemperaturetodayinLakeTahoewas 26 F.Thelowtemperaturetonightisexpected tobe )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 F.Howmanydegreesisthetemperatureexpectedtodrop? Weneedtondthedierencebetween26and )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 26 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(7=26+7=33 Thus,theexpectedtemperaturedropis 33 F. Example3.28 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10= )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6+5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(11

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141 3.5.5PracticeSetA Performthesubtractions. Exercise3.215 Solutiononp.184. 9 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise3.216 Solutiononp.184. 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 Exercise3.217 Solutiononp.184. 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 Exercise3.218 Solutiononp.184. 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Exercise3.219 Solutiononp.184. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise3.220 Solutiononp.184. )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise3.221 Solutiononp.184. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.222 Solutiononp.184. 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise3.223 Solutiononp.184. 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise3.224 Solutiononp.184. 86 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(32 Exercise3.225 Solutiononp.184. 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 Exercise3.226 Solutiononp.184. 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 Exercise3.227 Solutiononp.184. 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [( Exercise3.228 Solutiononp.184. 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise3.229 Solutiononp.184. 24 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 3.5.6Exercises Forthefollowingexercises,performtheindicatedoperations. Exercise3.230 Solutiononp.184. 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise3.231 12 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 Exercise3.232 Solutiononp.184. 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise3.233 14 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30

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142 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.234 Solutiononp.184. 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 Exercise3.235 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 Exercise3.236 Solutiononp.184. 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise3.237 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise3.238 Solutiononp.184. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise3.239 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Exercise3.240 Solutiononp.184. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise3.241 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise3.242 Solutiononp.184. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise3.243 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise3.244 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise3.245 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.246 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise3.247 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise3.248 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 Exercise3.249 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(16 Exercise3.250 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.251 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 Exercise3.252 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 Exercise3.253 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Exercise3.254 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.255 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 Exercise3.256 Solutiononp.185. 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6

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143 Exercise3.257 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 Exercise3.258 Solutiononp.185. 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 Exercise3.259 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise3.260 Solutiononp.185. 67 )]TJ/F8 9.9626 Tf 9.962 0 Td [(38 Exercise3.261 142 )]TJ/F8 9.9626 Tf 9.963 0 Td [(85 Exercise3.262 Solutiononp.185. 816 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1140 Exercise3.263 105 )]TJ/F8 9.9626 Tf 9.963 0 Td [(421 Exercise3.264 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(550 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(121 Exercise3.265 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 : 016 )]TJ/F8 9.9626 Tf 9.963 0 Td [( : 001 Exercise3.266 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(26+7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(52 Exercise3.267 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 )]TJ/F8 9.9626 Tf 9.962 0 Td [(21 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise3.268 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(104 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(216 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(52 Exercise3.269 )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 012 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 111 )]TJ/F8 9.9626 Tf 9.963 0 Td [( : 035 Exercise3.270 Solutiononp.185. [5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6] )]TJ/F8 9.9626 Tf 9.962 0 Td [([2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4] Exercise3.271 [2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(8] )]TJ/F8 9.9626 Tf 9.962 0 Td [([5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(7] Exercise3.272 Solutiononp.185. [4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(11] )]TJ/F8 9.9626 Tf 9.963 0 Td [([2+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(10] Exercise3.273 [9+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6] )]TJ/F8 9.9626 Tf 9.962 0 Td [([4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(12] Exercise3.274 Solutiononp.185. )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise3.275 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 Exercise3.276 Solutiononp.185. )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 Exercise3.277 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise3.278 Solutiononp.185. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4+7 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise3.279 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(11

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144 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.280 Solutiononp.185. [ )]TJ/F8 9.9626 Tf 7.748 0 Td [(8+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(5+3] )]TJ/F8 9.9626 Tf 9.963 0 Td [([9 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5] Exercise3.281 [ )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1+6] )]TJ/F8 9.9626 Tf 9.963 0 Td [([7 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1] Exercise3.282 Solutiononp.185. [2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+10] )]TJ/F8 9.9626 Tf 9.962 0 Td [([1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(11] Exercise3.283 [5 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5] )]TJ/F8 9.9626 Tf 9.962 0 Td [([2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4] Exercise3.284 Solutiononp.185. Whenaparticularmachineisoperatingproperly,itsmeterwillread34.Ifabrokenbearinginthe machinecausesthemeterreadingtodropby45units,whatisthemeterreading? Exercise3.285 ThelowtemperaturetodayinDenverwas )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Fandthehighwas 42 F.Whatisthetemperature dierence? 3.5.7ExercisesforReview Exercise3.286 Solutiononp.185. Section2.4 Usethedistributivepropertytoexpand 4 x y +11 Exercise3.287 Section2.7 Simplify 2 3 x 2 y 2 3 2 x 4 y 3 0 27 x 4 y 3 .Assume x 6 =0 ;y 6 =0 Exercise3.288 Solutiononp.185. Section3.3 Simplify j)]TJ/F1 9.9626 Tf 14.944 8.07 Td [()]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 2 +2 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 j Exercise3.289 Section3.4 Findthesum. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 Exercise3.290 Solutiononp.185. Section3.4 Findthesum. 3+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 3.6MultiplicationandDivisionofSignedNumbers 6 3.6.1Overview MultiplicationofSignedNumbers DivisionofSignedNumbers 3.6.2MultiplicationofSignedNumbers Letusconsiderrsttheproductoftwopositivenumbers. Multiply: 3 5 3 5 means 5+5+5=15 Thissuggeststhat positivenumber positivenumber = positivenumber. Morebriey, ++=+ Nowconsidertheproductofapositivenumberandanegativenumber. 6 Thiscontentisavailableonlineat.

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145 Multiply: )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 means )]TJ/F8 9.9626 Tf 7.749 0 Td [(5+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5+ )]TJ/F8 9.9626 Tf 7.748 0 Td [(5= )]TJ/F8 9.9626 Tf 7.748 0 Td [(15 Thissuggeststhat positivenumber negativenumber = negativenumber Morebriey, + )]TJ/F8 9.9626 Tf 7.749 0 Td [(= )]TJ/F15 9.9626 Tf 7.749 0 Td [(. Bythecommutativepropertyofmultiplication,weget negativenumber positivenumber = negativenumber Morebriey, )]TJ/F8 9.9626 Tf 7.748 0 Td [(+= )]TJ/F15 9.9626 Tf 7.749 0 Td [(. Thesignoftheproductoftwonegativenumberscanbedeterminedusingthefollowingillustration: Multiply )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 by,respectively, 4 ; 3 ; 2 ; 1 ; 0 ; )]TJ/F8 9.9626 Tf 9.483 0 Td [(1 ; )]TJ/F8 9.9626 Tf 9.483 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.484 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.483 0 Td [(4 .Noticethatwhenthemultiplierdecreasesby 1,theproductincreasesby2. 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 g! Asweknow ; + )]TJ/F8 9.9626 Tf 7.749 0 Td [(= )]TJ/F11 9.9626 Tf 7.749 0 Td [(: 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2=0 Asweknow ; 0 anynumber =0 : )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2=2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2=4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2=6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2=8 g! Thispatternsuggests )]TJ/F8 9.9626 Tf 7.749 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(=+ : Wehavethefollowingrulesformultiplyingsignednumbers. RulesforMultiplyingSignedNumbers Tomultiplytworealnumbersthathave 1.the samesign ,multiplytheirabsolutevalues.Theproductispositive. ++=+ )]TJ/F8 9.9626 Tf 7.748 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(=+ 2. oppositesigns ,multiplytheirabsolutevalues.Theproductisnegative. + )]TJ/F8 9.9626 Tf 7.749 0 Td [(= )]TJ/F8 9.9626 Tf -45.939 -16.737 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(+= )]TJ/F72 11.9552 Tf -75.827 -38.168 Td [(3.6.3SampleSetA Findthefollowingproducts. Example3.29 8 6 Multiplytheseabsolutevalues. j 8 j =8 j 6 j =6 g 8 6=48 Sincethenumbershavethesamesign,theproductispositive. 8 6=+48 or 8 6=48 Example3.30 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6

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146 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Multiplytheseabsolutevalues. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(8 j =8 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(6 j =6 g 8 6=48 Sincethenumbershavethesamesign,theproductispositive. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6=+48 or )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6=48 Example3.31 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Multiplytheseabsolutevalues. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(4 j =4 j 7 j =7 g 4 7=28 Sincethenumbershaveoppositesigns,theproductisnegative. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4= )]TJ/F8 9.9626 Tf 7.749 0 Td [(28 Example3.32 6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Multiplytheseabsolutevalues. j 6 j =6 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(3 j =3 g 6 3=18 Sincethenumbershaveoppositesigns,theproductisnegative. 6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3= )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 3.6.4PracticeSetA Findthefollowingproducts. Exercise3.291 Solutiononp.185. 3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.292 Solutiononp.186. 4 Exercise3.293 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise3.294 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.295 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise3.296 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.749 0 Td [(77

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147 3.6.5DivisionofSignedNumbers Wecandeterminethesignpatternfordivisionbyrelatingdivisiontomultiplication.Divisionisdenedin termsofmultiplicationinthefollowingway. If b c = a ,then a b = c;b 6 =0 Forexample,since 3 4=12 ,itfollowsthat 12 3 =4 Noticethepattern: Since 3 4 |{z} b c = a =12 ,itfollowsthat 12 3 |{z} a b = c =4 Thesignpatternfordivisionfollowsfromthesignpatternformultiplication. 1.Since ++ | {z } b c = a =+ ,itfollowsthat + + |{z} a b = c =+ ,thatis, positivenumber positivenumber = positivenumber 2.Since )]TJ/F8 9.9626 Tf 7.749 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [( | {z } b c = a =+ ,itfollowsthat + )]TJ/F8 9.9626 Tf 7.748 0 Td [( |{z} a b = c = )]TJ/F15 9.9626 Tf 7.749 0 Td [(,thatis, positivenumber negativenumber = negativenumber 3.Since + )]TJ/F8 9.9626 Tf 7.749 0 Td [( | {z } b c = a = )]TJ/F15 9.9626 Tf 7.748 0 Td [(,itfollowsthat )]TJ/F8 9.9626 Tf 7.748 0 Td [( + |{z} a b = c = )]TJ/F15 9.9626 Tf 7.749 0 Td [(,thatis, negativenumber positivenumber = negativenumber 4.Since )]TJ/F8 9.9626 Tf 7.749 0 Td [(+ | {z } b c = a = )]TJ/F15 9.9626 Tf 7.748 0 Td [(,itfollowsthat )]TJ/F8 9.9626 Tf 7.748 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [( |{z} a b = c =+ ,thatis negativenumber negativenumber = positivenumber Wehavethefollowingrulesfordividingsignednumbers. RulesforDividingSignedNumbers Todividetworealnumbersthathave 1.the samesign ,dividetheirabsolutevalues.Thequotientispositive. + + =+ )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [( =+ 2. oppositesigns ,dividetheirabsolutevalues.Thequotientisnegative. )]TJ/F7 6.9738 Tf 6.226 0 Td [( + = )]TJ/F7 6.9738 Tf 18.962 4.832 Td [(+ )]TJ/F7 6.9738 Tf 6.227 0 Td [( = )]TJ/F72 11.9552 Tf -103.855 -38.326 Td [(3.6.6SampleSetB Findthefollowingquotients. Example3.33 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 2 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(10 j =10 j 2 j =2 g Dividetheseabsolutevalues : 10 2 =5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Sincethenumbershaveoppositesigns,thequotientisnegative. Example3.34 )]TJ/F7 6.9738 Tf 6.226 0 Td [(35 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7

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148 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS j)]TJ/F8 9.9626 Tf 14.944 0 Td [(35 j =35 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(7 j =7 g Dividetheseabsolutevalues : 35 7 =5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(35 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 =5 Sincethenumbershavesamesigns,thequotientispositive. Example3.35 18 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 j 18 j =18 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(9 j =9 g Dividetheseabsolutevalues : 18 9 =2 18 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Sincethenumbershaveoppositesigns,thequotientisnegative. 3.6.7PracticeSetB Findthefollowingquotients. Exercise3.297 Solutiononp.186. )]TJ/F7 6.9738 Tf 6.227 0 Td [(24 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise3.298 Solutiononp.186. 30 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.299 Solutiononp.186. )]TJ/F7 6.9738 Tf 6.227 0 Td [(54 27 Exercise3.300 Solutiononp.186. 51 17 3.6.8SampleSetC Example3.36 Findthevalueof )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(+1+1 Usingtheorderofoperationsandwhatweknowaboutsignednumbers,weget )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(+1+1 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(+1 = 18+2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5+1 = 20 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Example3.37 Findthevalueof z = x )]TJ/F10 6.9738 Tf 6.226 0 Td [(u s if x =57 ;u =51 ; and s =2 Substitutingthesevaluesweget z = 57 )]TJ/F7 6.9738 Tf 6.227 0 Td [(51 2 = 6 2 =3

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149 3.6.9PracticeSetC Exercise3.301 Solutiononp.186. Findthevalueof )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(8+2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(11 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(17 Exercise3.302 Solutiononp.186. Findthevalueof P = n n )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 n ,if n =5 3.6.10Exercises Findthevalueofeachofthefollowingexpressions. Exercise3.303 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.304 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise3.305 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.306 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.307 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise3.308 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Exercise3.309 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.310 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.311 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise3.312 )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 Exercise3.313 Solutiononp.186. 8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.314 5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.315 Solutiononp.186. 9 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise3.316 7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.317 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(64 Exercise3.318 )]TJ/F8 9.9626 Tf 7.748 0 Td [(76 Exercise3.319 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(109 Exercise3.320 )]TJ/F8 9.9626 Tf 7.748 0 Td [(412

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150 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.321 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise3.322 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise3.323 Solutiononp.186. )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise3.324 )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise3.325 Solutiononp.186. 21 7 Exercise3.326 42 6 Exercise3.327 Solutiononp.186. )]TJ/F7 6.9738 Tf 6.227 0 Td [(39 3 Exercise3.328 )]TJ/F7 6.9738 Tf 6.227 0 Td [(20 10 Exercise3.329 Solutiononp.186. )]TJ/F7 6.9738 Tf 6.227 0 Td [(45 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.330 )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.331 Solutiononp.187. 25 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.332 36 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.333 Solutiononp.187. 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise3.334 14 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 Exercise3.335 Solutiononp.187. 20 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise3.336 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise3.337 Solutiononp.187. 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise3.338 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Exercise3.339 Solutiononp.187. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise3.340 15 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 Exercise3.341 Solutiononp.187. 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7+8 Exercise3.342 2+7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10+2 Exercise3.343 Solutiononp.187. 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6

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151 Exercise3.344 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 Exercise3.345 Solutiononp.187. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise3.346 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12+2 Exercise3.347 Solutiononp.187. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8+3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise3.348 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2+4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9+0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise3.349 Solutiononp.187. 6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6+9+4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise3.350 3+1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.351 Solutiononp.187. 4+1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.352 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1+2+5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.353 Solutiononp.187. )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2+ )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.354 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+2 Exercise3.355 Solutiononp.187. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise3.356 )]TJ/F8 9.9626 Tf 9.409 0 Td [(+21 Exercise3.357 Solutiononp.187. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Exercise3.358 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise3.359 Solutiononp.187. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise3.360 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise3.361 Solutiononp.187. )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise3.362 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+6 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.962 0 Td [(7] Exercise3.363 Solutiononp.187. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2[ )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(11] Exercise3.364 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+5+ )]TJ/F8 9.9626 Tf 9.962 0 Td [(8] Exercise3.365 Solutiononp.187. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.963 0 Td [(9+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8] Exercise3.366 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+6]

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152 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.367 Solutiononp.187. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10+11 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7] Exercise3.368 P = R )]TJ/F11 9.9626 Tf 9.963 0 Td [(C: Find P if R =2000 and C =2500 : Exercise3.369 Solutiononp.187. z = x )]TJ/F10 6.9738 Tf 6.226 0 Td [(u s : Find z if x =23 ;u =25 ; and s =1 : Exercise3.370 z = x )]TJ/F10 6.9738 Tf 6.226 0 Td [(u s : Find z if x =410 ;u =430 ; and s =2 : 5 : Exercise3.371 Solutiononp.187. m = 2 s +1 T : Find m if s = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 and T =5 : Exercise3.372 m = 2 s +1 T : Find m if s = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 and T = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 : Exercise3.373 Solutiononp.187. F = p 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(p 2 r 4 9 : Find F if p 1 =10 ;p 2 =8 ;r =3 : Exercise3.374 F = p 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(p 2 r 4 9 : Find F if p 1 =12 ;p 2 =7 ;r =2 : Exercise3.375 Solutiononp.187. P = n n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : Find P if n = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 : Exercise3.376 P = n n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 : Find P if n = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 : Exercise3.377 Solutiononp.187. P = n n )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 n : Find P if n = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 : 3.6.11ExercisesforReview Exercise3.378 Section2.3 Whatnaturalnumberscanreplace x sothatthestatement )]TJ/F8 9.9626 Tf 7.748 0 Td [(4
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153 3.7NegativeExponents 7 3.7.1Overview Reciprocals NegativeExponents WorkingwithNegativeExponents 3.7.2Reciprocals Reciprocals Tworealnumbersaresaidtobe reciprocals ofeachotheriftheirproductis1.Everynonzerorealnumber hasexactlyonereciprocal,asshownintheexamplesbelow.Zerohasnoreciprocal. Example3.38 4 1 4 =1 : Thismeansthat 4 and 1 4 arereciprocals : Example3.39 6 1 6 =1 : Hence, 6 and 1 6 arereciprocals : Example3.40 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 =1 : Hence, )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 and )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 arereciprocals : Example3.41 a 1 a =1 : Hence, a and 1 a arereciprocalsif a 6 =0 : Example3.42 x 1 x =1 : Hence, x and 1 x arereciprocalsif x 6 =0 : Example3.43 x 3 1 x 3 =1 : Hence, x 3 and 1 x 3 arereciprocalsif x 6 =0 : 3.7.3NegativeExponents Wecanusetheideaofreciprocalstondameaningfornegativeexponents. Considertheproductof x 3 and x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 .Assume x 6 =0 x 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = x 3+ )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = x 0 =1 Thus,sincetheproductof x 3 and x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 is1, x 3 and x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 mustbereciprocals. Wealsoknowthat x 3 1 x 3 =1 .Seeproblem6above.Thus, x 3 and 1 x 3 arealsoreciprocals. Then,since x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 and 1 x 3 arebothreciprocalsof x 3 andarealnumbercanhaveonlyonereciprocal,it mustbethat x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = 1 x 3 Wehaveused )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 astheexponent,buttheprocessworksaswellforallothernegativeintegers.Wemake thefollowingdenition. If n isanynaturalnumberand x isanynonzerorealnumber,then x )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = 1 x n 7 Thiscontentisavailableonlineat.

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154 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.7.4SampleSetA Writeeachofthefollowingsothatonlypositiveexponentsappear. Example3.44 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = 1 x 6 Example3.45 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 1 a 1 = 1 a Example3.46 7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 = 1 7 2 = 1 49 Example3.47 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = 1 a 6 Example3.48 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(24 = 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 24 Example3.49 k +2 z )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 = k +2 z 8 3.7.5PracticeSetA Writeeachofthefollowingusingonlypositiveexponents. Exercise3.383 Solutiononp.188. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.384 Solutiononp.188. m )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.385 Solutiononp.188. 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.386 Solutiononp.188. 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.387 Solutiononp.188. 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.388 Solutiononp.188. xy )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.389 Solutiononp.188. a +2 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Exercise3.390 Solutiononp.188. m )]TJ/F11 9.9626 Tf 9.962 0 Td [(n )]TJ/F7 6.9738 Tf 6.226 0 Td [( )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 CAUTION Itisimportanttonotethat a )]TJ/F10 6.9738 Tf 6.226 0 Td [(n isnotnecessarilyanegativenumber.Forexample, 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = 1 3 2 = 1 9 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 6 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 3.7.6WorkingwithNegativeExponents TheproblemsofSampleSetAsuggestthefollowingruleforworkingwithexponents: MovingFactorsUpandDown Inafraction,a factor canbemovedfromthenumeratortothedenominatororfromthedenominatorto thenumeratorbychangingthesignoftheexponent.

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155 3.7.7SampleSetB Writeeachofthefollowingsothatonlypositiveexponentsappear. Example3.50 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y 5 : The factorx )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 canbemovedfromthenumeratortothe denominatorbychangingtheexponent )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 to +2 : x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y 5 = y 5 x 2 Example3.51 a 9 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 : The factorb )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 canbemovedfromthenumeratortothe denominatorbychangingtheexponent )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 to +3 : a 9 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 = a 9 b 3 Example3.52 a 4 b 2 c )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 : Thisfractioncanbewrittenwithoutanynegativeexponents bymovingthe factorc )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 intothenumerator. Wemustchangethe )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 to +6 tomakethemovelegitimate. a 4 b 2 c )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 = a 4 b 2 c 6 Example3.53 1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 y )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 : Thisfractioncanbewrittenwithoutnegativeexponents bymovingallthe factors fromthedenominatorto thenumerator.Changethesignofeachexponent: )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 to +3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 to +2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 to +1 : 1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 y )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 = x 3 y 2 z 1 = x 3 y 2 z 3.7.8PracticeSetB Writeeachofthefollowingsothatonlypositiveexponentsappear. Exercise3.391 Solutiononp.188. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 y 7 Exercise3.392 Solutiononp.188. a 2 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 Exercise3.393 Solutiononp.188. x 3 y 4 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(8 Exercise3.394 Solutiononp.188. 6 m )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 n )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 7 k )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 Exercise3.395 Solutiononp.188. 1 a )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 c )]TJ/F6 4.9813 Tf 5.396 0 Td [(8 Exercise3.396 Solutiononp.188. 3 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 5 b a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 b 5

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156 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.7.9SampleSetC Example3.54 Rewrite 24 a 7 b 9 2 3 a 4 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 inasimplerform. Noticethatwearedividingpowerswiththesamebase.We'llproceedbyusingtherulesof exponents. 24 a 7 b 9 2 3 a 4 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 = 24 a 7 b 9 8 a 4 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 =3 a 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 9 )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 =3 a 3 b 9+6 =3 a 3 b 15 Example3.55 Write 9 a 5 b 3 5 x 3 y 2 sothatnodenominatorappears. Wecaneliminatethedenominatorbymovingallfactorsthatmakeupthedenominatortothe numerator. 9 a 5 b 3 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Example3.56 Findthevalueof 1 10 )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 + 3 4 )]TJ/F6 4.9813 Tf 5.397 0 Td [(3 Wecanevaluatethisexpressionbyeliminatingthenegativeexponents. 1 10 )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 + 3 4 )]TJ/F6 4.9813 Tf 5.397 0 Td [(3 =1 10 2 +3 4 3 =1 100+3 64 =100+192 =292 3.7.10PracticeSetC Exercise3.397 Solutiononp.188. Rewrite 36 x 8 b 3 3 2 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(5 inasimplerform. Exercise3.398 Solutiononp.188. Write 2 4 m )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 n 7 4 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 x 5 inasimplerformandoneinwhichnodenominatorappears. Exercise3.399 Solutiononp.188. Findthevalueof 2 5 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 +6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 3 3 2 3.7.11Exercises Writethefollowingexpressionsusingonlypositiveexponents.Assumeallvariablesarenonzero. Exercise3.400 Solutiononp.188. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.401 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.402 Solutiononp.188. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise3.403 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.404 Solutiononp.188. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(10

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157 Exercise3.405 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 Exercise3.406 Solutiononp.188. b )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 Exercise3.407 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.408 Solutiononp.188. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.409 x +1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.410 Solutiononp.188. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise3.411 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Exercise3.412 Solutiononp.188. a +9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 Exercise3.413 r +3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.414 Solutiononp.189. a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 Exercise3.415 x 3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.416 Solutiononp.189. x 7 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.417 a 4 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.418 Solutiononp.189. a 7 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.419 a 2 b 3 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.420 Solutiononp.189. x 3 y 2 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise3.421 x 3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 z 2 w Exercise3.422 Solutiononp.189. a 7 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 zw 3 Exercise3.423 a 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 zw 2 Exercise3.424 Solutiononp.189. x 5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.425 x 4 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 w )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.426 Solutiononp.189. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 d 4 Exercise3.427 x 9 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 w )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(2

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158 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.428 Solutiononp.189. 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 y 2 Exercise3.429 5 x 2 y 2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.430 Solutiononp.189. 7 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 2 c 2 Exercise3.431 4 x 3 x +1 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.432 Solutiononp.189. 7 a 2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise3.433 18 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 d 5 e )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.434 Solutiononp.189. 7 w +2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 w +1 3 Exercise3.435 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 5 Exercise3.436 Solutiononp.189. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 +3 3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.437 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 4 +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +5 4 Exercise3.438 Solutiononp.189. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x +11 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise3.439 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 y 2 +8 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise3.440 Solutiononp.189. 7 a )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.441 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 b 2 c 4 x +6 8 Exercise3.442 Solutiononp.189. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(y 3 +1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 5 y 3 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 w )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.443 5 x 3 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise3.444 Solutiononp.189. 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x Exercise3.445 6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise3.446 Solutiononp.189. 4 a 2 b 2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.447 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 c 9 Exercise3.448 Solutiononp.189. 2 3 x 2 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.449 7 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 5 a 6 bc )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 c 4

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159 Exercise3.450 Solutiononp.189. x +5 2 x +5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Exercise3.451 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 Exercise3.452 Solutiononp.189. 8 b +2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 b +2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b +2 3 Exercise3.453 3 a 5 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 2 +4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 +4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 +4 Exercise3.454 Solutiononp.189. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 a 2 b 7 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.455 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 z 4 )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 z Exercise3.456 Solutiononp.189. )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.457 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 3 Exercise3.458 Solutiononp.189. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.459 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.460 Solutiononp.190. 1 a )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 Exercise3.461 1 a )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 Exercise3.462 Solutiononp.190. 4 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 Exercise3.463 7 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(8 Exercise3.464 Solutiononp.190. 23 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 Exercise3.465 6 a 2 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 Exercise3.466 Solutiononp.190. 3 c 5 a 3 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(3 Exercise3.467 16 a )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 c 2 yz )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 w )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 Exercise3.468 Solutiononp.190. 24 y 2 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(8 6 a 2 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 c )]TJ/F6 4.9813 Tf 5.397 0 Td [(9 d 3 Exercise3.469 3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 b 5 b +7 )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 9 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 a )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 a +7 2 Exercise3.470 Solutiononp.190. 36 a 6 b 5 c 8 3 2 a 3 b 7 c 9 Exercise3.471 45 a 4 b 2 c 6 15 a 2 b 7 c 8

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160 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.472 Solutiononp.190. 3 3 x 4 y 3 z 3 2 xy 5 z 5 Exercise3.473 21 x 2 y 2 z 5 w 4 7 xyz 12 w 14 Exercise3.474 Solutiononp.190. 33 a )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(7 11 a 3 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 Exercise3.475 51 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 3 xy Exercise3.476 Solutiononp.190. 2 6 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 a )]TJ/F6 4.9813 Tf 5.396 0 Td [(7 b 5 2 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 b 6 Exercise3.477 x +3 3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 4 x +3 5 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F6 4.9813 Tf 5.397 0 Td [(8 Exercise3.478 Solutiononp.190. 4 x 3 y 7 Exercise3.479 5 x 4 y 3 a 3 Exercise3.480 Solutiononp.190. 23 a 4 b 5 c )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 y 5 Exercise3.481 2 3 b 5 c 2 d )]TJ/F6 4.9813 Tf 5.397 0 Td [(9 4 b 4 cx Exercise3.482 Solutiononp.190. 10 x 3 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(7 3 x 5 z 2 Exercise3.483 3 x 2 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 9 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 x +5 3 Exercise3.484 Solutiononp.190. 14 a 2 b 2 c )]TJ/F6 4.9813 Tf 5.397 0 Td [(12 a 2 +21 )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 4 )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 a 2 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 a +6 3 Forthefollowingproblems,evaluateeachnumericalexpression. Exercise3.485 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.486 Solutiononp.190. 7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.487 6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.488 Solutiononp.190. 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.489 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.490 Solutiononp.190. 6 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise3.491 4 9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.492 Solutiononp.190. 28 14 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1

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161 Exercise3.493 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.494 Solutiononp.190. 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.495 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 +3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.496 Solutiononp.190. )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.497 )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.498 Solutiononp.190. 3 2 )]TJ/F6 4.9813 Tf 5.397 0 Td [(3 Exercise3.499 4 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 5 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 Exercise3.500 Solutiononp.190. 2 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 4 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 Exercise3.501 2 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 +4 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 2 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 +4 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 Exercise3.502 Solutiononp.190. 21 0 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 6 2 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 Forthefollowingproblems,writeeachexpressionsothatonlypositiveexponentsappear. Exercise3.503 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.504 Solutiononp.190. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise3.505 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.506 Solutiononp.190. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.507 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 7 Exercise3.508 Solutiononp.190. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.509 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.510 Solutiononp.191. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise3.511 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.512 Solutiononp.191. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.513 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;a 6 =0

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162 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.514 Solutiononp.191. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;m 6 =0 Exercise3.515 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 y 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.516 Solutiononp.191. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 6 y 6 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 Exercise3.517 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 c 0 6 Exercise3.518 Solutiononp.191. y 3 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 5 Exercise3.519 a )]TJ/F6 4.9813 Tf 5.397 0 Td [(8 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 3 Exercise3.520 Solutiononp.191. )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(2 a b 3 4 Exercise3.521 )]TJ/F7 6.9738 Tf 6.13 -4.147 Td [(3 b a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.522 Solutiononp.191. 5 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 a 3 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 y 9 2 Exercise3.523 4 m )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 n 6 2 m )]TJ/F6 4.9813 Tf 5.396 0 Td [(5 n 3 Exercise3.524 Solutiononp.191. r 5 s )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 m )]TJ/F6 4.9813 Tf 5.397 0 Td [(8 n 7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.525 h )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 j )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 k )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 p )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 3.7.12ExercisesforReview Exercise3.526 Solutiononp.191. Section2.7 Simplify )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(4 x 5 y 3 z 0 3 Exercise3.527 Section3.4 Findthesum. )]TJ/F8 9.9626 Tf 7.749 0 Td [(15+3 Exercise3.528 Solutiononp.191. Section3.5 Findthedierence. 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise3.529 Section3.6 Simplify )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8+4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise3.530 Solutiononp.191. Section3.6 Findthevalueof m if m = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 k )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 t kt +6 when k =4 and t = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 .

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163 3.8ScienticNotation 8 3.8.1Overview StandardFormtoScienticForm ScienticFormtoStandardForm WorkingwithNumbersinScienticNotation 3.8.2StandardFormtoScienticForm Verylargenumberssuchas43,000,000,000,000,000,000thenumberofdierentpossiblecongurationsof Rubik'scubeandverysmallnumberssuchas 0 : 000000000000000000000340 themassoftheaminoacid tryptophanareextremelyinconvenienttowriteandread.Suchnumberscanbeexpressedmoreconveniently bywritingthemaspartofapowerof10. Toseehowthisisdone,letusstartwithasomewhatsmallernumbersuchas2480.Noticethat 2480 | {z } Standardform =248 : 0 10 1 =24 : 80 10 2 =2 : 480 10 3 | {z } Scienticform ScienticForm Thelastformiscalledthe scienticform ofthenumber.Thereis one nonzerodigittotheleftofthe decimalpointandtheabsolutevalueoftheexponenton10recordsthenumberofplacestheoriginaldecimal pointwasmovedtothe left 0 : 00059= 0 : 0059 10 = 0 : 0059 10 1 =0 : 0059 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 0 : 059 100 = 0 : 059 10 2 =0 : 059 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = 0 : 59 1000 = 0 : 59 10 3 =0 : 59 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = 5 : 9 10 ; 000 = 5 : 9 10 4 =5 : 9 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Thereis one nonzerodigittotheleftofthedecimalpointandtheabsolutevalueoftheexponentof10 recordsthenumberofplacestheoriginaldecimalpointwasmovedtothe right ScienticNotation Numberswritteninscienticformarealsosaidtobewrittenusingscienticnotation.In scienticnotation ,anumberiswrittenastheproductofanumberbetweenandincluding1and10 isincluded, 10 isnot andsomepowerof10. WritingaNumberinScienticNotation Towriteanumberinscienticnotation: 1.Movethedecimalpointsothatthereisonenonzerodigittoitsleft. 2.Multiplytheresultbyapowerof10usinganexponentwhoseabsolutevalueisthenumberofplaces thedecimalpointwasmoved.Maketheexponentpositiveifthedecimalpointwasmovedtotheleft andnegativeifthedecimalpointwasmovedtotheright. 3.8.3SampleSetA Writethenumbersinscienticnotation. Example3.57 981 8 Thiscontentisavailableonlineat.

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164 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Thenumber981isactually 981 : ,anditisfollowedbyadecimalpoint.Inintegers,thedecimal pointattheendisusuallyomitted. 981=981 : =9 : 81 10 2 Thedecimalpointisnowtwoplacestotheleftofitsoriginalposition,andthepowerof10is2. Example3.58 54 : 066=5 : 4066 10 1 =5 : 4066 10 Thedecimalpointisoneplacetotheleftofitsoriginalposition,andthepowerof10is1. Example3.59 0 : 000000000004632=4 : 632 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 Thedecimalpointistwelveplacestotherightofitsoriginalposition,andthepowerof10is )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Example3.60 0 : 027=2 : 7 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Thedecimalpointistwoplacestotherightofitsoriginalposition,andthepowerof10is )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 3.8.4PracticeSetA Writethefollowingnumbersinscienticnotation. Exercise3.531 Solutiononp.191. 346 Exercise3.532 Solutiononp.191. 72 : 33 Exercise3.533 Solutiononp.191. 5387 : 7965 Exercise3.534 Solutiononp.191. 87,000,000 Exercise3.535 Solutiononp.191. 179,000,000,000,000,000,000 Exercise3.536 Solutiononp.191. 100,000 Exercise3.537 Solutiononp.191. 1,000,000 Exercise3.538 Solutiononp.191. 0 : 0086 Exercise3.539 Solutiononp.191. 0 : 000098001 Exercise3.540 Solutiononp.191. 0 : 000000000000000054 Exercise3.541 Solutiononp.191. 0 : 0000001 Exercise3.542 Solutiononp.191. 0 : 00000001

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165 3.8.5ScienticFormtoStandardForm Anumberwritteninscienticnotationcanbeconvertedtostandardformbyreversingtheprocessshown inSampleSetA. ConvertingfromScienticNotation Toconvertanumberwritteninscienticnotationtoanumberinstandardform,movethedecimalpoint thenumberofplacesprescribedbytheexponentonthe10. PositiveExponentNegativeExponent Movethedecimalpointtotherightwhenyouhaveapositiveexponent,andmovethedecimalpointtothe leftwhenyouhaveanegativeexponent. 3.8.6SampleSetB Example3.61 4 : 673 10 4 Theexponentof10is4sowemustmovethedecimalpointtotheright4placesadding 0 'sif necessary. Example3.62 2 : 9 10 7 Theexponentof10is7sowemustmovethedecimalpointtotheright7placesadding 0 'sif necessary. 2 : 9 10 7 = 29000000 Example3.63 1 10 27 Theexponentof10is27sowemustmovethedecimalpointtotheright27placesadding 0 's withoutadoubt. 1 10 27 = 1,000,000,000,000,000,000,000,000,000 Example3.64 4 : 21 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Theexponentof10is )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 sowemustmovethedecimalpointtotheleft5placesadding 0 'sif necessary. 4 : 21 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = 0.0000421 Example3.65 1 : 006 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 Theexponentof10is )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 sowemustmovethedecimalpointtotheleft18placesadding 0 's ifnecessary. 1 : 006 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 =0 : 000000000000000001006 3.8.7PracticeSetB Convertthefollowingnumberstostandardform. Exercise3.543 Solutiononp.191. 9 : 25 10 2 Exercise3.544 Solutiononp.191. 4 : 01 10 5

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166 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.545 Solutiononp.192. 1 : 2 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.546 Solutiononp.192. 8 : 88 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 3.8.8WorkingwithNumbersinScienticNotation MultiplyingNumbersUsingScienticNotation Therearemanyoccasionsparticularlyinthescienceswhenitisnecessarytondtheproductoftwo numberswritteninscienticnotation.Thisisaccomplishedbyusingtwoofthebasicrulesofalgebra. Supposewewishtond a 10 n b 10 m .Sincetheonlyoperationismultiplication,wecanusethe commutativepropertyofmultiplicationtorearrangethenumbers. a 10 n b 10 m = a b n 10 m Then,bytherulesofexponents, 10 n 10 m =10 n + m .Thus, a 10 n b 10 m = a b 10 n + m Theproductof a b maynotbebetween1and10,so a b 10 n + m maynotbeinscienticform. Thedecimalpointin a b mayhavetobemoved.AnexampleofthissituationisinSampleSetC, problem2. 3.8.9SampleSetC Example3.66 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 10 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 10 8 = 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(10 3 10 8 =8 10 3+8 =8 10 11 Example3.67 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 10 17 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(8 : 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(22 = 8 : 1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(10 17 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(22 =40 : 5 10 17 )]TJ/F7 6.9738 Tf 6.226 0 Td [(22 =40 : 5 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Weneedtomovethedecimalpointoneplacetothe left toputthisnumberinscienticnotation. Thus,wemustalsochangetheexponentof10. 40 : 5 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 4 : 05 10 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 4 : 05 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(10 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 4 : 05 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(10 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 4 : 05 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Thus, )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 10 17 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(8 : 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(22 =4 : 05 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 3.8.10PracticeSetC Performeachmultiplication. Exercise3.547 Solutiononp.192. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 10 5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 10 12

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167 Exercise3.548 Solutiononp.192. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(6 10 24 Exercise3.549 Solutiononp.192. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 10 18 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 10 6 Exercise3.550 Solutiononp.192. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 : 1 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 3.8.11Exercises Convertthenumbersusedinthefollowingproblemstoscienticnotation. Exercise3.551 Solutiononp.192. MountKilimanjaroisthehighestmountaininAfrica.Itis5890metershigh. Exercise3.552 TheplanetMarsisabout222,900,000,000metersfromthesun. Exercise3.553 Solutiononp.192. Thereisanirregularlyshapedgalaxy,namedNGC4449,thatisabout 250,000,000,000,000,000,000,000metersfromearth. Exercise3.554 Thefarthestobjectastronomershavebeenabletoseeasof1981isaquasarnamed3C427.There seemstobeahazebeyondthisquasarthatappearstomarkthevisualboundaryoftheuniverse. Quasar3C427isatadistanceof110,000,000,000,000,000,000,000,000metersfromtheearth. Exercise3.555 Solutiononp.192. Thesmallestknowninsectsareaboutthesizeofatypicalgrainofsand.Theyareabout 0 : 0002 metersinlengthten-thousandthsofameter. Exercise3.556 Atomssuchashydrogen,carbon,nitrogen,andoxygenareabout 0 : 0000000001 meteracross. Exercise3.557 Solutiononp.192. TheislandofManhattan,inNewYork,isabout57,000squaremetersinarea. Exercise3.558 ThesecondlargestmoonofSaturnisRhea.Rheahasasurfaceareaofabout735,000square meters,roughlythesamesurfaceareaasAustralia. Exercise3.559 Solutiononp.192. Astar,namedEpsilonAurigaeB,hasadiameterdistanceacrossof2,800,000,000,000meters. Thisdiameterproducesasurfaceareaofabout24,630,000,000,000,000,000,000,000squaremeters. Thisstariswhatastronomerscallaredgiantanditisthelargestredgiantknown.IfEpsilon Aurigaewereplacedatthesun'sposition,itssurfacewouldextendouttotheplanetUranus. Exercise3.560 ThevolumeoftheplanetVenusis927,590,000,000,000,000,000cubicmeters. Exercise3.561 Solutiononp.192. TheaveragemassofanewbornAmericanfemaleisabout3360grams. Exercise3.562 Thelargestbrainevermeasuredwasthatofaspermwhale.Ithadamassof9200grams. Exercise3.563 Solutiononp.192. ThemassoftheEieltowerinParis,France,is8,000,000grams.

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168 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.564 In1981,aJapanesecompanybuiltthelargestoiltankertodate.Theshiphasamassofabout 510,000,000,000grams.Thisoiltankerismorethan6timesasmassiveastheU.S.aircraftcarrier, U.S.S. Nimitz Exercise3.565 Solutiononp.192. IntheconstellationofVirgo,thereisaclusterofabout2500galaxies.Thecombinedmassofthese galaxiesis150,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grams. Exercise3.566 Themassofanamoebaisabout 0 : 000004 gram. Exercise3.567 Solutiononp.192. Cellsinthehumanliverhavemassesofabout 0 : 000000008 gram. Exercise3.568 Thehumanspermcellhasamassofabout 0 : 000000000017 gram. Exercise3.569 Solutiononp.192. Theprincipalproteinofmuscleismyosin.Myosinhasamassof 0 : 00000000000000000103 gram. Exercise3.570 Aminoacidsaremoleculesthatcombinetomakeupproteinmolecules.Theaminoacidtryptophan hasamassof 0 : 000000000000000000000340 gram. Exercise3.571 Solutiononp.192. Anatomofthechemicalelementbrominehas35electrons.Themassofabromineatomis 0 : 000000000000000000000000031 gram. Exercise3.572 Physicistsareperformingexperimentsthattheyhopewilldeterminethemassofa smallparticlecalledaneutrino.Itissuspectedthatneutrinoshavemassesofabout 0 : 0000000000000000000000000000001 gram. Exercise3.573 Solutiononp.192. Theapproximatetimeittakesforahumanbeingtodieofasphyxiationis316seconds. Exercise3.574 Ontheaverage,themalehouseylives1,468,800secondsdays. Exercise3.575 Solutiononp.192. Aluminum-26hasahalf-lifeof740,000years. Exercise3.576 Manganese-53hasahalf-lifeof59,918,000,000,000seconds,900,000years. Exercise3.577 Solutiononp.192. Initsorbitaroundthesun,theearthmovesadistanceoneandonehalffeetinabout 0 : 0000316 second. Exercise3.578 Api-mesonisasubatomicparticlethathasahalf-lifeofabout 0 : 0000000261 second. Exercise3.579 Solutiononp.192. Asubatomicparticlecalledaneutralpionhasahalf-lifeofabout 0 : 0000000000000001 second. Exercise3.580 Nearthesurfaceoftheearth,thespeedofsoundis1195feetpersecond. Forthefollowingproblems,convertthenumbersfromscienticnotationtostandarddecimalform. Exercise3.581 Solutiononp.192. Thesunisabout 1 10 8 metersfromearth.

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169 Exercise3.582 Themassoftheearthisabout 5 : 98 10 27 grams. Exercise3.583 Solutiononp.192. Lighttravelsabout 5 : 866 10 12 milesinoneyear. Exercise3.584 Oneyearisabout 3 10 7 seconds. Exercise3.585 Solutiononp.192. Rubik'scubehasabout 4 : 3 10 19 dierentcongurations. Exercise3.586 Aphotonisaparticleoflight.A100-wattlightbulbemits 1 10 20 photonseverysecond. Exercise3.587 Solutiononp.192. Thereareabout 6 10 7 cellsintheretinaofthehumaneye. Exercise3.588 Acartravelingatanaveragespeedwilltraveladistanceaboutequaltothelengthofthesmallest ngernailin 3 : 16 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 seconds. Exercise3.589 Solutiononp.193. Aribosomeof E.coli hasamassofabout 4 : 7 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(19 grams. Exercise3.590 Amitochondrionistheenergy-producingelementofacell.Amitochondrionisabout 1 : 5 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 metersindiameter. Exercise3.591 Solutiononp.193. ThereisaspeciesoffrogsinCubathatattainalengthofatmost 1 : 25 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 meters. Performthefollowingoperations. Exercise3.592 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 10 4 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 10 5 Exercise3.593 Solutiononp.193. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 10 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(8 10 6 Exercise3.594 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6 10 14 )]TJ/F8 9.9626 Tf 10.792 -8.07 Td [(6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 Exercise3.595 Solutiononp.193. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(8 10 7 Exercise3.596 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise3.597 Solutiononp.193. )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(9 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 Exercise3.598 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(3 : 1 10 4 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(3 : 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise3.599 Solutiononp.193. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 : 2 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(20 Exercise3.600 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 : 1 10 6 2 Exercise3.601 Solutiononp.193. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 : 9 10 14 2 Exercise3.602 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 : 02 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(17 2

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170 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.603 Solutiononp.193. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(8 : 8 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(50 2 Exercise3.604 IfMountKilimanjarowas1,000,000timesashighasitreallyis,howhighwoulditbe?See problem1. Exercise3.605 Solutiononp.193. IftheplanetMarswas300,000timesasfarfromthesunasitreallyis,howfarfromthesunwould itbe?Seeproblem2. Exercise3.606 If800,000,000ofthesmallestinsectsknownwerelinedupheadtotail,howfarwouldtheystretch? Seeproblem5. Exercise3.607 Solutiononp.193. IfRhea,themoonofSaturn,hadasurfacearea 0 : 00000000002 ofitsrealsurfacearea,whatwould thatsurfaceareabe?Seeproblem8. Exercise3.608 IfthestarEpsilonAurigaeBhadasurfacearea 0 : 005 ofitsrealsurfacearea,whatwouldthat surfaceareabe?Seeproblem9. Exercise3.609 Solutiononp.193. IfthemassofallthegalaxiesintheconstellationVirgowasonly 0 : 0000000000000000000000003 ofitsrealmass,whatwouldthatmassbe?Seeproblem15. Exercise3.610 Whatisthemassof15,000,000,000,000bromineatoms?Seeproblem21. 3.8.12ExercisesforReview Exercise3.611 Solutiononp.193. Section2.3 Whatintegerscanreplace x sothatthestatement )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 .

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171 OppositesSection3.2 Oppositesarenumbersthatarethesamedistancefromzeroonthenumberlinebuthaveoppositesigns. Double-NegativePropertySection3.2 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F11 9.9626 Tf 7.748 0 Td [(a = a AbsoluteValueGeometricSection3.3 Theabsolutevalueofanumber a ,denoted j a j ,isthedistancefrom a to0onthenumberline. AbsoluteValueAlgebraicSection3.3 j a j = f a if a 0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a if a< 0 AdditionofSignedNumbersSection3.4 Toaddtwonumberswith likesigns ,addtheabsolutevaluesofthenumbersandassociatethecommonsignwiththesum. unlikesigns ,subtractthesmallerabsolutevaluefromthelargerabsolutevalueandassociatethesignof thelargerabsolutevaluewiththedierence. Additionwith0Section3.4 0+ anynumber = thatparticularnumber,thatis, 0+ a = a foranyrealnumber a AdditiveIdentitySection3.4 Sinceadding0toarealnumberleavesthatnumberunchanged,0iscalledtheadditiveidentity. DenitionofSubtractionSection3.5 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b = a + )]TJ/F11 9.9626 Tf 7.748 0 Td [(b SubtractionofSignedNumbersSection3.5 Toperformthesubtraction a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b ,addtheoppositeof b to a ,thatis,changethesignof b andadd. MultiplicationandDivisionofSignedNumbersSection3.6 ++=+ + + =+ + )]TJ/F7 6.9738 Tf 6.227 0 Td [( = )]TJ/F8 9.9626 Tf -140.971 -16.895 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(=+ + )]TJ/F8 9.9626 Tf 7.749 0 Td [(= )]TJ/F8 9.9626 Tf -49.293 -16.738 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(+= )]TJ/F7 6.9738 Tf 18.907 4.832 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [( =+ )]TJ/F7 6.9738 Tf 6.227 0 Td [( + = )]TJ/F39 9.9626 Tf -145.952 -13.645 Td [(ReciprocalsSection3.7 Twonumbersarereciprocalsofeachotheriftheirproductis1.Thenumbers4and 1 4 arereciprocalssince )]TJ/F7 6.9738 Tf 5.762 -4.148 Td [(1 4 =1 NegativeExponentsSection3.7 If n isanynaturalnumberand x isanynonzerorealnumber,then x )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = 1 x n WritingaNumberinScienticNotationSection3.8 Towriteanumberinscienticnotation: 1.Movethedecimalpointsothatthereisonenonzerodigittoitsleft. 2.Multiplytheresultbyapowerof10usinganexponentwhoseabsolutevalueisthenumberofplaces thedecimalpointwasmoved.Maketheexponentpositiveifthedecimalpointwasmovedtotheleft andnegativeifthedecimalpointwasmovedtotheright. ConvertingfromScienticNotation: positiveexponentSection3.8 Toconvertanumberwritteninscienticnotationtoanumberinstandardformwhenthereisa positive exponentasthepowerof10,movethedecimalpointtothe right thenumberofplacesprescribedbythe exponentonthe10. NegativeExponentSection3.8 Toconvertanumberwritteninscienticnotationtoanumberinstandardformwhenthereisa negative exponentasthepowerof10,movethedecimalpointtothe left thenumberofplacesprescribedbythe exponentonthe10.

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172 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS 3.10ExerciseSupplement 10 3.10.1ExerciseSupplement 3.10.1.1SignedNumbersSection3.2 Forthefollowingproblems,nd )]TJ/F11 9.9626 Tf 7.748 0 Td [(a if a is Exercise3.616 Solutiononp.193. 27 Exercise3.617 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 Exercise3.618 Solutiononp.193. )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(8 9 Exercise3.619 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise3.620 Solutiononp.193. k 3.10.1.2AbsoluteValueSection3.3 Simplifythefollowingproblems. Exercise3.621 j 8 j Exercise3.622 Solutiononp.193. j)]TJ/F8 9.9626 Tf 14.944 0 Td [(3 j Exercise3.623 j 16 j Exercise3.624 Solutiononp.193. )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 12 j Exercise3.625 j 0 j 3.10.1.3AddItionofSignedNumbersSection3.4-MultiplicationandDivisionofSigned NumbersSection3.6 Simplifythefollowingproblems. Exercise3.626 Solutiononp.193. 4+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise3.627 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 Exercise3.628 Solutiononp.193. 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 Exercise3.629 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 10 Thiscontentisavailableonlineat.

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173 Exercise3.630 Solutiononp.193. )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise3.631 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise3.632 Solutiononp.193. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise3.633 )]TJ/F7 6.9738 Tf 6.226 0 Td [(25 5 Exercise3.634 Solutiononp.193. )]TJ/F7 6.9738 Tf 6.226 0 Td [(100 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 Exercise3.635 16 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18+5 Exercise3.636 Solutiononp.193. )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4+10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.637 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8+4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 4+6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.638 Solutiononp.194. )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13+10 Exercise3.639 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 Exercise3.640 Solutiononp.194. 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 3.10.1.4MultiplicationandDivisionofSignedNumbersSection3.6 Findthevalueofeachexpressionforthefollowingproblems. Exercise3.641 P = R )]TJ/F11 9.9626 Tf 9.962 0 Td [(C .Find P if R =3000 and C =3800 Exercise3.642 Solutiononp.194. z = x )]TJ/F10 6.9738 Tf 6.227 0 Td [(u s .Find z if x =22 ;u =30 ,and s =8 Exercise3.643 P = n n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 .Find P if n = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 3.10.1.5NegativeExponentsSection3.7 Writetheexpressionsforthefollowingproblemsusingonlypositiveexponents. Exercise3.644 Solutiononp.194. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.645 c )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Exercise3.646 Solutiononp.194. a 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 c )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.647 x +5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2

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174 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.648 Solutiononp.194. x 3 y 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise3.649 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 c 5 Exercise3.650 Solutiononp.194. 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.651 x +9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 7 x 4 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 x +5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.652 Solutiononp.194. )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.653 1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 Exercise3.654 Solutiononp.194. 7 x y )]TJ/F6 4.9813 Tf 5.397 0 Td [(3 z )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 Exercise3.655 4 c )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 Exercise3.656 Solutiononp.194. 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 a )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(9 c 2 x 2 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 z )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 Exercise3.657 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 z +6 )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 Exercise3.658 Solutiononp.194. 16 a 5 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 3 b )]TJ/F6 4.9813 Tf 5.396 0 Td [(5 Exercise3.659 )]TJ/F7 6.9738 Tf 6.227 0 Td [(44 x 3 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 z )]TJ/F6 4.9813 Tf 5.396 0 Td [(8 )]TJ/F7 6.9738 Tf 6.226 0 Td [(11 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 y )]TJ/F6 4.9813 Tf 5.397 0 Td [(7 z )]TJ/F6 4.9813 Tf 5.397 0 Td [(8 Exercise3.660 Solutiononp.194. 8 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise3.661 9 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.662 Solutiononp.194. 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.663 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.664 Solutiononp.194. )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 2 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise3.665 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.666 Solutiononp.194. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise3.667 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise3.668 Solutiononp.194. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise3.669 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5

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175 Exercise3.670 Solutiononp.194. 2 b )]TJ/F6 4.9813 Tf 5.397 0 Td [(7 c )]TJ/F6 4.9813 Tf 5.396 0 Td [(8 d 4 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 y 3 z )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3.10.1.6ScienticNotationSection3.8 Writethefollowingproblemsusingscienticnotation. Exercise3.671 8739 Exercise3.672 Solutiononp.194. 73567 Exercise3.673 21,000 Exercise3.674 Solutiononp.194. 746,000 Exercise3.675 8866846 Exercise3.676 Solutiononp.194. 0 : 0387 Exercise3.677 0 : 0097 Exercise3.678 Solutiononp.194. 0 : 376 Exercise3.679 0 : 0000024 Exercise3.680 Solutiononp.194. 0 : 000000000000537 Exercise3.681 46,000,000,000,000,000 Convertthefollowingproblemsfromscienticformtostandardform. Exercise3.682 Solutiononp.194. 3 : 87 10 5 Exercise3.683 4 : 145 10 4 Exercise3.684 Solutiononp.194. 6 : 009 10 7 Exercise3.685 1 : 80067 10 6 Exercise3.686 Solutiononp.194. 3 : 88 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.687 4 : 116 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.688 Solutiononp.195. 8 : 002 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 Exercise3.689 7 : 36490 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(14

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176 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.690 Solutiononp.195. 2 : 101 10 15 Exercise3.691 6 : 7202 10 26 Exercise3.692 Solutiononp.195. 1 10 6 Exercise3.693 1 10 7 Exercise3.694 Solutiononp.195. 1 10 9 Findtheproductforthefollowingproblems.Writetheresultinscienticnotation. Exercise3.695 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 10 5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 10 3 Exercise3.696 Solutiononp.195. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 10 6 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(7 10 7 Exercise3.697 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 10 14 )]TJ/F8 9.9626 Tf 10.792 -8.07 Td [(8 10 19 Exercise3.698 Solutiononp.195. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(9 10 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 10 75 Exercise3.699 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 10 4 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(1 10 5 Exercise3.700 Solutiononp.195. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise3.701 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(9 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(2 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise3.702 Solutiononp.195. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(7 10 2 Exercise3.703 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 : 3 10 4 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 : 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise3.704 Solutiononp.195. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 : 06 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 : 815 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 Exercise3.705 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(9 : 3806 10 52 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(1 : 009 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(31 3.11ProciencyExam 11 3.11.1ProciencyExam Simplifytheexpressionsforthefollowingproblems. Exercise3.706 Solutiononp.195. Section3.2 f)]TJ/F8 9.9626 Tf 22.139 0 Td [([ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6] g Exercise3.707 Solutiononp.195. Section3.3 j)]TJ/F8 9.9626 Tf 22.693 0 Td [(15 j 11 Thiscontentisavailableonlineat.

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177 Exercise3.708 Solutiononp.195. Section3.5 )]TJ/F8 9.9626 Tf 7.749 0 Td [([ j)]TJ/F8 9.9626 Tf 14.944 0 Td [(12 j)]TJ/F8 9.9626 Tf 14.944 0 Td [(10] 2 Exercise3.709 Solutiononp.195. Section3.5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 )-222(j)]TJ/F8 9.9626 Tf 24.907 0 Td [(5 j Exercise3.710 Solutiononp.195. Section3.6 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise3.711 Solutiononp.195. Section3.6 j 7 j)]TJ/F8 9.9626 Tf 14.943 0 Td [( 2 + )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 2 Exercise3.712 Solutiononp.195. Section3.6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise3.713 Solutiononp.195. Section3.6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 f [ )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3][ )]TJ/F7 6.9738 Tf 6.227 0 Td [(2] g )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise3.714 Solutiononp.195. Section3.6 If z = x )]TJ/F10 6.9738 Tf 6.227 0 Td [(u s ,nd z if x =14 u =20 ,and s =2 Whensimplifyingthetermsforthefollowingproblems,writeeachsothatonlypositiveexponentsappear. Exercise3.715 Solutiononp.195. Section3.7 1 )]TJ/F7 6.9738 Tf 6.226 0 Td [( )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 Exercise3.716 Solutiononp.195. Section3.7 5 x 3 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 z )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 Exercise3.717 Solutiononp.195. Section3.7 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 m 6 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise3.718 Solutiononp.195. Section3.7 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise3.719 Solutiononp.195. Section3.7 6 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 x 3 y )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 y )]TJ/F6 4.9813 Tf 5.396 0 Td [(5 Exercise3.720 Solutiononp.195. Section3.7 k )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 2 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 3 Exercise3.721 Solutiononp.195. Section3.7 y +1 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 4 y +1 5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(8 Exercise3.722 Solutiononp.196. Section3.7 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(6 3 2 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(10 )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 )]TJ/F6 4.9813 Tf 5.397 0 Td [(9 Exercise3.723 Solutiononp.196. Section3.7 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise3.724 Solutiononp.196. Section3.7 h r 6 s )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 m )]TJ/F6 4.9813 Tf 5.396 0 Td [(5 n 4 i )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise3.725 Solutiononp.196. Section3.7 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(c 0 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 ;c 6 =0 Exercise3.726 Solutiononp.196. Section3.8 Write 0 : 000271 usingscienticnotation. Exercise3.727 Solutiononp.196. Section3.8 Write 8 : 90 10 5 instandardform. Exercise3.728 Solutiononp.196. Section3.8 Findthevalueof )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 10 5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 .

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178 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS Exercise3.729 Solutiononp.196. Section3.8 Findthevalueof )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(4 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 2 Exercise3.730 Solutiononp.196. Section3.8 If k isanegativeinteger,is )]TJ/F11 9.9626 Tf 7.748 0 Td [(k apositiveornegativeinteger?

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179 SolutionstoExercisesinChapter3 SolutiontoExercise3.1p.123 fourplusten SolutiontoExercise3.2p.123 sevenplusnegativefour SolutiontoExercise3.3p.123 negativenineplustwo SolutiontoExercise3.4p.123 negativesixteenminuspositiveeight SolutiontoExercise3.5p.123 negativeoneminusnegativenine SolutiontoExercise3.6p.123 zeroplusnegativeseven SolutiontoExercise3.7p.124 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.8p.124 )]TJ/F8 9.9626 Tf 7.749 0 Td [(17 SolutiontoExercise3.9p.124 6 SolutiontoExercise3.10p.124 15 SolutiontoExercise3.11p.124 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,since )]TJ/F8 9.9626 Tf 9.41 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(1=1 SolutiontoExercise3.12p.124 7 SolutiontoExercise3.13p.124 If a ispositive, )]TJ/F11 9.9626 Tf 7.749 0 Td [(a isnegative. SolutiontoExercise3.14p.124 If a isnegative, )]TJ/F11 9.9626 Tf 7.749 0 Td [(a ispositive. SolutiontoExercise3.15p.124 Wemustsaythatwedonotknow. SolutiontoExercise3.16p.124 aplussignornosignatall SolutiontoExercise3.18p.124 anegativeve SolutiontoExercise3.20p.124 twelve SolutiontoExercise3.22p.125 negativenegativefour SolutiontoExercise3.24p.125 veplusseven SolutiontoExercise3.26p.125 elevenplusnegativetwo SolutiontoExercise3.28p.125 sixminusnegativeeight SolutiontoExercise3.30p.125 )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(8=8 SolutiontoExercise3.32p.125 2 SolutiontoExercise3.34p.125 1

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180 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.36p.125 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.38p.125 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.40p.125 26 SolutiontoExercise3.42p.125 31 SolutiontoExercise3.44p.126 12 SolutiontoExercise3.46p.126 17 SolutiontoExercise3.48p.126 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(2=5+2=7 SolutiontoExercise3.50p.126 16 SolutiontoExercise3.52p.126 32 SolutiontoExercise3.54p.126 13 SolutiontoExercise3.56p.126 26 SolutiontoExercise3.58p.126 0 SolutiontoExercise3.60p.126 x n +8 SolutiontoExercise3.62p.126 16 a 4 b 2 9 x 2 y 6 SolutiontoExercise3.63p.128 7 SolutiontoExercise3.64p.128 9 SolutiontoExercise3.65p.128 12 SolutiontoExercise3.66p.128 5 SolutiontoExercise3.67p.128 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.68p.128 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise3.69p.128 )]TJ/F8 9.9626 Tf 7.749 0 Td [(52 SolutiontoExercise3.70p.128 )]TJ/F8 9.9626 Tf 7.749 0 Td [(31 SolutiontoExercise3.71p.129 5 SolutiontoExercise3.73p.129 6 SolutiontoExercise3.75p.129 8

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181 SolutiontoExercise3.77p.129 16 SolutiontoExercise3.79p.129 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.81p.129 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 SolutiontoExercise3.83p.129 1 SolutiontoExercise3.85p.129 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.87p.129 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 SolutiontoExercise3.89p.129 )]TJ/F8 9.9626 Tf 7.749 0 Td [(28 SolutiontoExercise3.91p.129 )]TJ/F8 9.9626 Tf 7.749 0 Td [(68 SolutiontoExercise3.93p.130 26 SolutiontoExercise3.95p.130 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.97p.130 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise3.99p.130 4 SolutiontoExercise3.101p.130 6 SolutiontoExercise3.103p.130 3 SolutiontoExercise3.105p.130 12 SolutiontoExercise3.107p.130 4 SolutiontoExercise3.109p.130 4 SolutiontoExercise3.111p.130 5 SolutiontoExercise3.113p.130 3 SolutiontoExercise3.115p.131 6 SolutiontoExercise3.117p.131 100 SolutiontoExercise3.119p.131 92 SolutiontoExercise3.121p.131 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise3.123p.131 j $ )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ; 400 ; 000 j SolutiontoExercise3.125p.131 4 a + b

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182 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.127p.131 commutativepropertyofaddition SolutiontoExercise3.129p.131 4 SolutiontoExercise3.130p.133 14 SolutiontoExercise3.131p.133 52 SolutiontoExercise3.132p.133 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.133p.133 )]TJ/F8 9.9626 Tf 7.749 0 Td [(45 SolutiontoExercise3.134p.133 )]TJ/F8 9.9626 Tf 7.749 0 Td [(34 SolutiontoExercise3.135p.133 )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(7 3 SolutiontoExercise3.136p.133 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 : 4 SolutiontoExercise3.137p.135 1 SolutiontoExercise3.138p.135 2 SolutiontoExercise3.139p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.140p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.141p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 SolutiontoExercise3.142p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(43 SolutiontoExercise3.143p.135 5 SolutiontoExercise3.144p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 5 SolutiontoExercise3.145p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.146p.135 0 : 57 SolutiontoExercise3.147p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(425 SolutiontoExercise3.148p.135 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7993 : 7 SolutiontoExercise3.149p.136 16 SolutiontoExercise3.151p.136 8 SolutiontoExercise3.153p.136 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 SolutiontoExercise3.155p.136 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12

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183 SolutiontoExercise3.157p.136 )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 SolutiontoExercise3.159p.136 11 SolutiontoExercise3.161p.136 8 SolutiontoExercise3.163p.136 2 SolutiontoExercise3.165p.136 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 SolutiontoExercise3.167p.136 8 SolutiontoExercise3.169p.136 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.171p.137 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 SolutiontoExercise3.173p.137 )]TJ/F8 9.9626 Tf 7.749 0 Td [(25 SolutiontoExercise3.175p.137 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.177p.137 24 SolutiontoExercise3.179p.137 )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 SolutiontoExercise3.181p.137 0 SolutiontoExercise3.183p.137 0 SolutiontoExercise3.185p.137 23 SolutiontoExercise3.187p.137 328 SolutiontoExercise3.189p.137 876 SolutiontoExercise3.191p.137 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1265 SolutiontoExercise3.193p.138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 : 084 SolutiontoExercise3.195p.138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 SolutiontoExercise3.197p.138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(17 SolutiontoExercise3.199p.138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 SolutiontoExercise3.201p.138 14 SolutiontoExercise3.203p.138 16 SolutiontoExercise3.205p.138 4

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184 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.207p.138 )]TJ/F8 9.9626 Tf 7.749 0 Td [($28 : 50 SolutiontoExercise3.209p.138 $3 : 00 SolutiontoExercise3.211p.139 5 a 6 c SolutiontoExercise3.213p.139 8 SolutiontoExercise3.215p.141 3 SolutiontoExercise3.216p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.217p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise3.218p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 SolutiontoExercise3.219p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 SolutiontoExercise3.220p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(27 SolutiontoExercise3.221p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise3.222p.141 18 SolutiontoExercise3.223p.141 13 SolutiontoExercise3.224p.141 118 SolutiontoExercise3.225p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 SolutiontoExercise3.226p.141 16 SolutiontoExercise3.227p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.228p.141 10 SolutiontoExercise3.229p.141 0 SolutiontoExercise3.230p.141 5 SolutiontoExercise3.232p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise3.234p.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 SolutiontoExercise3.236p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.238p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 SolutiontoExercise3.240p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13

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185 SolutiontoExercise3.242p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 SolutiontoExercise3.244p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise3.246p.142 5 SolutiontoExercise3.248p.142 11 SolutiontoExercise3.250p.142 5 SolutiontoExercise3.252p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise3.254p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise3.256p.142 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.258p.143 7 SolutiontoExercise3.260p.143 29 SolutiontoExercise3.262p.143 )]TJ/F8 9.9626 Tf 7.749 0 Td [(324 SolutiontoExercise3.264p.143 )]TJ/F8 9.9626 Tf 7.749 0 Td [(429 SolutiontoExercise3.266p.143 )]TJ/F8 9.9626 Tf 7.749 0 Td [(71 SolutiontoExercise3.268p.143 164 SolutiontoExercise3.270p.143 1 SolutiontoExercise3.272p.143 1 SolutiontoExercise3.274p.143 8 SolutiontoExercise3.276p.143 4 SolutiontoExercise3.278p.143 6 SolutiontoExercise3.280p.144 )]TJ/F8 9.9626 Tf 7.749 0 Td [(27 SolutiontoExercise3.282p.144 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.284p.144 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 SolutiontoExercise3.286p.144 20 xy +44 x SolutiontoExercise3.288p.144 11 SolutiontoExercise3.290p.144 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3

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186 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.291p.146 )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 SolutiontoExercise3.292p.146 64 SolutiontoExercise3.293p.146 30 SolutiontoExercise3.294p.146 14 SolutiontoExercise3.295p.146 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise3.296p.146 )]TJ/F8 9.9626 Tf 7.749 0 Td [(49 SolutiontoExercise3.297p.148 4 SolutiontoExercise3.298p.148 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.299p.148 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise3.300p.148 3 SolutiontoExercise3.301p.149 1 SolutiontoExercise3.302p.149 1 SolutiontoExercise3.303p.149 16 SolutiontoExercise3.305p.149 32 SolutiontoExercise3.307p.149 54 SolutiontoExercise3.309p.149 32 SolutiontoExercise3.311p.149 )]TJ/F8 9.9626 Tf 7.749 0 Td [(36 SolutiontoExercise3.313p.149 )]TJ/F8 9.9626 Tf 7.749 0 Td [(32 SolutiontoExercise3.315p.149 )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 SolutiontoExercise3.317p.149 )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 SolutiontoExercise3.319p.149 )]TJ/F8 9.9626 Tf 7.749 0 Td [(90 SolutiontoExercise3.321p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(60 SolutiontoExercise3.323p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.325p.150 3 SolutiontoExercise3.327p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13

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187 SolutiontoExercise3.329p.150 9 SolutiontoExercise3.331p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise3.333p.150 11 SolutiontoExercise3.335p.150 28 SolutiontoExercise3.337p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise3.339p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.341p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise3.343p.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.345p.151 15 SolutiontoExercise3.347p.151 49 SolutiontoExercise3.349p.151 )]TJ/F8 9.9626 Tf 7.749 0 Td [(140 SolutiontoExercise3.351p.151 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise3.353p.151 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.355p.151 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise3.357p.151 13 SolutiontoExercise3.359p.151 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.361p.151 4 SolutiontoExercise3.363p.151 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise3.365p.151 15 SolutiontoExercise3.367p.152 2 SolutiontoExercise3.369p.152 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise3.371p.152 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.373p.152 1458 SolutiontoExercise3.375p.152 )]TJ/F8 9.9626 Tf 7.749 0 Td [(120 SolutiontoExercise3.377p.152 40

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188 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.379p.152 x +2 y 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 SolutiontoExercise3.381p.152 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 SolutiontoExercise3.383p.154 1 y 5 SolutiontoExercise3.384p.154 1 m 2 SolutiontoExercise3.385p.154 1 9 SolutiontoExercise3.386p.154 1 5 SolutiontoExercise3.387p.154 1 16 SolutiontoExercise3.388p.154 1 xy 4 SolutiontoExercise3.389p.154 1 a +2 b 12 SolutiontoExercise3.390p.154 m )]TJ/F11 9.9626 Tf 9.962 0 Td [(n 4 SolutiontoExercise3.391p.155 y 7 x 4 SolutiontoExercise3.392p.155 a 2 b 4 SolutiontoExercise3.393p.155 x 3 y 4 z 8 SolutiontoExercise3.394p.155 6 k 7 m 3 n 2 SolutiontoExercise3.395p.155 a 2 b 6 c 8 SolutiontoExercise3.396p.155 3 a 5 b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 5 SolutiontoExercise3.397p.156 4 x 10 b 8 SolutiontoExercise3.398p.156 64 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 n 7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 SolutiontoExercise3.399p.156 52 SolutiontoExercise3.400p.156 1 x 2 SolutiontoExercise3.402p.156 1 x 7 SolutiontoExercise3.404p.156 1 a 10 SolutiontoExercise3.406p.157 1 b 14 SolutiontoExercise3.408p.157 1 y 5 SolutiontoExercise3.410p.157 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 3

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189 SolutiontoExercise3.412p.157 1 a +9 10 SolutiontoExercise3.414p.157 1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 12 SolutiontoExercise3.416p.157 x 7 y 5 SolutiontoExercise3.418p.157 a 7 b 8 SolutiontoExercise3.420p.157 x 3 y 2 z 6 SolutiontoExercise3.422p.157 a 7 zw 3 b 9 SolutiontoExercise3.424p.157 x 5 y 5 z 2 SolutiontoExercise3.426p.157 d 4 a 4 b 6 c SolutiontoExercise3.428p.158 4 y 2 x 6 SolutiontoExercise3.430p.158 7 b 2 c 2 a 2 SolutiontoExercise3.432p.158 7 a 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 b 6 c 7 SolutiontoExercise3.434p.158 7 w +1 3 w +2 2 SolutiontoExercise3.436p.158 x 2 +3 3 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 SolutiontoExercise3.438p.158 1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 9 x +11 3 SolutiontoExercise3.440p.158 7 a a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 SolutiontoExercise3.442p.158 5 y 3 y 3 +1 z 4 w 2 y 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 SolutiontoExercise3.444p.158 27 x y 3 SolutiontoExercise3.446p.158 4 a 3 SolutiontoExercise3.448p.158 1 SolutiontoExercise3.450p.158 1 x +5 4 SolutiontoExercise3.452p.159 8 b +2 9 SolutiontoExercise3.454p.159 )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 a 5 b 2 c 2 SolutiontoExercise3.456p.159 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5

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190 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.458p.159 1 SolutiontoExercise3.460p.159 a 4 SolutiontoExercise3.462p.159 4 x 6 SolutiontoExercise3.464p.159 23 y SolutiontoExercise3.466p.159 3 b 3 c 5 a 3 SolutiontoExercise3.468p.159 4 bc 9 y 2 a 2 d 3 z 8 SolutiontoExercise3.470p.159 4 a 3 b 2 c SolutiontoExercise3.472p.159 3 x 3 y 2 z 4 SolutiontoExercise3.474p.160 3 a 7 b 5 SolutiontoExercise3.476p.160 128 a 7 bx SolutiontoExercise3.478p.160 4 x 3 y 7 SolutiontoExercise3.480p.160 23 a 4 b 5 x 6 c 2 y 5 SolutiontoExercise3.482p.160 10 3 x 2 y 7 z 2 SolutiontoExercise3.484p.160 224 b 3 c 12 a 2 +21 4 a +6 3 SolutiontoExercise3.486p.160 1 7 SolutiontoExercise3.488p.160 1 32 SolutiontoExercise3.490p.160 2 9 SolutiontoExercise3.492p.160 2 SolutiontoExercise3.494p.161 1 24 SolutiontoExercise3.496p.161 1 9 SolutiontoExercise3.498p.161 24 SolutiontoExercise3.500p.161 36 SolutiontoExercise3.502p.161 63 SolutiontoExercise3.504p.161 1 a 15 SolutiontoExercise3.506p.161 1 x 32

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191 SolutiontoExercise3.508p.161 b 4 SolutiontoExercise3.510p.161 y 27 SolutiontoExercise3.512p.161 b SolutiontoExercise3.514p.161 1 SolutiontoExercise3.516p.162 x 12 y 12 z 2 SolutiontoExercise3.518p.162 x 20 y 15 SolutiontoExercise3.520p.162 16 a 4 b 12 SolutiontoExercise3.522p.162 a 6 x 4 25 b 12 y 18 SolutiontoExercise3.524p.162 n 28 s 16 m 32 r 20 SolutiontoExercise3.526p.162 64 x 15 y 9 SolutiontoExercise3.528p.162 20 SolutiontoExercise3.530p.162 1 SolutiontoExercise3.531p.164 3 : 46 10 2 SolutiontoExercise3.532p.164 7 : 233 10 SolutiontoExercise3.533p.164 5 : 3877965 10 3 SolutiontoExercise3.534p.164 8 : 7 10 7 SolutiontoExercise3.535p.164 1 : 79 10 20 SolutiontoExercise3.536p.164 1 : 0 10 5 SolutiontoExercise3.537p.164 1 : 0 10 6 SolutiontoExercise3.538p.164 8 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise3.539p.164 9 : 8001 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 SolutiontoExercise3.540p.164 5 : 4 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(17 SolutiontoExercise3.541p.164 1 : 0 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 SolutiontoExercise3.542p.164 1 : 0 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 SolutiontoExercise3.543p.165 925

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192 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.544p.165 401000 SolutiontoExercise3.545p.165 0 : 12 SolutiontoExercise3.546p.166 0 : 0000888 SolutiontoExercise3.547p.166 6 10 17 SolutiontoExercise3.548p.166 6 10 20 SolutiontoExercise3.549p.167 1 : 5 10 25 SolutiontoExercise3.550p.167 6 : 3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(20 SolutiontoExercise3.551p.167 5 : 89 10 3 SolutiontoExercise3.553p.167 2 : 5 10 23 SolutiontoExercise3.555p.167 2 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 SolutiontoExercise3.557p.167 5 : 7 10 4 SolutiontoExercise3.559p.167 2 : 8 10 12 ; 2 : 463 10 25 SolutiontoExercise3.561p.167 3 : 36 10 3 SolutiontoExercise3.563p.167 8 10 6 SolutiontoExercise3.565p.168 1 : 5 10 62 SolutiontoExercise3.567p.168 8 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 SolutiontoExercise3.569p.168 1 : 03 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 SolutiontoExercise3.571p.168 3 : 1 10 )]TJ/F7 6.9738 Tf 6.725 -0.342 Td [(26 SolutiontoExercise3.573p.168 3 : 16 10 2 SolutiontoExercise3.575p.168 7 : 4 10 5 SolutiontoExercise3.577p.168 3 : 16 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 SolutiontoExercise3.579p.168 1 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 SolutiontoExercise3.581p.168 100 ; 000 ; 000 SolutiontoExercise3.583p.169 5 ; 866 ; 000 ; 000 ; 000 SolutiontoExercise3.585p.169 43 ; 000 ; 000 ; 000 ; 000 ; 000 ; 000

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193 SolutiontoExercise3.587p.169 60 ; 000 ; 000 SolutiontoExercise3.589p.169 0 : 00000000000000000047 SolutiontoExercise3.591p.169 0 : 0125 SolutiontoExercise3.593p.169 3 : 2 10 9 SolutiontoExercise3.595p.169 2 : 4 10 3 SolutiontoExercise3.597p.169 9 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 SolutiontoExercise3.599p.169 1 : 512 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(31 SolutiontoExercise3.601p.169 3 : 481 10 29 SolutiontoExercise3.603p.169 7 : 744 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(99 SolutiontoExercise3.605p.170 6 : 687 10 16 SolutiontoExercise3.607p.170 1 : 47 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 SolutiontoExercise3.609p.170 4 : 5 10 37 SolutiontoExercise3.611p.170 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise3.613p.170 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise3.615p.170 1 z +1 2 SolutiontoExercise3.616p.172 )]TJ/F8 9.9626 Tf 7.749 0 Td [(27 SolutiontoExercise3.618p.172 8 9 SolutiontoExercise3.620p.172 )]TJ/F11 9.9626 Tf 7.749 0 Td [(k SolutiontoExercise3.622p.172 3 SolutiontoExercise3.624p.172 12 SolutiontoExercise3.626p.172 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise3.628p.172 17 SolutiontoExercise3.630p.172 18 SolutiontoExercise3.632p.173 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise3.634p.173 10

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194 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.636p.173 )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(18 5 SolutiontoExercise3.638p.173 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise3.640p.173 )]TJ/F8 9.9626 Tf 7.749 0 Td [(48 SolutiontoExercise3.642p.173 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise3.644p.173 1 a SolutiontoExercise3.646p.173 a 3 b 2 c 5 SolutiontoExercise3.648p.173 x 3 y 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 7 SolutiontoExercise3.650p.174 1 2 x SolutiontoExercise3.652p.174 1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise3.654p.174 7 xy 3 z 2 SolutiontoExercise3.656p.174 c 2 y 4 z 9 a 5 b 9 x 2 SolutiontoExercise3.658p.174 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 a 2 b 3 SolutiontoExercise3.660p.174 1 64 SolutiontoExercise3.662p.174 1 32 SolutiontoExercise3.664p.174 1 a 6 b 3 SolutiontoExercise3.666p.174 c 4 SolutiontoExercise3.668p.174 y 24 z 12 x 18 SolutiontoExercise3.670p.174 b 28 c 32 y 12 z 4 16 d 16 x 8 SolutiontoExercise3.672p.175 7 : 3567 10 4 SolutiontoExercise3.674p.175 7 : 46 10 5 SolutiontoExercise3.676p.175 3 : 87 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise3.678p.175 3 : 76 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise3.680p.175 5 : 37 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(13 SolutiontoExercise3.682p.175 387 ; 000 SolutiontoExercise3.684p.175 60 ; 090 ; 000

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195 SolutiontoExercise3.686p.175 0 : 0000388 SolutiontoExercise3.688p.175 0 : 000000000008002 SolutiontoExercise3.690p.175 2 ; 101 ; 000 ; 000 ; 000 ; 000 SolutiontoExercise3.692p.176 1 ; 000 ; 000 SolutiontoExercise3.694p.176 1 ; 000 ; 000 ; 000 SolutiontoExercise3.696p.176 2 : 1 10 14 SolutiontoExercise3.698p.176 2 : 7 10 78 SolutiontoExercise3.700p.176 2 : 4 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 SolutiontoExercise3.702p.176 2 : 1 10 1 SolutiontoExercise3.704p.176 2 : 9839 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(28 SolutiontoExercise3.706p.176 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise3.707p.176 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 SolutiontoExercise3.708p.176 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise3.709p.177 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise3.710p.177 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise3.711p.177 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise3.712p.177 3 SolutiontoExercise3.713p.177 5 SolutiontoExercise3.714p.177 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise3.715p.177 125 SolutiontoExercise3.716p.177 5 x 3 z 4 y 2 SolutiontoExercise3.717p.177 m 6 4 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 SolutiontoExercise3.718p.177 8 a 11 SolutiontoExercise3.719p.177 1 6 SolutiontoExercise3.720p.177 1 k )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 5

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196 CHAPTER3.BASICOPERATIONSWITHREALNUMBERS SolutiontoExercise3.721p.177 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 12 y +1 2 SolutiontoExercise3.722p.177 1 SolutiontoExercise3.723p.177 1 a 12 SolutiontoExercise3.724p.177 n 16 s 8 m 20 r 24 SolutiontoExercise3.725p.177 1 SolutiontoExercise3.726p.177 2 : 71 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 SolutiontoExercise3.727p.177 890 ; 000 SolutiontoExercise3.728p.177 6000 SolutiontoExercise3.729p.178 1 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(31 SolutiontoExercise3.730p.178 apositiveinteger

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Chapter4 AlgebraicExpressionsandEquations 4.1Objectives 1 Aftercompletingthischapter,youshould AlgebraicExpressionsSection4.2 befamiliarwithalgebraicexpressions understandthedierencebetweenatermandafactor befamiliarwiththeconceptofcommonfactors knowthefunctionofacoecient EquationsSection4.3 understandthemeaningofanequation beabletoperformnumericalevaluations ClassicationofExpressionsandEquationsSection4.4 befamiliarwithpolynomials beabletoclassifypolynomialsandpolynomialequations CombiningPolynomialsUsingAdditionandSubtractionSection4.5 understandtheconceptofliketerms beabletocombineliketerms beabletosimplifyexpressionscontainingparentheses CombiningPolynomialsUsingMultiplicationSection4.6 beabletomultiplyapolynomialbyamonomial beabletosimplify + a + b and )]TJ/F8 9.9626 Tf 9.41 0 Td [( a + b beabletomultiplyapolynomialbyapolynomial SpecialBinomialProductsSection4.7 beabletoexpand a + b 2 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 ,and a + b a + b TerminologyAssociatedwithEquationsSection4.8 beabletoidentifytheindependentanddependentvariablesofanequation beabletospecifythedomainofanequation 1 Thiscontentisavailableonlineat. 197

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198 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.2AlgebraicExpressions 2 4.2.1Overview AlgebraicExpressions TermsandFactors CommonFactors Coecients 4.2.2AlgebraicExpressions AlgebraicExpression An algebraicexpression isanumber,aletter,oracollectionofnumbersandlettersalongwithmeaningful signsofoperation. Expressions Algebraicexpressionsareoftenreferredtosimplyas expressions ,asinthefollowingexamples: Example4.1 x +4 isanexpression. Example4.2 7 y isanexpression. Example4.3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x 2 y 7+9 x isanexpression. Example4.4 Thenumber8isanexpression.8canbewrittenwithexplicitsignsofoperationbywritingitas 8+0 or 8 1 3 x 2 +6=4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 is not anexpression,itisan equation .Wewillstudyequationsinthenextsection. 4.2.3TermsandFactors Terms Inanalgebraicexpression,thequantitiesjoinedby "+" signsarecalled terms. Insomeexpressionsitwillappearthattermsarejoinedby )]TJ/F8 9.9626 Tf 10.555 0 Td [(" signs.Wemustkeepinmindthat subtractionisadditionofthenegative,thatis, a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b = a + )]TJ/F11 9.9626 Tf 7.748 0 Td [(b Animportantconceptthatallstudentsofalgebramustbeawareofisthedierencebetween terms and factors Factors Anynumbersorsymbolsthataremultipliedtogetherare factors oftheirproduct. Termsarepartsof sums andarethereforejoinedbyadditionorsubtractionsigns. Factorsarepartsof products andarethereforejoinedbymultiplicationsigns. 4.2.4SampleSetA Identifythetermsinthefollowingexpressions. Example4.5 3 x 4 +6 x 2 +5 x +8 Thisexpressionhasfourterms: 3 x 4 ; 6 x 2 ; 5 x; and8. 2 Thiscontentisavailableonlineat.

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199 Example4.6 15 y 8 Inthisexpressionthereisonlyoneterm.Thetermis 15 y 8 Example4.7 14 x 5 y + a +3 2 Inthisexpressiontherearetwoterms:thetermsare 14 x 5 y and a +3 2 .Noticethattheterm a +3 2 isitselfcomposedoftwolikefactors,eachofwhichiscomposedofthetwoterms, a and3. Example4.8 m 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Usingourdenitionofsubtraction,thisexpressioncanbewrittenintheform m 3 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 .Now wecanseethatthetermsare m 3 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Ratherthanrewritingtheexpressionwhenasubtractionoccurs,wecanidentifytermsmore quicklybyassociatingthe + or )]TJ/F15 9.9626 Tf 11.069 0 Td [(signwiththeindividualquantity. Example4.9 p 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 p 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Associatingthesignwiththeindividualquantitiesweseethatthetermsofthisexpressionare p 4 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 p 3 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p; and )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 4.2.5PracticeSetA Exercise4.1 Solutiononp.259. Let'ssayitagain.Thedierencebetweentermsandfactorsisthattermsarejoinedby signsandfactorsarejoinedby signs. Listthetermsinthefollowingexpressions. Exercise4.2 Solutiononp.259. 4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x +7 Exercise4.3 Solutiononp.259. 2 xy +6 x 2 + x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 4 Exercise4.4 Solutiononp.259. 5 x 2 +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 xy 7 + x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 4.2.6SampleSetB Identifythefactorsineachterm. Example4.10 9 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 containsthreeterms.Someofthefactorsineachtermare rstterm: 9 and a 2 ; or ; 9 and a and a secondterm: )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 and a thirdterm: )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 and 1 ; or ; 12 and )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Example4.11 14 x 5 y + a +3 2 containstwoterms.Someofthefactorsofthesetermsare rstterm: 14 ;x 5 ;y secondterm: a +3 and a +3

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200 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.2.7PracticeSetB Exercise4.5 Solutiononp.259. Intheexpression 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +6 ,listthefactorsofthe rstterm: secondterm: thirdterm: Exercise4.6 Solutiononp.259. Intheexpression 10+2 b +6 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 2 ,listthefactorsofthe rstterm: secondterm: 4.2.8CommonFactors CommonFactors Sometimes,whenweobserveanexpressioncarefully,wewillnoticethatsomeparticularfactorappearsin everyterm.Whenweobservethis,wesayweareobserving commonfactors .Weusethephrase common factors sincetheparticularfactorweobserveiscommontoallthetermsintheexpression.Thefactor appearsineachandeverytermintheexpression. 4.2.9SampleSetC Namethecommonfactorsineachexpression. Example4.12 5 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x 3 +14 x 3 Thefactor x 3 appearsineachandeveryterm.Theexpression x 3 isacommonfactor. Example4.13 4 x 2 +7 x Thefactor x appearsineachterm.Theterm 4 x 2 isactually 4 xx .Thus, x isacommonfactor. Example4.14 12 xy 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 xy +15 Theonlyfactorcommontoallthreetermsisthenumber3.Noticethat 12=3 4 ; 9=3 3 ; 15= 3 5 Example4.15 3 x +5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x +5 Thefactor x +5 appearsineachterm.So, x +5 isacommonfactor. Example4.16 45 x 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 2 +15 x 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 x 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 5 Thenumber5,the x 2 ,andthe x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 appearineachterm.Also, 5 x 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 isafactorsince eachoftheindividualquantitiesisjoinedbyamultiplicationsign.Thus, 5 x 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 isacommon factor. Example4.17 10 x 2 +9 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Thereisnofactorthatappearsineachandeveryterm.Hence,therearenocommonfactorsin thisexpression.

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201 4.2.10PracticeSetC List,ifanyappear,thecommonfactorsinthefollowingexpressions. Exercise4.7 Solutiononp.259. x 2 +5 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 x 2 Exercise4.8 Solutiononp.259. 4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x 3 +16 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 x 5 Exercise4.9 Solutiononp.259. 4 a +1 3 +10 a +1 Exercise4.10 Solutiononp.259. 9 ab a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 2 Exercise4.11 Solutiononp.259. 14 a 2 b 2 c c )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 c +5+28 c c +5 Exercise4.12 Solutiononp.259. 6 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 +19 x )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 + y 2 4.2.11Coecients Coecient Inalgebra,aswenowknow,aletterisoftenusedtorepresentsomequantity.Supposewerepresentsome quantitybytheletter x .Thenotation 5 x means x + x + x + x + x .Wecannowseethatwehaveveof thesequantities.Intheexpression 5 x ,thenumber5iscalledthe numericalcoecient ofthequantity x .Often,thenumericalcoecientisjustcalledthecoecient.The coecient ofaquantityrecordshow manyofthatquantitythereare. 4.2.12SampleSetD Example4.18 12 x meansthereare 12 x 's. Example4.19 4 ab meanstherearefour ab 's. Example4.20 10 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 meansthereareten x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 's. Example4.21 1 y meansthereisone y .Weusuallywritejust y ratherthan 1 y sinceitisclearjustbylooking thatthereisonlyone y Example4.22 7 a 3 meansthereareseven a 3 s. Example4.23 5 ax meansthereareve ax 's.Itcouldalsomeanthereare 5 ax 's.Thisexampleshowsusthatit isimportantforustobeveryclearastowhichquantityweareworkingwith.Whenweseethe expression 5 ax wemustaskourselves"Areweworkingwiththequantity ax orthequantity x ?". Example4.24 6 x 2 y 9 meanstherearesix x 2 y 9 s.Itcouldalsomeanthereare 6 x 2 y 9 s.Itcouldevenmeanthere are 6 y 9 x 2 s.

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202 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Example4.25 5 x 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 meansthereareve x 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 's.Itcouldalsomeanthereare 5 x 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 's.Itcould alsomeanthereare 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x 3 's. 4.2.13PracticeSetD Exercise4.13 Solutiononp.259. Whatdoesthe coecient ofaquantitytellus? TheDierenceBetweenCoecientsandExponents Itisimportanttokeepinmindthedierencebetween coecients and exponents Coecients recordthenumberoflike terms inanalgebraicexpression. x + x + x + x | {z } 4 terms =4 x coecientis 4 Exponents recordthenumberoflike factors inaterm. x x x x | {z } 4 factors = x 4 exponentis 4 Inaterm,the coecient ofaparticulargroupoffactorsistheremaininggroupoffactors. 4.2.14SampleSetE Example4.26 3 x Thecoecientof x is3. Example4.27 6 a 3 Thecoecientof a 3 is6. Example4.28 9 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a Thecoecientof )]TJ/F11 9.9626 Tf 9.963 0 Td [(a is9. Example4.29 3 8 xy 4 Thecoecientof xy 4 is 3 8 Example4.30 3 x 2 y Thecoecientof x 2 y is3;thecoecientof y is 3 x 2 ;andthecoecientof3is x 2 y Example4.31 4 x + y 2 Thecoecientof x + y 2 is4;thecoecientof4is x + y 2 ;andthecoecientof x + y is 4 x + y since 4 x + y 2 canbewrittenas 4 x + y x + y .

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203 4.2.15PracticeSetE Exercise4.14 Solutiononp.259. Determinethecoecients. Intheterm 6 x 3 ,thecoecientof a x 3 is b6is Exercise4.15 Solutiononp.259. Intheterm 3 x y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ,thecoecientof a x y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 is b y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 is c 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 is d x is e3is fThenumericalcoecientis Exercise4.16 Solutiononp.259. Intheterm 10 ab 4 ,thecoecientof a ab 4 is b b 4 is c a is d10is e 10 ab 3 is 4.2.16Exercises Exercise4.17 Solutiononp.259. Whatisanalgebraicexpression? Exercise4.18 Whyisthenumber14consideredtobeanexpression? Exercise4.19 Solutiononp.259. Whyisthenumber x consideredtobeanexpression? Fortheexpressionsinthefollowingproblems,writethenumberoftermsthatappearandthenlisttheterms. Exercise4.20 2 x +1 Exercise4.21 Solutiononp.259. 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 Exercise4.22 2 x 3 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 Exercise4.23 Solutiononp.259. 5 x 2 +6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise4.24 3 x Exercise4.25 Solutiononp.259. 5 cz Exercise4.26 2

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204 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.27 Solutiononp.259. 61 Exercise4.28 x Exercise4.29 Solutiononp.259. 4 y 3 Exercise4.30 17 ab 2 Exercise4.31 Solutiononp.259. a +1 Exercise4.32 a +1 Exercise4.33 Solutiononp.259. 2 x + x +7 Exercise4.34 2 x + x +7 Exercise4.35 Solutiononp.260. a +1+ a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise4.36 a +1+ a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Forthefollowingproblems,list,ifanyshouldappear,thecommonfactorsintheexpressions. Exercise4.37 Solutiononp.260. x 2 +5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 Exercise4.38 11 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(33 y 3 Exercise4.39 Solutiononp.260. 45 ab 2 +9 b 2 Exercise4.40 6 x 2 y 3 +18 x 2 Exercise4.41 Solutiononp.260. 2 a + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 a + b Exercise4.42 8 a 2 b +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a 2 b +1 Exercise4.43 Solutiononp.260. 14 ab 2 c 2 c +8+12 ab 2 c 2 Exercise4.44 4 x 2 y +5 a 2 b Exercise4.45 Solutiononp.260. 9 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 +10 b a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.46 15 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 xy 2 Exercise4.47 Solutiononp.260. 12 a 3 b 2 c )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 b +1 c )]TJ/F11 9.9626 Tf 9.963 0 Td [(a Exercise4.48 0 : 06 ab 2 +0 : 03 a

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205 Exercise4.49 Solutiononp.260. 5 : 2 a +7 2 +17 : 1 a +7 Exercise4.50 3 4 x 2 y 2 z 2 + 3 8 x 2 z 2 Exercise4.51 Solutiononp.260. 9 16 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 + 9 32 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 2 Forthefollowingproblems,notehowmany: Exercise4.52 a 'sin 4 a ? Exercise4.53 Solutiononp.260. z 'sin 12 z ? Exercise4.54 x 2 'sin 5 x 2 ? Exercise4.55 Solutiononp.260. y 3 'sin 6 y 3 ? Exercise4.56 xy 'sin 9 xy ? Exercise4.57 Solutiononp.260. a 2 b 'sin 10 a 2 b ? Exercise4.58 a +1 'sin 4 a +1? Exercise4.59 Solutiononp.260. + y 'sin 8+ y ? Exercise4.60 y 2 'sin 3 x 3 y 2 ? Exercise4.61 Solutiononp.260. 12 x 'sin 12 x 2 y 5 ? Exercise4.62 a +5 'sin 2 a +5? Exercise4.63 Solutiononp.260. x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 'sin 5 x x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y ? Exercise4.64 x +1 'sin 8 x +1? Exercise4.65 Solutiononp.260. 2 'sin 2 x 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7? Exercise4.66 3 a +8 'sin 6 x 2 a +8 3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8? Forthefollowingproblems,atermwillbegivenfollowedbyagroupofitsfactors.Listthecoecientofthe givengroupoffactors. Exercise4.67 Solutiononp.260. 7 y ; y Exercise4.68 10 x ; x Exercise4.69 Solutiononp.260. 5 a ;5

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206 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.70 12 a 2 b 3 c 2 r 7 ; a 2 c 2 r 7 Exercise4.71 Solutiononp.260. 6 x 2 b 2 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(1; c )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.72 10 x x +7 2 ;10 x +7 Exercise4.73 Solutiononp.260. 9 a 2 b 5 ;3 ab 3 Exercise4.74 15 x 4 y 4 z +9 a 3 ; z +9 a Exercise4.75 Solutiononp.260. )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 a 5 b 2 ; ab Exercise4.76 )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 a a +8 3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1; a +8 2 4.2.17ExercisesforReview Exercise4.77 Solutiononp.260. Section2.7 Simplify h 2 x 8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 5 x 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 i 4 Exercise4.78 Section3.3 Supplythemissingphrase.Absolutevaluespeakstothequestionof and not"whichway." Exercise4.79 Solutiononp.260. Section3.6 Findthevalueof )]TJ/F8 9.9626 Tf 9.41 0 Td [([ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+7 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3+5] Exercise4.80 Section3.7 Findthevalueof 2 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 Exercise4.81 Solutiononp.260. Section3.8 Express 0 : 0000152 usingscienticnotation. 4.3Equations 3 4.3.1Overview Equations NumericalEvaluation 3 Thiscontentisavailableonlineat.

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207 4.3.2Equations Equation An equation isastatementthattwoalgebraicexpressionsareequal. Anequationiscomposedofthreeparts. = Eachoftheboxesrepresentsanalgebraicexpression.Anequationconsistsoftwoexpressionsseparated byanequalsign.Theequalsignmakesthestatementthatthetwoexpressionsareequivalent,thatis,they representthesamevalue.Forexample: Example4.32 f =32 a Theequationexpressestherelationshipbetweenthevariables f and a .Itstatesthatthevalueof f isalways32timesthatof a Example4.33 y =6 x +8 Theequationexpressestherelationshipbetweenthevariables x and y .Itstatesthatthevalueof y isalways8morethan6timesthevalueof x 4.3.3NumericalEvaluation Numericalevaluation Numericalevaluation istheprocessofdeterminingavaluebysubstitutingnumbersforletters. Formulas Invariousareasbusiness,statistics,physics,chemistry,astronomy,sociology,psychology,etc.,particularequationsoccurquitefrequently.Suchequationsarecalled formulas .Numericalevaluationisused frequentlywithformulas. 4.3.4SampleSetA Example4.34 f =32 a: Determinethevalueof f if a =2 : f =32 Replace a by 2 : =64 Example4.35 p = 10 ; 000 v Thischemistryequationexpressestherelationshipbetweenthepressure p ofagasandthe volume v ofthegas.Determinethevalueof p if v =500 p = 10 ; 000 500 Replace v by 500 : =20 OntheCalculator

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208 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Type 10000 Press Type 500 Press = Displayreads: 20 Example4.36 z = x )]TJ/F10 6.9738 Tf 6.227 0 Td [(u s Thisstatisticsequationexpressestherelationshipbetweenthevariables z;x;u and s .Determine thevalueof z if x =41 ;u =45 ; and s =1 : 3 .Roundtotwodecimalplaces. z = 41 )]TJ/F7 6.9738 Tf 6.227 0 Td [(45 1 : 3 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 1 : 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 : 08 OntheCalculator Type 41 Press )]TJETq1 0 0 1 216.615 425.312 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQq1 0 0 1 198.306 425.113 cm[]0 d 0 J 0.398 w 0 0 m 18.508 0 l SQBT/F15 9.9626 Tf 96.907 413.198 Td [(Type 45 Press = Press Type 1 : 3 Press = Displayreads: )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 : 076923 We'llroundto )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 : 08 Example4.37 p =5 w 3 + w 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(w )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Thisequationexpressestherelationshipbetween p and w .Determinethevalueof p if w =5 p =5 3 + 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =5+25 )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =625+25 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 =644 OntheCalculator

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209 Type 5 Press y x Type 3 Press = Press Type 5 Press = Press + Type 5 Press x 2 Press )]TJETq1 0 0 1 184.973 480.811 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQq1 0 0 1 166.664 480.612 cm[]0 d 0 J 0.398 w 0 0 m 18.508 0 l SQBT/F15 9.9626 Tf 96.907 468.697 Td [(Type 5 Press )]TJETq1 0 0 1 184.973 446.082 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQq1 0 0 1 166.664 445.882 cm[]0 d 0 J 0.398 w 0 0 m 18.508 0 l SQBT/F15 9.9626 Tf 96.907 433.967 Td [(Type 1 Press = Displayreads: 644 4.3.5PracticeSetA Exercise4.82 Solutiononp.260. f =32 a: Determinethevalueof f if a =6 : Exercise4.83 Solutiononp.261. p = 10 ; 000 v : Determinethevalueof p if v =250 : Exercise4.84 Solutiononp.261. F = 9 5 C +32 : Determinethevalueof F if C =10 : Exercise4.85 Solutiononp.261. y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 : Determinethevalueof y if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 : Exercise4.86 Solutiononp.261. m =5 p 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p +7 : Determinethevalueof m if p = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 : 4.3.6Exercises Forthefollowingproblems,observetheequationsandstatetherelationshipbeingexpressed. Exercise4.87 Solutiononp.261. x =6 y Exercise4.88 y = x +4

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210 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.89 Solutiononp.261. e = g )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 Exercise4.90 y = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise4.91 Solutiononp.261. 3 t =6 s Exercise4.92 u = v 5 Exercise4.93 Solutiononp.261. r = 2 9 s Exercise4.94 b = 3 4 a Exercise4.95 Solutiononp.261. f =0 : 97 k +55 Exercise4.96 w =4 z 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Exercise4.97 Solutiononp.261. q 2 =9 x 8 +2 y Exercise4.98 I = m 2 qb 5 +3 : 115 p Usenumericalevaluationontheequationsforthefollowingproblems. Exercise4.99 Solutiononp.261. Geometrycircumferenceofacircle C =2 r: Find C if isapproximatedby 3 : 14 and r =5 : Exercise4.100 Geometryareaofarectangle A = lw: Find A if l =15 and w =9 : Exercise4.101 Solutiononp.261. Electricitycurrentinacircuit I = E R : Find I if E =21 and R =7 : Exercise4.102 Electricitycurrentinacircuit I = E R : Find I if E =106 and R =8 : Exercise4.103 Solutiononp.261. Businesssimpleinterest I = prt: Find I if p =3000 ;r = : 12 and t =1 : Exercise4.104 Businesssimpleinterest I = prt: Find I if p =250 ;r =0 : 07 and t =6 : Exercise4.105 Solutiononp.261. Geometryareaofaparallelogram A = 1 2 bh: Find A if b =16 and h =6 :

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211 Exercise4.106 Geometryareaofatriangle A = 1 2 bh: Find A if b =25 and h =10 : Exercise4.107 Solutiononp.261. Geometryperimeterofarectangle P =2 l +2 w: Find P if l =3 and w =1 : Exercise4.108 Geometryperimeterofarectangle P =2 l +2 w: Find P if l =74 and w =16 : Exercise4.109 Solutiononp.261. Geometryperimeterofarectangle P =2 l +2 w: Find P if l =8 1 4 and w =12 8 9 : Exercise4.110 Physicsforce F =32 m: Find F if m =6 : Exercise4.111 Solutiononp.261. Physicsforce F =32 m: Find F if m =14 : Exercise4.112 Physicsforce F =32 m: Find F if m = 1 16 : Exercise4.113 Solutiononp.261. Physicsforce F =32 m: Find F if m =6 : 42 : Exercise4.114 Physicsmomentum p = mv: Find p if m =18 and v =5 : Exercise4.115 Solutiononp.261. Physicsmomentum p = mv: Find p if m =44 and v =9 : Exercise4.116 Physicsmomentum p = mv: Find p if m =9 : 18 and v =16 : 5 : Exercise4.117 Solutiononp.261. Physicsenergy E = 1 2 mv 2 : Find E if m =12 and v =5 : Exercise4.118 Physicsenergy E = 1 2 mv 2 : Find E if m =8 and v =15 : Exercise4.119 Solutiononp.261. Physicsenergy E = 1 2 mv 2 : Find E if m =24 : 02 and v =7 :

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212 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.120 AstronomyKepler'slawofplanetarymotion P 2 = ka 3 : Find P 2 if k =1 and a =4 : Exercise4.121 Solutiononp.261. AstronomyKepler'slawofplanetarymotion P 2 = ka 3 : Find P 2 if k =8 and a =31 : Exercise4.122 AstronomyKepler'slawofplanetarymotion P 2 = ka 3 : Find P 2 if k =4 and a =5 : 1 : Hint: Onthecalculator,Type5.1,Press y x ,Type3,Press = ,Press ,Type4,Press = Exercise4.123 Solutiononp.261. AstronomyKepler'slawofplanetarymotion P 2 = ka 3 : Find P 2 if k =53 : 7 and a =0 : 7 : Exercise4.124 Businessprot,revenue,andcost P = R )]TJ/F11 9.9626 Tf 9.963 0 Td [(C: Find P if R =3100 and C =2500 : Exercise4.125 Solutiononp.261. Businessprot,revenue,andcost P = R )]TJ/F11 9.9626 Tf 9.963 0 Td [(C: Find P if R =4240 and C =3590 : Exercise4.126 Geometryareaofacircle A = r 2 : Find A if isapproximately 3 : 14 and r =3 : Exercise4.127 Solutiononp.261. Geometryareaofacircle A = r 2 : Find A if isapproximately 3 : 14 and r =11 : Exercise4.128 t =21 x +6 : Find t if x =3 : Exercise4.129 Solutiononp.262. t =21 x +6 : Find t if x =97 : Exercise4.130 E = mc 2 : Find E if m =2 and c =186 ; 000 : Hint: Thenumber10thatoccursonthedisplayafewspacesawayfromtheothernumberonthe displayistheexponentof10inthescienticnotationformofthenumber. Exercise4.131 Solutiononp.262. E = mc 2 : Find E if m =5 and c =186 ; 000 : Exercise4.132 Anobjecttravelsonahorizontalline.Thedistanceittravelsisrepresentedby d andismeasured inmeters.Theequationrelatingtimeoftravel, t ,anddistanceoftravel, d ,is

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213 d = t 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 t +20 Determinethedistancetraveledbytheobjectifithasbeeninmotionfor6seconds. Exercise4.133 Solutiononp.262. Inmedicine,thereareseveralrulesofthumbusedbyphysicianstodetermineachild'sdose, D c ,of aparticulardrug.Onesuchrule,Young'sRule,relatesachild'sdoseofadrugtoanadult'sdose ofthatdrug, D a .Young'sRuleis D c = t t +12 D a where t isthechild'sageinyears.Whatdoseshouldbegiventoachild8yearsoldifthe correspondingadultdosageis15units? Exercise4.134 Ahemisphericalwatertankofradius6feethaswaterdrippingintoit.Theequationrelatingthe volume, V ,ofwaterinthetankatanytimeis V =6 h 2 )]TJ/F10 6.9738 Tf 11.461 3.922 Td [( 3 h 3 ,where h representsthedepthof thewater.Using 3 : 14 toapproximatetheirrationalnumber ,determinethevolumeofwaterin thetankwhenthedepthofthewateris3feet. Exercise4.135 Solutiononp.262. Theequation W =3 : 51 L )]TJ/F8 9.9626 Tf 9.542 0 Td [(192 hasbeenestablishedbytheInternationalWhalingCommissionto relatetheweight, W inlongtons,ofamaturebluewhaletoitslength, L infeet.Theequation isonlyusedwhen L 70 .When 0
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214 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.137 Solutiononp.262. Thereisarelationshipbetweenthelengthofasuspensionbridgecablethatissecuredbetween twoverticalsupportsandtheamountofsagofthecable.Ifwerepresentthelengthofthecable by c ,thehorizontaldistancebetweentheverticalsupportsby d ,andtheamountofsagby s ,the equationis c = d + 8 s 2 3 d )]TJ/F7 6.9738 Tf 11.248 3.923 Td [(32 s 4 5 d 3 .Ifthehorizontaldistancebetweenthetwoverticalsupportsis190 feetandtheamountofsaginacablethatissuspendedbetweenthetwosupportsis20feet,what isthelengthofthecable? 4.3.7ExercisesforReview Exercise4.138 Section2.6 simplify )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 x 3 y 8 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 x 2 y Exercise4.139 Solutiononp.262. Section3.3 simplify j)]TJ/F8 9.9626 Tf 22.693 0 Td [(8 j Exercise4.140 Section3.7 Findthevalueof 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 8 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 Exercise4.141 Solutiononp.262. Section4.2 Fortheexpression 5 a + b +2 x 2 ,writethenumberoftermsthatappearandthen writethetermsthemselves. Exercise4.142 Section4.2 Howmany xy 3 s aretherein 5 x 2 y 5 ? 4.4ClassicationofExpressionsandEquations 4 4.4.1Overview Polynomials ClassicationofPolynomials ClassicationofPolynomialEquations 4.4.2Polynomials Polynomials Letusconsiderthecollectionofallalgebraicexpressionsthatdonotcontainvariablesinthedenominatorsof fractionsandwhereallexponentsonthevariablequantitiesarewholenumbers.Expressionsinthiscollection arecalled polynomials. Someexpressionsthat are polynomialsare 4 Thiscontentisavailableonlineat.

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215 Example4.38 3 x 4 Example4.39 2 5 x 2 y 6 Afractionoccurs,butnovariableappearsinthedenominator. Example4.40 5 x 3 +3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +1 Someexpressionsthat arenot polynomialsare Example4.41 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 Avariableappearsinthedenominator. Example4.42 4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Anegativeexponentappearsonavariable. 4.4.3ClassicationofPolynomials Polynomialscanbeclassiedusingtwocriteria:thenumberoftermsanddegreeofthepolynomial. NumberofTerms Name Example Comment One Monomial 4 x 2 mono meansoneinGreek. Two Binomial 4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x bi meanstwoinLatin. Three Trinomial 4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x +3 tri meansthreeinGreek. Fourormore Polynomial 4 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x 2 +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 poly meansmanyinGreek. Table4.1 DegreeofaTermContainingOneVariable The degreeofaterm containingonly one variableisthevalueoftheexponentofthevariable.Exponents appearingonnumbersdonotaectthedegreeoftheterm.Weconsideronlytheexponentofthevariable. Forexample: Example4.43 5 x 3 isamonomialofdegree3. Example4.44 60 a 5 isamonomialofdegree5. Example4.45 21 b 2 isamonomialofdegree2. Example4.46 8isamonomialofdegree0.Wesaythatanonzeronumberisatermof0degreesinceitcouldbe writtenas 8 x 0 .Since x 0 =1 x 6 =0 8 x 0 =8 .Theexponentonthevariableis0soitmustbeof degree0.Byconvention,thenumber0hasnodegree.

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216 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Example4.47 4 x isamonomialoftherstdegree. 4 x couldbewrittenas 4 x 1 .Theexponentonthevariableis1 soitmustbeoftherstdegree. DegreeofaTermContainingSeveralVariables Thedegreeofatermcontaining more thanonevariableisthe sum oftheexponentsofthevariables,as shownbelow. Example4.48 4 x 2 y 5 isamonomialofdegree 2+5=7 .Thisisa7thdegreemonomial. Example4.49 37 ab 2 c 6 d 3 isamonomialofdegree 1+2+6+3=12 .Thisisa12thdegreemonomial. Example4.50 5 xy isamonomialofdegree 1+1=2 .Thisisa2nddegreemonomial. DegreeofaPolynomial The degreeofapolynomial isthedegreeofthe term ofhighestdegree;forexample: Example4.51 2 x 3 +6 x )]TJ/F8 9.9626 Tf 10.579 0 Td [(1 isatrinomialofdegree3.Therstterm, 2 x 3 ,isthetermofthehighestdegree. Therefore,itsdegreeisthedegreeofthepolynomial. Example4.52 7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 y 4 isabinomialofdegree4. Example4.53 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4+5 a 2 isatrinomialofdegree2. Example4.54 2 x 6 +9 x 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x 3 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 isapolynomialofdegree7. Example4.55 4 x 3 y 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 xy 3 isabinomialofdegree8.Thedegreeofthersttermis8. Example4.56 3 x +10 isabinomialofdegree1. LinearQuadraticCubic Polynomialsoftherstdegreearecalled linear polynomials. Polynomialsoftheseconddegreearecalled quadratic polynomials. Polynomialsofthethirddegreearecalled cubic polynomials. Polynomialsofthefourthdegreearecalled fourthdegree polynomials. Polynomialsofthe n thdegreearecalled n thdegree polynomials. Nonzeroconstants arepolynomialsofthe 0th degree. Someexamplesofthesepolynomialsfollow: Example4.57 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 isalinearpolynomial. Example4.58 3 x 2 +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 isaquadraticpolynomial. Example4.59 8 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 3 isacubicpolynomial. Example4.60 16 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(32 a 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(64 isa5thdegreepolynomial. Example4.61 x 12 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 12 isa12thdegreepolynomial.

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217 Example4.62 7 x 5 y 7 z 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 4 y 7 z + x 3 y 7 isa15thdegreepolynomial.Thersttermisofdegree 5+7+3=15 Example4.63 43isa0thdegreepolynomial. 4.4.4ClassicationofPolynomialEquations Asweknow,anequationiscomposedoftwoalgebraicexpressionsseparatedbyanequalsign.Ifthetwo expressionshappentobepolynomialexpressions,thenwecanclassifytheequationaccordingtoitsdegree. Classicationofequationsbydegreeisusefulsinceequationsofthesamedegreehavethesametypeofgraph. WewillstudygraphsofequationsinChapter6. Thedegreeofanequationisthedegreeofthehighestdegreeexpression. 4.4.5SampleSetA Example4.64 x +7=15 Thisisalinearequationsinceitisofdegree1,thedegreeoftheexpressionontheleftof the "=" sign. Example4.65 5 x 2 +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=4 isaquadraticequationsinceitisofdegree2. Example4.66 9 x 3 )]TJ/F8 9.9626 Tf 10.522 0 Td [(8=5 x 2 +1 isacubicequationsinceitisofdegree3.Theexpressionontheleftofthe "=" signisofdegree3. Example4.67 y 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 4 =0 isa4thdegreeequation. Example4.68 a 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 a 4 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(a 3 +6 a 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 isa5thdegreeequation. Example4.69 y = 2 3 x +3 isalinearequation. Example4.70 y =3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 isaquadraticequation. Example4.71 x 2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4=0 isa4thdegreeequation.Thedegreeof x 2 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 is 2+2=4 4.4.6PracticeSetA Classifythefollowingequationsintermsoftheirdegree. Exercise4.143 Solutiononp.262. 3 x +6=0 Exercise4.144 Solutiononp.262. 9 x 2 +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6=3 Exercise4.145 Solutiononp.262. 25 y 3 + y =9 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(17 y +4

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218 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.146 Solutiononp.262. x =9 Exercise4.147 Solutiononp.262. y =2 x +1 Exercise4.148 Solutiononp.262. 3 y =9 x 2 Exercise4.149 Solutiononp.262. x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9=0 Exercise4.150 Solutiononp.262. y = x Exercise4.151 Solutiononp.262. 5 x 7 =3 x 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 8 +11 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 4.4.7Exercises Forthefollowingproblems,classifyeachpolynomialasamonomial,binomial,ortrinomial.Statethedegree ofeachpolynomialandwritethenumericalcoecientofeachterm. Exercise4.152 Solutiononp.262. 5 x +7 Exercise4.153 16 x +21 Exercise4.154 Solutiononp.262. 4 x 2 +9 Exercise4.155 7 y 3 +8 Exercise4.156 Solutiononp.262. a 4 +1 Exercise4.157 2 b 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise4.158 Solutiononp.262. 5 x Exercise4.159 7 a Exercise4.160 Solutiononp.262. 5 x 3 +2 x +3 Exercise4.161 17 y 4 + y 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Exercise4.162 Solutiononp.262. 41 a 3 +22 a 2 + a Exercise4.163 6 y 2 +9 Exercise4.164 Solutiononp.262. 2 c 6 +0 Exercise4.165 8 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(0

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219 Exercise4.166 Solutiononp.262. 9 g Exercise4.167 5 xy +3 x Exercise4.168 Solutiononp.263. 3 yz )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 y +11 Exercise4.169 7 ab 2 c 2 +2 a 2 b 3 c 5 + a 14 Exercise4.170 Solutiononp.263. x 4 y 3 z 2 +9 z Exercise4.171 5 a 3 b Exercise4.172 Solutiononp.263. 6+3 x 2 y 5 b Exercise4.173 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9+3 x 2 +2 xy 6 z 2 Exercise4.174 Solutiononp.263. 5 Exercise4.175 3 x 2 y 0 z 4 +12 z 3 ;y 6 =0 Exercise4.176 Solutiononp.263. 4 xy 3 z 5 w 0 ;w 6 =0 Classifyeachoftheequationsforthefollowingproblemsbydegree.Ifthetermlinear,quadratic,orcubic applies,stateit. Exercise4.177 4 x +7=0 Exercise4.178 Solutiononp.263. 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(15=9 Exercise4.179 y =5 s +6 Exercise4.180 Solutiononp.263. y = x 2 +2 Exercise4.181 4 y =8 x +24 Exercise4.182 Solutiononp.263. 9 z =12 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 Exercise4.183 y 2 +3=2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise4.184 Solutiononp.263. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5+ y 3 =3 y 2 +2 Exercise4.185 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4=7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x +9 Exercise4.186 Solutiononp.263. 2 y +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+4 xy =5 xy +2 y Exercise4.187 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 y =9

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220 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.188 Solutiononp.263. 8 a +2 b =4 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise4.189 2 x 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x 2 +9 x +4=12 x 4 +3 x 3 +4 x 2 +1 Exercise4.190 Solutiononp.263. x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y =0 Exercise4.191 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(25=0 Exercise4.192 Solutiononp.263. x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(64=0 Exercise4.193 x 12 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 12 =0 Exercise4.194 Solutiononp.263. x +3 x 5 = x +2 x 5 Exercise4.195 3 x 2 y 4 +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 y =14 Exercise4.196 Solutiononp.263. 10 a 2 b 3 c 6 d 0 e 4 +27 a 3 b 2 b 4 b 3 b 2 c 5 =1 ;d 6 =0 Exercise4.197 Theexpression 4 x 3 9 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 isnotapolynomialbecause Exercise4.198 Solutiononp.263. Theexpression a 4 7 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a isnotapolynomialbecause Exercise4.199 Iseveryalgebraicexpressionapolynomialexpression?Ifnot,giveanexampleofanalgebraic expressionthatisnotapolynomialexpression. Exercise4.200 Solutiononp.263. Iseverypolynomialexpressionanalgebraicexpression?Ifnot,giveanexampleofapolynomial expressionthatisnotanalgebraicexpression. Exercise4.201 Howdowendthedegreeofatermthatcontainsmorethanonevariable? 4.4.8ExercisesforReview Exercise4.202 Solutiononp.263. Section2.2 Usealgebraicnotationtowriteelevenminusthreetimesanumberisve. Exercise4.203 Section2.7 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 4 y 2 z 3 5 Exercise4.204 Solutiononp.263. Section3.6 Findthevalueof z if z = x )]TJ/F10 6.9738 Tf 6.226 0 Td [(u s and x =55 ;u =49 ; and s =3 Exercise4.205 Section4.2 List,ifanyshouldappear,thecommonfactorsintheexpression 3 x 4 +6 x 3 )]TJ/F8 9.9626 Tf 9.578 0 Td [(18 x 2 Exercise4.206 Solutiononp.263. Section4.3 Statebywritingittherelationshipbeingexpressedbytheequation y =3 x +5 .

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221 4.5CombiningPolynomialsUsingAdditionandSubtraction 5 4.5.1Overview LikeTerms CombiningLikeTerms SimplifyingExpressionsContainingParentheses 4.5.2LikeTerms LikeTerms Termswhosevariableparts,includingtheexponents,areidenticalarecalled liketerms .Liketermsis anappropriatenamesincetermswithidenticalvariablepartsanddierentnumericalcoecientsrepresent dierentamountsofthesamequantity.Aslongaswearedealingwithquantitiesofthesametypewecan combinethemusingadditionandsubtraction. SimplifyinganAlgebraicExpression Analgebraicexpressioncanbe simplied bycombiningliketerms. 4.5.3SampleSetA Combinetheliketerms. Example4.72 6 houses +4 houses =10 houses.6and4ofthesametypegive10ofthattype. Example4.73 6 houses +4 houses +2 motels =10 houses +2 motels.6and4ofthesametypegive10ofthat type.Thus,wehave10ofonetypeand2ofanothertype. Example4.74 Supposewelettheletter x represent"house."Then, 6 x +4 x =10 x .6and4ofthesametype give10ofthattype. Example4.75 Supposewelet x represent"house"and y represent"motel." 6 x +4 x +2 y =10 x +2 y .1 4.5.4PracticeSetA Liketermswiththesamenumericalcoecientrepresentequalamountsofthesamequantity. Exercise4.207 Solutiononp.263. Liketermswithdierentnumericalcoecientsrepresent 4.5.5CombiningLikeTerms Sinceliketermsrepresentamountsofthesamequantity,theymaybecombined,thatis,liketermsmaybe addedtogether. 5 Thiscontentisavailableonlineat.

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222 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.5.6SampleSetB Simplifyeachofthefollowingpolynomialsbycombiningliketerms. Example4.76 2 x +5 x +3 x Thereare 2 x 's,then5more,then3more.Thismakesatotalof 10 x 's. 2 x +5 x +3 x =10 x Example4.77 7 x +8 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x From 7 x 's,welose 3 x 's.Thismakes 4 x 's.The 8 y 'srepresentaquantitydierentfromthe x 'sand thereforewillnotcombinewiththem. 7 x +8 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x =4 x +8 y Example4.78 4 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a 2 +8 a 3 + a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a 3 4 a 3 ; 8 a 3 ; and )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 a 3 representquantitiesofthesametype. 4 a 3 +8 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a 3 =10 a 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a 2 and a 2 representquantitiesofthesametype. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a 2 + a 2 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(a 2 Thus, 4 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a 2 +8 a 3 + a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a 3 =10 a 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 2 4.5.7PracticeSetB Simplifyeachofthefollowingexpressions. Exercise4.208 Solutiononp.263. 4 y +7 y Exercise4.209 Solutiononp.263. 3 x +6 x +11 x Exercise4.210 Solutiononp.263. 5 a +2 b +4 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 b Exercise4.211 Solutiononp.264. 10 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 3 +3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x 3 +5 x 2 +2 x + x 3 +8 x Exercise4.212 Solutiononp.264. 2 a 5 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 5 +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ab )]TJ/F8 9.9626 Tf 9.962 0 Td [(9+9 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 5 4.5.8SimplifyingExpressionsContainingParentheses SimplifyingExpressionsContainingParentheses Whenparenthesesoccurinexpressions,theymustberemovedbeforetheexpressioncanbesimplied. Parenthesescanberemovedusingthedistributiveproperty. DistributiveProperty

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223 4.5.9SampleSetC Simplifyeachofthefollowingexpressionsbyusingthedistributivepropertyandcombiningliketerms. Example4.79 Example4.80 4 x +9 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 +5 Removeparentheses. 4 x +9 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(54 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(18+5 Combineliketerms. )]TJ/F8 9.9626 Tf 7.748 0 Td [(50 x +9 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(13 Byconvention,thetermsinanexpressionareplacedindescendingorderwiththehighest degreetermappearingrst.Numericaltermsareplacedattherightendoftheexpression.The commutativepropertyofadditionallowsustochangetheorderoftheterms. 9 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(50 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(13 Example4.81 2+2[5+4+ a ] Eliminatetheinnermostsetofparenthesesrst. 2+2[5+4+4 a ] Bytheorderofoperations,simplifyinsidetheparenthesesbeforemultiplyingbythe2. 2+2[9+4 a ] Removethissetofparentheses. 2+18+8 a Combineliketerms. 20+8 a Writeindescendingorder. 8 a +20 Example4.82 x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3+6 x x +3 Usetheruleformultiplyingpowerswiththesamebase. x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +12 x 2 +18 x Combineliketerms. 13 x 2 +15 x 4.5.10PracticeSetC Simplifyeachofthefollowingexpressionsbyusingthedistributivepropertyandcombiningliketerms. Exercise4.213 Solutiononp.264. 4 x +6+3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2+ x +3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 Exercise4.214 Solutiononp.264. 7 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x + x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x +1+4 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 3 +7 Exercise4.215 Solutiononp.264. 5 a +2+6 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7++4 a +3 a +2 Exercise4.216 Solutiononp.264. x x +3+4 x 2 +2 x Exercise4.217 Solutiononp.264. a 3 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 + a +5 + a )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 4 +3 a 2 +4 +1 Exercise4.218 Solutiononp.264. 2[8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3]

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224 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.219 Solutiononp.264. x 2 +3 x +7 x +4 x 2 +3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x + x 2 4.5.11Exercises Forthefollowingproblems,simplifyeachofthealgebraicexpressions. Exercise4.220 Solutiononp.264. x +3 x Exercise4.221 4 x +7 x Exercise4.222 Solutiononp.264. 9 a +12 a Exercise4.223 5 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 m Exercise4.224 Solutiononp.264. 10 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x Exercise4.225 7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 y Exercise4.226 Solutiononp.264. 6 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 k Exercise4.227 3 a +5 a +2 a Exercise4.228 Solutiononp.264. 9 y +10 y +2 y Exercise4.229 5 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m Exercise4.230 Solutiononp.264. h )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 h )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 h Exercise4.231 a +8 a +3 a Exercise4.232 Solutiononp.264. 7 ab +4 ab Exercise4.233 8 ax +2 ax +6 ax Exercise4.234 Solutiononp.264. 3 a 2 +6 a 2 +2 a 2 Exercise4.235 14 a 2 b +4 a 2 b +19 a 2 b Exercise4.236 Solutiononp.264. 10 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 y Exercise4.237 7 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 ab +4 ab Exercise4.238 Solutiononp.264. 210 ab 4 +412 ab 4 +100 a 4 b Lookcloselyattheexponents.

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225 Exercise4.239 5 x 2 y 0 +3 x 2 y +2 x 2 y +1 ;y 6 =0 Lookcloselyattheexponents. Exercise4.240 Solutiononp.264. 8 w 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 w 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 w 2 Exercise4.241 6 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 xy +7 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 xy Exercise4.242 Solutiononp.264. 7 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x +1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12+ x Exercise4.243 21 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 x +40 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 y +7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(xy Exercise4.244 Solutiononp.264. 1 x +1 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 y + x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y Exercise4.245 5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7+2 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x Exercise4.246 Solutiononp.264. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 z 3 +15 z +4 z 3 + z 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 z 2 + z Exercise4.247 18 x 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 x 2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 x 2 y Exercise4.248 Solutiononp.264. )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 w 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 w 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 w 5 +10 w 4 Exercise4.249 2 x 4 +4 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x 2 +12 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x +2 Exercise4.250 Solutiononp.264. 17 d 3 r +3 d 3 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 d 3 r +6 d 2 r + d 3 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 d 2 r +3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7+2 Exercise4.251 a 0 +2 a 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 a 0 ;a 6 =0 Exercise4.252 Solutiononp.265. 4 x 0 +3 x 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x 0 +7 x 0 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 0 ;x 6 =0 Exercise4.253 2 a 3 b 2 c +3 a 2 b 2 c 0 +4 a 2 b 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 3 b 2 c;c 6 =0 Exercise4.254 Solutiononp.265. 3 z )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 z +8 z Exercise4.255 3 z 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(z +3 z 3 Exercise4.256 Solutiononp.265. 6 x 3 +12 x +5 Exercise4.257 3 x +5+2 x Exercise4.258 Solutiononp.265. 7 a +2+4 Exercise4.259 y +5 y +6 Exercise4.260 Solutiononp.265. 2 b +6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 b

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226 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.261 5 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 c +3 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(c Exercise4.262 Solutiononp.265. 8 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +4 x +5+3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise4.263 2 z +4 ab +5 z )]TJ/F11 9.9626 Tf 9.962 0 Td [(ab +12 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ab )]TJ/F11 9.9626 Tf 9.963 0 Td [(z Exercise4.264 Solutiononp.265. a +54+6 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 Exercise4.265 a +5 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(23+3 a +5 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise4.266 Solutiononp.265. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(10 x +3 y 2 4+4 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(10 x +3 y 2 Exercise4.267 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6+5 Exercise4.268 Solutiononp.265. 1 x +15+2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 Exercise4.269 1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2+9 a +4 a 2 + a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 a Exercise4.270 Solutiononp.265. 1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 b +6 a 2 b +8 b 2 +1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 x +2 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a 2 b Exercise4.271 Afterobservingthefollowingproblems,canyoumakeaconjectureabout 1 a + b ? 1 a + b = Exercise4.272 Solutiononp.265. Usingtheresultofproblem52,isitcorrecttowrite a + b = a + b ? Exercise4.273 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a +2 a 2 +8 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 a +3 a 2 Exercise4.274 Solutiononp.265. x x +2+2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise4.275 A A +7+4 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(A 2 +3 a +1 Exercise4.276 Solutiononp.265. b )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 b 3 +5 b 2 + b +6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 b +2 Exercise4.277 4 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(a a +5 Exercise4.278 Solutiononp.265. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise4.279 ab a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a 2 b +2 ab )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise4.280 Solutiononp.265. xy xy +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x 2 y +4 xy 2 Exercise4.281 3 h [2 h +5 h +2] Exercise4.282 Solutiononp.265. 2 k [5 k +3+7 k ]

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227 Exercise4.283 8 a [2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ab +9 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ab ] Exercise4.284 Solutiononp.265. 6 f m +5 n [ n +3 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]+2 n 2 g)]TJ/F8 9.9626 Tf 17.158 0 Td [(4 n 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 m Exercise4.285 5[4 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 s )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 r )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 s ]+12 s Exercise4.286 Solutiononp.265. 8 f 9 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a +6 c c +4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 c 2 +4 a + b g)]TJ/F8 9.9626 Tf 17.158 0 Td [(3 b Exercise4.287 5[4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+ x ] )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(25 x +4 Exercise4.288 Solutiononp.265. 3 xy 2 xy +5 y +2 xy 3 +6 x 2 y 3 +4 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 xy 3 Exercise4.289 9 a 3 b 7 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 3 b 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a 2 b 2 +6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 b 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a 5 b 12 +3 a 4 b 9 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 3 b 7 Exercise4.290 Solutiononp.265. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 a +2 Exercise4.291 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y Exercise4.292 Solutiononp.265. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 xy 2 7 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 )]TJ/F11 9.9626 Tf 9.962 0 Td [(xy 2 +3 )]TJ/F11 9.9626 Tf 7.749 0 Td [(xy +1+1 4.5.12ExercisesforReview Exercise4.293 Section2.6 Simplify x 10 y 8 z 2 x 2 y 6 3 Exercise4.294 Solutiononp.265. Section3.6 Findthevalueof )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 3 Exercise4.295 Section3.7 Writetheexpression 42 x 2 y 5 z 3 21 x 4 y 7 sothatnodenominatorappears. Exercise4.296 Solutiononp.265. Section4.2 Howmany a +5 'saretherein 3 x a +5 ? Exercise4.297 Section4.4 Simplify 3 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 n +6 m 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 n +4 m 2 4.6CombiningPolynomialsUsingMultiplication 6 4.6.1Overview MultiplyingaPolynomialbyaMonomial Simplifying + a + b and )]TJ/F8 9.9626 Tf 9.409 0 Td [( a + b MultiplyingaPolynomialbyaPolynomial 6 Thiscontentisavailableonlineat.

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228 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.6.2MultiplyingaPolynomialbyaMonomial Multiplyingapolynomialbyamonomialisadirectapplicationofthedistributiveproperty. DistributiveProperty Thedistributivepropertysuggeststhefollowingrule. MultiplyingaPolynomialbyaMonomial Tomultiplyapolynomialbyamonomial,multiply every termofthepolynomialbythemonomialandthen addtheresultingproductstogether. 4.6.3SampleSetA Example4.83 Example4.84 Example4.85 Example4.86 Example4.87 Example4.88 10 ab 2 c )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(125 a 2 =1250 a 3 b 2 c Example4.89

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229 4.6.4PracticeSetA Determinethefollowingproducts. Exercise4.298 Solutiononp.265. 3 x +8 Exercise4.299 Solutiononp.265. + a 4 Exercise4.300 Solutiononp.266. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b +6 2 a Exercise4.301 Solutiononp.266. 8 a 2 b 3 a +7 b +3 Exercise4.302 Solutiononp.266. 4 x )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x 5 +6 x 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 2 +9 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise4.303 Solutiononp.266. )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(3 a 2 b )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(2 ab 2 +4 b 3 Exercise4.304 Solutiononp.266. 5 mn )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(m 2 n 2 + m + n 0 ;n 6 =0 Exercise4.305 Solutiononp.266. 6 : 03 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 : 11 a 3 +8 : 00 a 2 b 4.6.5Simplifying + a + b and )]TJ/F89 11.9552 Tf 11.291 0 Td [( a + b + a + b and )]TJ/F8 9.9626 Tf 9.409 0 Td [( a + b Oftentimes,wewillencountermultiplicationsoftheform +1 a + b or )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 a + b Thesetermswillactuallyappearas + a + b and )]TJ/F8 9.9626 Tf 9.41 0 Td [( a + b Usingthedistributiveproperty,wecanremovetheparentheses. Theparentheseshavebeenremovedandthesignofeachtermhasremainedthesame. Theparentheseshavebeenremovedandthesignofeachtermhasbeenchangedtoitsopposite. 1.Toremoveasetofparenthesesprecededbya" + "sign,simplyremovetheparenthesesandleavethe signofeachtermthesame. 2.Toremoveasetofparenthesesprecededbya )]TJ/F15 9.9626 Tf 7.748 0 Td [(sign,removetheparenthesesandchangethesignof eachtermtoitsoppositesign.

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230 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.6.6SampleSetB Simplifytheexpressions. Example4.90 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Thissetofparenthesesisprecededbya + signimplied.Wesimplydroptheparentheses. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Example4.91 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(14 a 2 b 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 a 3 b 2 + ab 4 =14 a 2 b 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a 3 b 2 + ab 4 Example4.92 )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(21 a 2 +7 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 Thissetofparenthesesisprecededbya )]TJ/F15 9.9626 Tf 7.749 0 Td [(sign.Wecandroptheparenthesesaslongas wechangethesignof every terminsidetheparenthesestoitsoppositesign. )]TJ/F1 9.9626 Tf 9.409 8.069 Td [()]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(21 a 2 +7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 a +18 Example4.93 )]TJ/F1 9.9626 Tf 9.41 8.07 Td [()]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 +9 y +1 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 y 3 +2 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4.6.7PracticeSetB Simplifybyremovingtheparentheses. Exercise4.306 Solutiononp.266. a +3 b Exercise4.307 Solutiononp.266. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a +10 Exercise4.308 Solutiononp.266. )]TJ/F8 9.9626 Tf 9.409 0 Td [( x +2 y Exercise4.309 Solutiononp.266. )]TJ/F8 9.9626 Tf 9.409 0 Td [( m )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 n Exercise4.310 Solutiononp.266. )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(3 s 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 s +9 4.6.8MultiplyingaPolynomialbyaPolynomial Sincewecanconsideranexpressionenclosedwithinparenthesesasasinglequantity,wehave,bythe distributiveproperty,

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231 Forconveniencewewillusethecommutativepropertyofadditiontowritethisexpressionsothatthe rsttwotermscontain a andthesecondtwocontain b a + b c + d = ac + ad + bc + bd Thismethodiscommonlycalledthe FOILmethod F Firstterms O Outerterms I Innerterms L Lastterms a + b +3= a + b + a + b | {z } 2 terms + a + b + a + b + a + b | {z } 3 terms Rearranging, = a + a + b + b + a + a + a + b + b + b =2 a +2 b +3 a +3 b Combiningliketerms, =5 a +5 b Thisuseofthedistributivepropertysuggeststhefollowingrule. MultiplyingaPolynomialbyaPolynomial Tomultiplytwopolynomialstogether,multiply every termofonepolynomialby every termoftheother polynomial. 4.6.9SampleSetC Performthefollowingmultiplicationsandsimplify. Example4.94 Withsomepractice,thesecondandthirdtermscanbecombinedmentally. Example4.95 Example4.96

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232 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Example4.97 Example4.98 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 = m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 = m m + m )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m +9 = m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 m +9 Example4.99 x +5 3 = x +5 x +5 x +5 Associatethersttwofactors. =[ x +5 x +5] x +5 = x 2 +5 x +5 x +25 x +5 = x 2 +10 x +25 x +5 = x 2 x + x 2 5+10 x x +10 x 5+25 x +25 5 = x 3 +5 x 2 +10 x 2 +50 x +25 x +125 = x 3 +15 x 2 +75 x +125 4.6.10PracticeSetC Findthefollowingproductsandsimplify. Exercise4.311 Solutiononp.266. a +1 a +4 Exercise4.312 Solutiononp.266. m )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise4.313 Solutiononp.266. x +4 x +5 Exercise4.314 Solutiononp.266. x + y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y Exercise4.315 Solutiononp.266. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(5 a 2 + a Exercise4.316 Solutiononp.266. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x 2 y 3 + xy 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(5 x 3 y 2 + x 2 y Exercise4.317 Solutiononp.266. a +3 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 +3 a +6 Exercise4.318 Solutiononp.266. a +4 a +4 Exercise4.319 Solutiononp.266. r )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(7

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233 Exercise4.320 Solutiononp.266. x +6 2 Exercise4.321 Solutiononp.266. y )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 2 4.6.11SampleSetD Performthefollowingadditionsandsubtractions. Example4.100 3 x +7+ x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 : Wemustrstremovetheparentheses.Theyareprecededby a "+" sign,soweremovethemandleavethesignofeach termthesame. 3 x +7+ x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Combineliketerms. 4 x +4 Example4.101 5 y 3 +11 )]TJ/F1 9.9626 Tf 9.963 8.07 Td [()]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(12 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : Werstremovetheparentheses.Theyareprecededbya "-"sign,soweremovethemandchangethesignofeach terminsidethem. 5 y 3 +11 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 y 3 +2 Combineliketerms. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 y 3 +13 Example4.102 Add 4 x 2 +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 to 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 x 2 +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 + )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 4 x 2 +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8+3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 7 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 Example4.103 Subtract 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +2 from 3 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 )]TJ/F1 9.9626 Tf 9.963 8.07 Td [()]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +2 3 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x 2 +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 2 +6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Beverycareful not towritethisproblemas 3 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +2 Thisformhasussubtractingonlytheveryrstterm, 8 x 2 ,ratherthantheentireexpression. Useparentheses. Anotherincorrectformis 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +2 )]TJ/F1 9.9626 Tf 9.962 8.069 Td [()]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(3 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12

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234 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Thisformhasusperformingthesubtractioninthewrongorder. 4.6.12PracticeSetD Performthefollowingadditionsandsubtractions. Exercise4.322 Solutiononp.266. 6 y 2 +2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+ )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 Exercise4.323 Solutiononp.266. m )]TJ/F11 9.9626 Tf 9.963 0 Td [(n )]TJ/F8 9.9626 Tf 9.963 0 Td [( m +12 n Exercise4.324 Solutiononp.266. Add 2 r 2 +4 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 to 3 r 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(r )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 Exercise4.325 Solutiononp.267. Subtract 4 s )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 from 7 s +8 4.6.13Exercises Forthefollowingproblems,performthemultiplicationsandcombineanyliketerms. Exercise4.326 Solutiononp.267. 7 x +6 Exercise4.327 4 y +3 Exercise4.328 Solutiononp.267. 6 y +4 Exercise4.329 8 m +7 Exercise4.330 Solutiononp.267. 5 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise4.331 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 Exercise4.332 Solutiononp.267. 3 x +2 Exercise4.333 6 x +4 Exercise4.334 Solutiononp.267. 9 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.335 5 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.336 Solutiononp.267. )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 a +7 Exercise4.337 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 b +8 Exercise4.338 Solutiononp.267. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x +2 Exercise4.339 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 y +7

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235 Exercise4.340 Solutiononp.267. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.341 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise4.342 Solutiononp.267. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 a +1 Exercise4.343 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 x +2 Exercise4.344 Solutiononp.267. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise4.345 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise4.346 Solutiononp.267. x x +6 Exercise4.347 y y +7 Exercise4.348 Solutiononp.267. m m )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise4.349 k k )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise4.350 Solutiononp.267. 3 x x +2 Exercise4.351 4 y y +7 Exercise4.352 Solutiononp.267. 6 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise4.353 9 x x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.354 Solutiononp.267. 3 x x +4 Exercise4.355 4 m m +7 Exercise4.356 Solutiononp.267. 2 b b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.357 7 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise4.358 Solutiononp.267. 3 x 2 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(5 x 2 +4 Exercise4.359 9 y 3 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(3 y 2 +2 Exercise4.360 Solutiononp.267. 4 a 4 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 a 3 +3 a 2 +2 a Exercise4.361 2 x 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +3 Exercise4.362 Solutiononp.267. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 2 x +2

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236 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.363 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 y 3 y +5 Exercise4.364 Solutiononp.267. 2 x 2 y )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 2 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x Exercise4.365 8 a 3 b 2 c )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 ab 3 +3 b Exercise4.366 Solutiononp.267. b 5 x 2 bx )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise4.367 4 x )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x +10 Exercise4.368 Solutiononp.267. 9 y 3 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y 3 +8 y 2 + y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise4.369 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a 2 b 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6 ab 4 +5 ab 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 b 2 +7 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise4.370 Solutiononp.267. a +4 a +2 Exercise4.371 x +1 x +7 Exercise4.372 Solutiononp.267. y +6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.373 t +8 t )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise4.374 Solutiononp.268. i )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 i +5 Exercise4.375 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y x + y Exercise4.376 Solutiononp.268. a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.377 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise4.378 Solutiononp.268. y +11 y +10 Exercise4.379 t +6 t +4 Exercise4.380 Solutiononp.268. + x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x Exercise4.381 + a + a Exercise4.382 Solutiononp.268. )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 +2 x +1 Exercise4.383 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 +5 x +4 Exercise4.384 Solutiononp.268. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 x 2 +1 Exercise4.385 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 a 2 b 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(5 a 2 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b

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237 Exercise4.386 Solutiononp.268. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 x 3 y 4 +6 x )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 x 2 y 3 +5 y Exercise4.387 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.388 Solutiononp.268. 4 a +1 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise4.389 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a +5 Exercise4.390 Solutiononp.268. x x +1 x +4 Exercise4.391 x 2 x +5 x +7 Exercise4.392 Solutiononp.268. y 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise4.393 2 a 2 a +4 a +3 Exercise4.394 Solutiononp.268. 5 y 6 y +7 y +1 Exercise4.395 ab 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(a + b 4 Exercise4.396 Solutiononp.268. x 3 y 2 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(5 x 2 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.397 6 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 2 +5 a +3 Exercise4.398 Solutiononp.268. 8 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c 3 +5 c +11 Exercise4.399 3 a 2 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a +9 Exercise4.400 Solutiononp.268. 6 a 3 b 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 a 2 b 6 +7 ab 8 +2 b 10 +14 Exercise4.401 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a 2 + a )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Exercise4.402 Solutiononp.268. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.403 x +1 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 x 3 +6 x 2 +8 Exercise4.404 Solutiononp.268. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 a 2 +2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 a 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 a 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise4.405 x + y )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 x 2 +3 xy +5 y 2 Exercise4.406 Solutiononp.268. a + b )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 a 2 +4 a 2 b )]TJ/F11 9.9626 Tf 9.962 0 Td [(b )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise4.407 x +3 2 Exercise4.408 Solutiononp.268. x +1 2

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238 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.409 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 2 Exercise4.410 Solutiononp.268. a +2 2 Exercise4.411 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 2 Exercise4.412 Solutiononp.268. )]TJ/F8 9.9626 Tf 7.749 0 Td [( x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 2 Exercise4.413 )]TJ/F8 9.9626 Tf 7.749 0 Td [( t +7 2 Forthefollowingproblems,performtheindicatedoperationsandcombineliketerms. Exercise4.414 Solutiononp.268. 3 x 2 +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+ )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Exercise4.415 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x 3 +4 x 2 +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8+ )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 x +1 Exercise4.416 Solutiononp.268. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 xy +4 y 2 + )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(7 x +7 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y 2 Exercise4.417 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a +7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a 2 +2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise4.418 Solutiononp.268. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 + x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 x +14 Exercise4.419 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x 2 +2 + )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 +6 Exercise4.420 Solutiononp.268. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(9 a 2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 ab +12 ab 2 + ab 2 +2 ab Exercise4.421 6 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 x + )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 +4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise4.422 Solutiononp.268. 5 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(26+ )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(4 a 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 a 2 +2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a +8 a 3 +20 Exercise4.423 2 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 )]TJ/F8 9.9626 Tf 9.962 0 Td [( xy +4 Exercise4.424 Solutiononp.269. Add 4 x +6 to 8 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 Exercise4.425 Add 5 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y +1 to )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 y 2 +4 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise4.426 Solutiononp.269. Add 3 x +6 to 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 Exercise4.427 Add )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 to 5 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise4.428 Solutiononp.269. Addfourtimes 5 x +2 tothreetimes 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.429 Addvetimes )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +2 toseventimes 4 x +3 Exercise4.430 Solutiononp.269. Add )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 times 9 x +6 to )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 times )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 .

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239 Exercise4.431 Subtract 6 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x +4 from 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +5 Exercise4.432 Solutiononp.269. Substract a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 from a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 4.6.14ExercisesforReview Exercise4.433 Section2.7 Simplify 15 x 2 y 6 5 xy 2 4 Exercise4.434 Solutiononp.269. Section3.8 Expressthenumber198,000usingscienticnotation. Exercise4.435 Section4.2 Howmany 4 a 2 x 3 'saretherein )]TJ/F8 9.9626 Tf 7.748 0 Td [(16 a 4 x 5 ? Exercise4.436 Solutiononp.269. Section4.4 Statethedegreeofthepolynomial 4 xy 3 +3 x 5 y )]TJ/F8 9.9626 Tf 10.093 0 Td [(5 x 3 y 3 ,andwritethenumerical coecientofeachterm. Exercise4.437 Section4.5 Simplify 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5+2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 4.7SpecialBinomialProducts 7 4.7.1Overview Expanding a + b 2 and a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 Expanding a + b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b Threebinomialproductsoccursofrequentlyinalgebrathatwedesignatethemas specialbinomialproducts .WehaveseenthembeforeSectionsSection3.8andSection4.6,butwewillstudythemagainbecause oftheirimportanceastimesavingdevicesandinsolvingequationswhichwewillstudyinalaterchapter. Thesespecialproductscanbeshownasthe squaresofabinomial a + b 2 and a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 andasthe sumanddierenceoftwoterms a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b Therearetwosimplerulesthatallowustoeasilyexpandmultiplyoutthesebinomials.Theyarewell worthmemorizing,astheywillsavealotoftimeinthefuture. 4.7.2Expanding a + b 2 and a )]TJ/F24 11.9552 Tf 11.956 0 Td [(b 2 SquaringaBinomial Tosquareabinomial: 1.Squaretherstterm. 2.Taketheproductofthetwotermsanddoubleit. 3.Squarethelastterm. 4.Addthethreeresultstogether. 7 Thiscontentisavailableonlineat.

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240 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS a + b 2 = a 2 +2 ab + b 2 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 = a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 ab + b 2 4.7.3Expanding a + b a )]TJ/F24 11.9552 Tf 11.955 0 Td [(b SumandDierenceofTwoTerms Toexpandthesumanddierenceoftwoterms: y 1.Squarethersttermandsquarethesecondterm. 2.Subtractthesquareofthesecondtermfromthesquareoftherstterm. a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b = a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 Seeproblems56and57attheendofthissection. y Seeproblem58. 4.7.4SampleSetA Example4.104 x +4 2 Squaretherstterm: x 2 : Theproductofbothtermsis 4 x: Doubleit: 8 x: Squarethelastterm:16 : Addthemtogether: x 2 +8 x +16 : x +4 2 = x 2 +8 x +16 Notethat x +4 2 6 = x 2 +4 2 .The 8 x termismissing! Example4.105 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 2 Squaretherstterm: a 2 : Theproductofbothtermsis )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 a: Doubleit: )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 a: Squarethelastterm: 64 : Addthemtogether: a 2 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 a +64 : a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 2 = a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 a +64 Noticethatthesignofthelastterminthisexpressionis + .Thiswillalwayshappensince thelasttermresultsfromanumberbeing squared .Anynonzeronumbertimesitselfisalways positive. ++=+ and )]TJ/F8 9.9626 Tf 7.749 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(=+ Thesignofthesecondterminthetrinomialwillalwaysbethesignthatoccurs inside the parentheses. Example4.106 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 Squaretherstterm: y 2 : Theproductofbothtermsis )]TJ/F11 9.9626 Tf 9.963 0 Td [(y: Doubleit: )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y: Squarethelastterm: +1 : Addthemtogether: y 2 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y +1 :

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241 Example4.107 x +3 2 Squaretherstterm: 25 x 2 : Theproductofbothtermsis 15 x: Doubleit: 30 x: Squarethelastterm:9 : Addthemtogether: 25 x 2 +30 x +9 : Example4.108 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 2 Squaretherstterm: 49 b 2 : Theproductofbothtermsis )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 b: Doubleit: )]TJ/F8 9.9626 Tf 9.963 0 Td [(28 b: Squarethelastterm:4 : Addthemtogether: 49 b 2 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(28 b +4 : Example4.109 x +6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Squaretherstterm: x 2 : Subtractthesquareofthesecondterm from thesquareoftherstterm: x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 : x +6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6= x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(36 Example4.110 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a +12 Squaretherstterm: 16 a 2 : Subtractthesquareofthesecondterm from thesquareoftherstterm: 16 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(144 : a )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a +12=16 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(144 Example4.111 x +8 y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 y Squaretherstterm: 36 x 2 : Subtractthesquareofthesecondterm )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(64 y 2 from thesquareoftherstterm: 36 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(64 y 2 : x +8 y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 y =36 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(64 y 2

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242 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.7.5PracticeSetA Findthefollowingproducts. Exercise4.438 Solutiononp.269. x +5 2 Exercise4.439 Solutiononp.269. x +7 2 Exercise4.440 Solutiononp.269. y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 2 Exercise4.441 Solutiononp.269. a + b 2 Exercise4.442 Solutiononp.269. m )]TJ/F11 9.9626 Tf 9.963 0 Td [(n 2 Exercise4.443 Solutiononp.269. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 Exercise4.444 Solutiononp.269. a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 b 2 Exercise4.445 Solutiononp.269. h )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 k 2 4.7.6Exercises Forthefollowingproblems,ndtheproducts. Exercise4.446 Solutiononp.269. x +3 2 Exercise4.447 x +5 2 Exercise4.448 Solutiononp.269. x +8 2 Exercise4.449 x +6 2 Exercise4.450 Solutiononp.269. y +9 2 Exercise4.451 y +1 2 Exercise4.452 Solutiononp.269. a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 Exercise4.453 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 2 Exercise4.454 Solutiononp.269. a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 2 Exercise4.455 b +10 2 Exercise4.456 Solutiononp.269. b +15 2

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243 Exercise4.457 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 2 Exercise4.458 Solutiononp.269. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 2 Exercise4.459 x +20 2 Exercise4.460 Solutiononp.269. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 2 Exercise4.461 x +5 2 Exercise4.462 Solutiononp.269. x +2 2 Exercise4.463 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 2 Exercise4.464 Solutiononp.269. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 2 Exercise4.465 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 2 Exercise4.466 Solutiononp.270. a )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 2 Exercise4.467 w )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z 2 Exercise4.468 Solutiononp.270. a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b 2 Exercise4.469 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 s 2 Exercise4.470 Solutiononp.270. h )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 k 2 Exercise4.471 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a + 1 2 2 Exercise4.472 Solutiononp.270. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(a + 1 3 2 Exercise4.473 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x + 3 4 2 Exercise4.474 Solutiononp.270. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x + 2 5 2 Exercise4.475 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 3 2 Exercise4.476 Solutiononp.270. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(y )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(5 6 2 Exercise4.477 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(y + 2 3 2 Exercise4.478 Solutiononp.270. x +1 : 3 2

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244 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.479 x +5 : 2 2 Exercise4.480 Solutiononp.270. a +0 : 5 2 Exercise4.481 a +0 : 08 2 Exercise4.482 Solutiononp.270. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 : 1 2 Exercise4.483 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 : 2 2 Exercise4.484 Solutiononp.270. b )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 : 04 2 Exercise4.485 f )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : 006 2 Exercise4.486 Solutiononp.270. x +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise4.487 x +6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.488 Solutiononp.270. x +1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.489 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 t +1 Exercise4.490 Solutiononp.270. f +9 f )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Exercise4.491 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 y +7 Exercise4.492 Solutiononp.270. y +3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise4.493 x +6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise4.494 Solutiononp.270. a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 b a +7 b Exercise4.495 x +3 t x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 t Exercise4.496 Solutiononp.270. h )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 k h +2 k Exercise4.497 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x + 1 3 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(1 3 Exercise4.498 Solutiononp.270. )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(a + 2 9 )]TJ/F11 9.9626 Tf 10.793 -8.069 Td [(a )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(2 9 Exercise4.499 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x + 7 3 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(7 3 Exercise4.500 Solutiononp.270. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 b + 6 7 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 b )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(6 7 Exercise4.501 Expand a + b 2 toproveitisequalto a 2 +2 ab + b 2 .

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245 Exercise4.502 Solutiononp.270. Expand a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 toproveitisequalto a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ab + b 2 Exercise4.503 Expand a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b toproveitisequalto a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 Exercise4.504 Solutiononp.270. Fillinthemissinglabelintheequationbelow. Exercise4.505 Labelthepartsoftheequationbelow. Exercise4.506 Solutiononp.270. Labelthepartsoftheequationbelow.

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246 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.7.7ExercisesforReview Exercise4.507 Section2.6 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 y 0 z 4 5 Exercise4.508 Solutiononp.270. Section3.7 Findthevalueof 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise4.509 Section4.6 Findtheproduct. x +6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise4.510 Solutiononp.270. Section4.6 Findtheproduct. m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m +3 Exercise4.511 Section4.6 Findtheproduct. a +4 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a +3 4.8TerminologyAssociatedwithEquations 8 4.8.1Overview IndependentandDependentVariables TheDomainofanEquation 4.8.2IndependentandDependentVariables IndependentandDependentVariables Inanequation,anyvariablewhosevaluecanbefreelyassignedissaidtobean independentvariable. Anyvariablewhosevalueisdeterminedoncetheothervalueshavebeenassignedissaidtobea dependent variable. Twoexampleswillhelpillustratetheseconcepts. 1.Considertheequation y =2 x )]TJ/F8 9.9626 Tf 10.228 0 Td [(7 .Ifwearefreetochoosevaluesfor x ,then x wouldbeconsidered theindependentvariable.Sincethevalueof y dependsonthevalueof x y wouldbethedependent variable. 2.Considertheequation m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 gk 2 .Ifwearefreetochoosevaluesforboth g and k ,then g and k wouldbeconsideredindependentvariables.Sincethevalueof m dependsonthevalueschosenfor g and k m wouldbethedependentvariable. 4.8.3TheDomainofanEquation Domain Theprocessofreplacingletterswithnumbersiscallednumericalevaluation.Thecollectionofnumbersthat canreplacetheindependentvariableinanequationandyieldameaningfulresultiscalledthe domain of theequation.Thedomainofanequationmaybetheentirecollectionofrealnumbersormayberestricted tosomesubcollectionoftherealnumbers.Therestrictionsmaybeduetoparticularapplicationsofthe equationortoproblemsofcomputability. 8 Thiscontentisavailableonlineat.

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247 4.8.4SampleSetA Findthedomainofeachofthefollowingequations. Example4.112 y = 2 x ,where x istheindependentvariable. Anynumberexcept0canbesubstitutedfor x andyieldameaningfulresult.Hence,thedomain isthecollectionofallrealnumbersexcept0. Example4.113 d =55 t ,where t istheindependentvariableandtheequationrelatestime, t ,anddistance, d Itmakeslittlesensetoreplace t byanegativenumber,sothedomainisthecollectionofallreal numbersgreaterthanorequalto0. Example4.114 k = 2 w w )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 ,wheretheindependentvariableis w Theletter w canbereplacedbyanyrealnumberexcept4sincethatwillproduceadivisionby 0.Hence,thedomainisthecollectionofallrealnumbersexcept4. Example4.115 a =5 b 2 +2 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 ,wheretheindependentvariableis b Wecanreplace b byanyrealnumberandtheexpression 5 b 2 +2 b )]TJ/F8 9.9626 Tf 9.436 0 Td [(6 iscomputable.Hence,the domainisthecollectionofallrealnumbers. 4.8.5PracticeSetA Findthedomainofeachofthefollowingequations.Assumethattheindependentvariableisthevariable thatappearsintheexpressionontherightsideofthe" = "sign. Exercise4.512 Solutiononp.270. y =5 x +10 Exercise4.513 Solutiononp.270. y = 5 x Exercise4.514 Solutiononp.271. y = 3+ x x Exercise4.515 Solutiononp.271. y = 9 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise4.516 Solutiononp.271. m = 1 n +2 Exercise4.517 Solutiononp.271. s = 4 9 t 2 ,wherethisequationrelatesthedistanceanobjectfalls, s ,tothetime, t ,ithashadto fall. Exercise4.518 Solutiononp.271. g = 4 h )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 21

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248 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.8.6Exercises Forthefollowingproblems,ndthedomainoftheequations.Assumethattheindependentvariableisthe variablethatappearsintheexpressiontotherightoftheequalsign. Exercise4.519 Solutiononp.271. y =4 x +7 Exercise4.520 y =3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Exercise4.521 Solutiononp.271. y = x 2 +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 Exercise4.522 y =8 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.523 Solutiononp.271. y =11 x Exercise4.524 s =7 t Exercise4.525 Solutiononp.271. y = 3 x Exercise4.526 y = 2 x Exercise4.527 Solutiononp.271. m = )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 h Exercise4.528 k = 4 t 2 t )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise4.529 Solutiononp.271. t = 5 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise4.530 y = 12 x +7 4.8.7ExercisesforReview Exercise4.531 Solutiononp.271. Section2.4 Namethepropertyofrealnumbersthatmakes 4 yx 2 =4 x 2 y atruestatement. Exercise4.532 Section2.6 Simplify x 5 n +6 x 4 Exercise4.533 Solutiononp.271. Section3.3 Supplythemissingphrase.Absolutevaluespeakstothequestionof and not"whichway." Exercise4.534 Section4.7 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 2 Exercise4.535 Solutiononp.271. Section4.7 Findtheproduct. x +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 .

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249 4.9SummaryofKeyConcepts 9 4.9.1SummaryofKeyConcepts AlgebraicExpressionsSection4.2 An algebraicexpression oftencalledsimplyanexpressionisanumber,aletter,oracollectionofnumbers andlettersalongwithmeaningfulsignsofoperation. 5 0 isnotmeaningful. TermsSection4.2 Inanalgebraicexpression,thequantitiesjoinedby" + "signsare terms DistinctionBetweenTermsandFactorsSection4.2 Terms arepartsofsumsandarethereforeseparatedbyadditionsigns. Factors arepartsofproductsand arethereforeseparatedbymultiplicationsigns. CommonFactorsSection4.2 Inanalgebraicexpression,afactorthatappearsin every term,thatis,afactorthatiscommontoeach term,iscalleda commonfactor CoecientsSection4.2 The coecient ofaquantityrecordshowmanyofthatquantitythereare.Thecoecientofagroupof factorsistheremaininggroupoffactors. DistinctionBetweenCoecientsandExponentsSection4.2 Coecients recordthenumberofliketermsinanexpression. x + x + x | {z } 3 terms =3 x coecientis 3 Exponents recordthenumberoflikefactorsinanexpression x x x | {z } 3 factors = x 3 exponentis 3 EquationSection4.3 An equation isastatementthattwoexpressionsareequal. NumericalEvaluationSection4.3 Numericalevaluation istheprocessofdeterminingavaluebysubstitutingnumbersforletters. PolynomialsSection4.4 Apolynomialisanalgebraicexpressionthatdoesnotcontainvariablesinthedenominatorsoffractions andinwhichallexponentsonvariablequantitiesarewholenumbers. A monomial isapolynomialconsistingofonlyoneterm. A binomial isapolynomialconsistingoftwoterms. A trinomial isapolynomialconsistingofthreeterms. DegreeofaPolynomialSection4.4 Thedegreeofatermcontainingonevariableisthevalueoftheexponentonthevariable. Thedegreeofatermcontainingmorethanonevariableisthesumoftheexponentsonthevariables. Thedegreeofapolynomialisthedegreeofthetermofthehighestdegree. LinearQuadraticCubicPolynomialsSection4.4 Polynomialsoftherstdegreeare linear polynomials. Polynomialsoftheseconddegreeare quadratic polynomials. Polynomialsofthethirddegreeare cubic polynomials. LikeTermsSection4.5 Liketerms aretermsinwhichthevariableparts,includingtheexponents,areidentical. 9 Thiscontentisavailableonlineat.

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250 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS DescendingOrderSection4.5 Byconvention,andwhenpossible,thetermsofanexpressionareplacedindescendingorderwiththehighest degreetermappearingrst. 5 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 +10 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 isindescendingorder. MultiplyingaPolynomialbyaMonomialSection4.6 Tomultiplyapolynomialbyamonomial,multiplyeverytermofthepolynomialbythemonomialandthen addtheresultingproductstogether. 7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 3=7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Simplifying + a + b and )]TJ/F8 9.9626 Tf 9.409 0 Td [( a + b Section4.6 + a + b = a + b )]TJ/F8 9.9626 Tf 9.409 0 Td [( a + b = )]TJ/F11 9.9626 Tf 7.749 0 Td [(a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b MultiplyingaPolynomialbyaPolynomialSection4.6 Tomultiplypolynomialstogether,multiplyeverytermofonepolynomialbyeverytermoftheotherpolynomial. x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4= x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 = x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 SpecialProductsSection4.7 a + b 2 = a 2 +2 ab + b 2 Note : a + b 2 6 = a 2 + b 2 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 = a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 ab + b 2 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 6 = a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 a + b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b = a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 IndependentandDependentVariablesSection4.8 Inanequation,anyvariablewhosevaluecanbefreelyassignedissaidtobean independentvariable Anyvariablewhosevalueisdeterminedoncetheothervalueshavebeenassignedissaidtobea dependent variable DomainSection4.8 Thecollectionofnumbersthatcanbeusedasreplacementsfortheindependentvariableinanexpressionor equationandyieldameaningfulresultiscalledthe domain oftheexpressionorequation. 4.10ExerciseSupplement 10 4.10.1ExerciseSupplement 4.10.1.1AlgebraicExpressionsSection4.2 Forthefollowingproblems,writethenumberoftermsthatappear,thenwritetheterms. Exercise4.536 Solutiononp.271. 4 x 2 +7 x +12 Exercise4.537 14 y 6 Exercise4.538 Solutiononp.271. c +8 Exercise4.539 8 10 Thiscontentisavailableonlineat.

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251 List,ifanyshouldappear,thecommonfactorsforthefollowingproblems. Exercise4.540 Solutiononp.271. a 2 +4 a 2 +6 a 2 Exercise4.541 9 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 y 4 Exercise4.542 Solutiononp.271. 12 x 2 y 3 +36 y 3 Exercise4.543 6 a +4+12 a +4 Exercise4.544 Solutiononp.271. 4 a +2 b +6 a +2 b Exercise4.545 17 x 2 y z +4+51 y z +4 Exercise4.546 Solutiononp.271. 6 a 2 b 3 c +5 x 2 y Forthefollowingproblems,answerthequestionofhowmany. Exercise4.547 x 'sin 9 x ? Exercise4.548 Solutiononp.271. a + b 'sin12 a + b ? Exercise4.549 a 4 'sin 6 a 4 ? Exercise4.550 Solutiononp.271. c 3 'sin 2 a 2 bc 3 ? Exercise4.551 x +3 y 2 'sin 5 x +2 y x +3 y 3 ? Forthefollowingproblems,atermwillbegivenfollowedbyagroupofitsfactors.Listthecoecientofthe givengroupoffactors. Exercise4.552 Solutiononp.271. 8 z;z Exercise4.553 16 a 3 b 2 c 4 ;c 4 Exercise4.554 Solutiononp.271. 7 y y +3 ; 7 y Exercise4.555 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 a 5 b 5 c 5 ;bc

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252 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS 4.10.1.2EquationsSection4.3 Forthefollowingproblems,observetheequationsandwritetherelationshipbeingexpressed. Exercise4.556 Solutiononp.271. a =3 b Exercise4.557 r =4 t +11 Exercise4.558 Solutiononp.272. f = 1 2 m 2 +6 g Exercise4.559 x =5 y 3 +2 y +6 Exercise4.560 Solutiononp.272. P 2 = ka 3 Usenumericalevaluationtoevaluatetheequationsforthefollowingproblems. Exercise4.561 C =2 r: Find C if isapproximatedby 3 : 14 and r =6 : Exercise4.562 Solutiononp.272. I = E R : Find I if E =20 and R =2 : Exercise4.563 I = prt: Find I if p =1000 ;r =0 : 06 ; and t =3 : Exercise4.564 Solutiononp.272. E = mc 2 : Find E if m =120 and c =186 ; 000 : Exercise4.565 z = x )]TJ/F10 6.9738 Tf 6.226 0 Td [(u s : Find z if x =42 ;u =30 ; and s =12 : Exercise4.566 Solutiononp.272. R = 24 C P n +1 : Find R if C =35 ;P =300 ; and n =19 : 4.10.1.3ClassicationofExpressionsandEquationsSection4.4 Forthefollowingproblems,classifyeachofthepolynomialsasamonomial,binomial,ortrinomial.State thedegreeofeachpolynomialandwritethenumericalcoecientofeachterm. Exercise4.567 2 a +9 Exercise4.568 Solutiononp.272. 4 y 3 +3 y +1 Exercise4.569 10 a 4 Exercise4.570 Solutiononp.272. 147 Exercise4.571 4 xy +2 yz 2 +6 x

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253 Exercise4.572 Solutiononp.272. 9 ab 2 c 2 +10 a 3 b 2 c 5 Exercise4.573 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 xy 3 0 ;xy 3 6 =0 Exercise4.574 Solutiononp.272. Whyistheexpression 4 x 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 notapolynomial? Exercise4.575 Whyistheexpression 5 a 3 = 4 notapolynomial? Forthefollowingproblems,classifyeachoftheequationsbydegree.Ifthetermlinear,quadratic,orcubic applies,useit. Exercise4.576 Solutiononp.272. 3 y +2 x =1 Exercise4.577 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a +8=0 Exercise4.578 Solutiononp.272. y )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F11 9.9626 Tf 9.962 0 Td [(z +4 w =21 Exercise4.579 5 x 2 +2 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +1=19 Exercise4.580 Solutiononp.272. )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 x 3 0 +5 x 2 =7 4.10.1.4CombiningPolynomialsUsingAdditionandSubtractionSection4.5-SpecialBinomialProductsSection4.7 Simplifythealgebraicexpressionsforthefollowingproblems. Exercise4.581 4 a 2 b +8 a 2 b )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 2 b Exercise4.582 Solutiononp.272. 21 x 2 y 3 +3 xy + x 2 y 3 +6 Exercise4.583 7 x +1+2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.584 Solutiononp.272. 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 y 2 +4 y +4 +5 y 2 +3 y +2 Exercise4.585 5 3 x +7 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 x 2 +3 x +2 +5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x 2 +4 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(3 x 2 + x Exercise4.586 Solutiononp.272. 8 f 3 4 y 3 + y +2 +6 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 3 +2 y 2 g)]TJ/F8 9.9626 Tf 17.158 0 Td [(24 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.587 4 a 2 bc 3 +5 abc 3 +9 abc 3 +7 a 2 bc 2 Exercise4.588 Solutiononp.272. x x +5+3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +3 Exercise4.589 4 k )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 k 2 +2 k +6 + k )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 k 2 + k +16 Exercise4.590 Solutiononp.272. 2 f 5 6 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b +2 a + c 2 g

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254 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.591 9 x 2 y xy +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x 3 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 x 3 y +5 y )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 y +2 x Exercise4.592 Solutiononp.272. 3 m 5+2 m )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(m +6 m 2 + m )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m 2 +4 m +1 Exercise4.593 2 r [4 r +5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(10]+6 r r +2 Exercise4.594 Solutiononp.272. abc abc + c + b +6 a )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 bc + bc 2 Exercise4.595 s 10 )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(2 s 5 +3 s 4 +4 s 3 +5 s 2 +2 s +2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 15 +2 s 14 +3 s )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(s 12 +4 s 11 )]TJ/F11 9.9626 Tf 9.962 0 Td [(s 10 Exercise4.596 Solutiononp.272. 6 a 4 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 2 +5 Exercise4.597 2 x 2 y 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 2 y +4 xy +3 y Exercise4.598 Solutiononp.272. 5 m 6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 m 7 +3 m 4 + m 2 + m +1 Exercise4.599 a 3 b 3 c 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 a +2 b +3 c + ab + ac + bc 2 Exercise4.600 Solutiononp.272. x +2 x +3 Exercise4.601 y +4 y +5 Exercise4.602 Solutiononp.272. a +1 a +3 Exercise4.603 x +4 x +6 Exercise4.604 Solutiononp.272. 4 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 xy Exercise4.605 5 ab 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 ab 2 +4 Exercise4.606 Solutiononp.272. 7 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 x 4 Exercise4.607 5 x 2 +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise4.608 Solutiononp.273. 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise4.609 7 x )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x +3 Exercise4.610 Solutiononp.273. )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.611 4 x 2 y 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 x 3 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 2 y 3 Exercise4.612 Solutiononp.273. )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 y )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 y 2 +7 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 Exercise4.613 )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(4]

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255 Exercise4.614 Solutiononp.273. )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( f)]TJ/F8 9.9626 Tf 22.14 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [(] g ] Exercise4.615 x 2 +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(9+2 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise4.616 Solutiononp.273. 4 a 2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 q 2 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a 2 b )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 Exercise4.617 2 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x )]TJ/F1 9.9626 Tf 9.963 8.069 Td [()]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise4.618 Solutiononp.273. 3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 a +6 Exercise4.619 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 a +2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4+6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.620 Solutiononp.273. Add )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +4 to 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise4.621 Add 4 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 to )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise4.622 Solutiononp.273. Subtract3times x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 from8times x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise4.623 x +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.624 Solutiononp.273. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise4.625 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.626 Solutiononp.273. b +2 c b )]TJ/F11 9.9626 Tf 9.962 0 Td [(c Exercise4.627 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 Exercise4.628 Solutiononp.273. )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 2 Exercise4.629 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 Exercise4.630 Solutiononp.273. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 Exercise4.631 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 b 2 Exercise4.632 Solutiononp.273. )]TJ/F11 9.9626 Tf 7.748 0 Td [(x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 Exercise4.633 k +6 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise4.634 Solutiononp.273. m +1 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise4.635 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a +2 Exercise4.636 Solutiononp.273. c +10 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(10

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256 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.637 a +3 b a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b Exercise4.638 Solutiononp.273. +2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 b Exercise4.639 y +5 y +5 Exercise4.640 Solutiononp.273. y +3 a y + a Exercise4.641 + a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a Exercise4.642 Solutiononp.273. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 +2 )]TJ/F11 9.9626 Tf 10.792 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise4.643 6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a +8 Exercise4.644 Solutiononp.273. 8 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 y +8 Exercise4.645 x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x +4 Exercise4.646 Solutiononp.273. m 2 n m + n m +2 n Exercise4.647 b +2 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b +3 Exercise4.648 Solutiononp.273. 3 p )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(p 2 +5 p +4 )]TJ/F11 9.9626 Tf 10.793 -8.069 Td [(p 2 +2 p +7 Exercise4.649 a +6 2 Exercise4.650 Solutiononp.273. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 Exercise4.651 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 Exercise4.652 Solutiononp.273. )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 + y 2 Exercise4.653 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 n 2 Exercise4.654 Solutiononp.273. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 2 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 4 y 2 Exercise4.655 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 4

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257 4.10.1.5TerminologyAssociatedwithEquationsSection4.8 Findthedomainoftheequationsforthefollowingproblems. Exercise4.656 Solutiononp.273. y =8 x +7 Exercise4.657 y =5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +6 Exercise4.658 Solutiononp.274. y = 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise4.659 m = )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x h Exercise4.660 Solutiononp.274. z = 4 x +5 y +10 4.11ProciencyExam 11 4.11.1ProciencyExam Exercise4.661 Solutiononp.274. Section4.2 Intheexpressionbelow,specifythenumberoftermsthatarepresent,thenlist them. 3 a a +1 )]TJ/F8 9.9626 Tf 9.962 0 Td [( a +2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise4.662 Solutiononp.274. Section4.2 List,ifthereareany,thecommonfactorsof 20 x 3 y 2 +15 x 3 y 2 z 2 +10 x 3 z 2 Exercise4.663 Solutiononp.274. Section4.2 Howmany y 2 b +2 s in 8 xy 2 b +2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 ? Exercise4.664 Solutiononp.274. Section4.2 Writethecoecientof x 3 in 8 x 3 y 3 z Exercise4.665 Solutiononp.274. Section4.3 Findthevalueof P 2 if k =4 and a =3 P 2 = ka 3 Exercise4.666 Solutiononp.274. Section4.4 Classifythepolynomialgivenbelowasamonomial,binomial,trinomial,ornone ofthese.Specifythedegreeofthepolynomialandwritethenumericalcoecientofeachterm. 3 x 3 y +4 xy 4 +8 x 2 y 2 z 0 w;z 6 =0 Simplifythealgebraicexpressionsforthefollowingproblems. Exercise4.667 Solutiononp.274. Section4.5 4 x 2 +3 x +2 x +11 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise4.668 Solutiononp.274. Section4.5 3 a [2 a +1+4] )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 a Exercise4.669 Solutiononp.274. Section4.6 x +2 x +4 Exercise4.670 Solutiononp.274. Section4.6 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a +10 11 Thiscontentisavailableonlineat.

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258 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS Exercise4.671 Solutiononp.274. Section4.7 y +3 2 Exercise4.672 Solutiononp.274. Section4.7 a +7 y 2 Exercise4.673 Solutiononp.274. Section4.7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 y 2 Exercise4.674 Solutiononp.274. Section4.5-Section4.7 3 x 2 x +5 x +1 Exercise4.675 Solutiononp.274. Section4.5-Section4.7 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b Exercise4.676 Solutiononp.274. Section4.5-Section4.7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 y 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 y +3 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise4.677 Solutiononp.274. Section4.5-Section4.7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 b 3 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 Exercise4.678 Solutiononp.274. Section4.5-Section4.7 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a 3 +3 b 2 2 Exercise4.679 Solutiononp.274. Section4.5-Section4.7 6 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F1 9.9626 Tf 9.963 8.069 Td [()]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 a 2 + a )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 Exercise4.680 Solutiononp.274. Section4.5-Section4.7 h +2 k h )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 k Exercise4.681 Solutiononp.274. Section4.5-Section4.7 Subtract 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 from 2 a 2 +6 a +1 Exercise4.682 Solutiononp.274. Section4.5-Section4.7 Addthreetimes 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 totwotimes )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x +5 Exercise4.683 Solutiononp.274. Section4.5-Section4.7 Evaluate 6 k 2 +2 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 if k = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise4.684 Solutiononp.275. Section4.5-Section4.7 Evaluate )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 m m )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 if m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise4.685 Solutiononp.275. Section4.8 Whatisthedomainof y = 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 x +3 ?

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259 SolutionstoExercisesinChapter4 SolutiontoExercise4.1p.199 addition,multiplication SolutiontoExercise4.2p.199 4 x 2 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x; 7 SolutiontoExercise4.3p.199 2 xy; 6 x 2 ; x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 4 SolutiontoExercise4.4p.199 5 x 2 ; 3 x; )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 xy 7 ; x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 SolutiontoExercise4.5p.200 8, x x ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x ;6and1or3and2 SolutiontoExercise4.6p.200 10and1or5and2;2, b +6 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 SolutiontoExercise4.7p.201 x 2 SolutiontoExercise4.8p.201 4 x 2 SolutiontoExercise4.9p.201 2 a +1 SolutiontoExercise4.10p.201 3 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 SolutiontoExercise4.11p.201 14 c c +5 SolutiontoExercise4.12p.201 nocommonfactor SolutiontoExercise4.13p.202 howmanyofthatquantitythereare SolutiontoExercise4.14p.203 a6b x 3 SolutiontoExercise4.15p.203 a3b 3 x c x d 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 e x y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 f3 SolutiontoExercise4.16p.203 a10b 10 a c 10 b 4 d ab 4 e b SolutiontoExercise4.17p.203 Analgebraicexpressionisanumber,aletter,oracollectionofnumbersandlettersalongwithmeaningful signsofoperation. SolutiontoExercise4.19p.203 x isanexpressionbecauseitisaletterseethedenition. SolutiontoExercise4.21p.203 two :6 x; )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 SolutiontoExercise4.23p.203 three :5 x 2 ; 6 x; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise4.25p.203 one :5 cz SolutiontoExercise4.27p.204 one :61 SolutiontoExercise4.29p.204 one :4 y 3 SolutiontoExercise4.31p.204 two : a; 1

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260 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.33p.204 three :2 x;x; 7 SolutiontoExercise4.35p.204 two : a + b ; a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise4.37p.204 x 2 SolutiontoExercise4.39p.204 9 b 2 SolutiontoExercise4.41p.204 a + b SolutiontoExercise4.43p.204 2 ab 2 c 2 SolutiontoExercise4.45p.204 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise4.47p.204 nocommomfactors SolutiontoExercise4.49p.204 a +7 SolutiontoExercise4.51p.205 9 32 SolutiontoExercise4.53p.205 12 SolutiontoExercise4.55p.205 6 SolutiontoExercise4.57p.205 10 SolutiontoExercise4.59p.205 8 SolutiontoExercise4.61p.205 xy 5 SolutiontoExercise4.63p.205 5 x SolutiontoExercise4.65p.205 x 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 SolutiontoExercise4.67p.205 7 SolutiontoExercise4.69p.205 a SolutiontoExercise4.71p.206 6 x 2 b 2 SolutiontoExercise4.73p.206 3 ab 2 SolutiontoExercise4.75p.206 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a 5 b SolutiontoExercise4.77p.206 16 x 16 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 12 SolutiontoExercise4.79p.206 )]TJ/F8 9.9626 Tf 7.749 0 Td [(50 SolutiontoExercise4.81p.206 1 : 52 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5

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261 SolutiontoExercise4.82p.209 192 SolutiontoExercise4.83p.209 40 SolutiontoExercise4.84p.209 50 SolutiontoExercise4.85p.209 13 SolutiontoExercise4.86p.209 )]TJ/F8 9.9626 Tf 7.749 0 Td [(29 SolutiontoExercise4.87p.209 Thevalueof x isequaltosixtimesthevalueof y SolutiontoExercise4.89p.209 e isequalto9lessthenthevalueof g SolutiontoExercise4.91p.210 Thevalueofthreetimes t isequaltosixtimes s SolutiontoExercise4.93p.210 Thevalueof r isequaltotwoninthtimesthevalueof s SolutiontoExercise4.95p.210 Thevalueof f isequalto 55 morethen 97 100 timesthevalueof k SolutiontoExercise4.97p.210 Thevalueof q 2 isequaltoninetimesthevalueof x 8 plustwotimesthevalueof y SolutiontoExercise4.99p.210 31 : 4 SolutiontoExercise4.101p.210 3 SolutiontoExercise4.103p.210 360 SolutiontoExercise4.105p.210 48 SolutiontoExercise4.107p.211 8 SolutiontoExercise4.109p.211 42 5 18 SolutiontoExercise4.111p.211 448 SolutiontoExercise4.113p.211 205 : 44 SolutiontoExercise4.115p.211 396 SolutiontoExercise4.117p.211 150 SolutiontoExercise4.119p.211 588 : 49 SolutiontoExercise4.121p.212 238 ; 328 SolutiontoExercise4.123p.212 18 : 4191 SolutiontoExercise4.125p.212 650

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262 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.127p.212 379 : 94 SolutiontoExercise4.129p.212 2 ; 043 SolutiontoExercise4.131p.212 1 : 7298 10 11 SolutiontoExercise4.133p.213 6units SolutiontoExercise4.135p.213 99 : 33 tons SolutiontoExercise4.137p.214 195 : 46474 feet SolutiontoExercise4.139p.214 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise4.141p.214 2;5 a + b ; 2 x 2 SolutiontoExercise4.143p.217 rst,orlinear SolutiontoExercise4.144p.217 quadratic SolutiontoExercise4.145p.217 cubic SolutiontoExercise4.146p.217 linear SolutiontoExercise4.147p.218 linear SolutiontoExercise4.148p.218 quadratic SolutiontoExercise4.149p.218 quadratic SolutiontoExercise4.150p.218 linear SolutiontoExercise4.151p.218 eighthdegree SolutiontoExercise4.152p.218 binomial;rstlinear; 5 ; 7 SolutiontoExercise4.154p.218 binomial;secondquadratic; 4 ; 9 SolutiontoExercise4.156p.218 binomial;fourth; 1 ; 1 SolutiontoExercise4.158p.218 monomial;rstlinear; 5 SolutiontoExercise4.160p.218 trinomial;thirdcubic; 5 ; 2 ; 3 SolutiontoExercise4.162p.218 trinomial;thirdcubic; 41 ; 22 ; 1 SolutiontoExercise4.164p.218 monomial;sixth; 2

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263 SolutiontoExercise4.166p.218 monomial;rstlinear; 9 SolutiontoExercise4.168p.219 trinomial;secondquadratic; 3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; 11 SolutiontoExercise4.170p.219 binomial;ninth; 1 ; 9 SolutiontoExercise4.172p.219 binomial;eighth; 6 ; 3 SolutiontoExercise4.174p.219 monomial;zero; 5 SolutiontoExercise4.176p.219 monomial;ninth; 4 SolutiontoExercise4.178p.219 linear SolutiontoExercise4.180p.219 quadratic SolutiontoExercise4.182p.219 linear SolutiontoExercise4.184p.219 cubic SolutiontoExercise4.186p.219 quadratic SolutiontoExercise4.188p.220 linear SolutiontoExercise4.190p.220 linear SolutiontoExercise4.192p.220 cubic SolutiontoExercise4.194p.220 fthdegree SolutiontoExercise4.196p.220 19thdegree SolutiontoExercise4.198p.220 ...thereisavariableinthedenominator SolutiontoExercise4.200p.220 yes SolutiontoExercise4.202p.220 11 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x =5 SolutiontoExercise4.204p.220 z =2 SolutiontoExercise4.206p.220 Thevalueof y is5morethenthreetimesthevalueof x SolutiontoExercise4.207p.221 dierentamountsofthesamequantity SolutiontoExercise4.208p.222 11 y SolutiontoExercise4.209p.222 20 x

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264 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.210p.222 9 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 b SolutiontoExercise4.211p.222 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 3 +8 x 2 +10 x SolutiontoExercise4.212p.222 5 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(13 SolutiontoExercise4.213p.223 7 x 2 +7 x +30 SolutiontoExercise4.214p.223 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 3 +4 x 2 +6 x +29 SolutiontoExercise4.215p.223 59 a +27 SolutiontoExercise4.216p.223 5 x 2 +5 x SolutiontoExercise4.217p.223 2 a 5 + a 4 +8 a 3 +4 a +1 SolutiontoExercise4.218p.223 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x +34 SolutiontoExercise4.219p.224 50 x 2 +31 x SolutiontoExercise4.220p.224 4 x SolutiontoExercise4.222p.224 21 a SolutiontoExercise4.224p.224 3 x SolutiontoExercise4.226p.224 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 k SolutiontoExercise4.228p.224 21 y SolutiontoExercise4.230p.224 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 h SolutiontoExercise4.232p.224 11 ab SolutiontoExercise4.234p.224 11 a 2 SolutiontoExercise4.236p.224 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y SolutiontoExercise4.238p.224 622 ab 4 +100 a 4 b SolutiontoExercise4.240p.225 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 w 2 SolutiontoExercise4.242p.225 4 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 SolutiontoExercise4.244p.225 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y SolutiontoExercise4.246p.225 2 z 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 z 2 +16 z SolutiontoExercise4.248p.225 )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 w 5 + w 4

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265 SolutiontoExercise4.250p.225 16 d 3 r )]TJ/F8 9.9626 Tf 9.962 0 Td [(24 d 2 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise4.252p.225 8 SolutiontoExercise4.254p.225 5 z SolutiontoExercise4.256p.225 6 x 3 +12 x +5 SolutiontoExercise4.258p.225 7 a +18 SolutiontoExercise4.260p.225 )]TJ/F8 9.9626 Tf 7.749 0 Td [(28 b +18 SolutiontoExercise4.262p.226 31 x +8 SolutiontoExercise4.264p.226 10 a SolutiontoExercise4.266p.226 80 x +24 y 2 SolutiontoExercise4.268p.226 5 x +3 SolutiontoExercise4.270p.226 3 a 2 b +8 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 b +7 x SolutiontoExercise4.272p.226 yes SolutiontoExercise4.274p.226 3 x 2 +8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 SolutiontoExercise4.276p.226 2 b 4 +5 b 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 b 2 +2 b +2 SolutiontoExercise4.278p.226 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x 3 +21 x 2 +4 x SolutiontoExercise4.280p.226 x 2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x 2 y )]TJ/F11 9.9626 Tf 9.962 0 Td [(xy 2 SolutiontoExercise4.282p.226 52 k 2 +6 k SolutiontoExercise4.284p.227 128 n 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(90 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m SolutiontoExercise4.286p.227 144 c 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(112 a +77 b +1728 c SolutiontoExercise4.288p.227 18 x 2 y 3 +5 xy 3 +4 y 3 SolutiontoExercise4.290p.227 )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 SolutiontoExercise4.292p.227 )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 x 2 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 x 2 y 3 +104 xy 2 SolutiontoExercise4.294p.227 4 SolutiontoExercise4.296p.227 3 x SolutiontoExercise4.298p.229 3 x +24

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266 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.299p.229 4 a +8 SolutiontoExercise4.300p.229 2 a 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 ab +12 a SolutiontoExercise4.301p.229 16 a 3 b 3 +56 a 2 b 4 +24 a 2 b 3 SolutiontoExercise4.302p.229 8 x 6 +24 x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 3 +36 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(44 x SolutiontoExercise4.303p.229 6 a 3 b 3 +12 a 2 b 4 SolutiontoExercise4.304p.229 5 m 3 n 3 +5 m 2 n +5 mn SolutiontoExercise4.305p.229 12 : 7233 a 3 +48 : 24 a 2 b SolutiontoExercise4.306p.230 2 a +3 b SolutiontoExercise4.307p.230 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a +10 SolutiontoExercise4.308p.230 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y SolutiontoExercise4.309p.230 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 m +2 n SolutiontoExercise4.310p.230 3 s 2 +7 s )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 SolutiontoExercise4.311p.232 a 2 +5 a +4 SolutiontoExercise4.312p.232 m 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 m +18 SolutiontoExercise4.313p.232 2 x 2 +14 x +20 SolutiontoExercise4.314p.232 2 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y 2 SolutiontoExercise4.315p.232 15 a 4 +3 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a SolutiontoExercise4.316p.232 10 x 5 y 5 +7 x 4 y 4 + x 3 y 3 SolutiontoExercise4.317p.232 a 3 +6 a 2 +15 a +18 SolutiontoExercise4.318p.232 a 2 +8 a +16 SolutiontoExercise4.319p.232 r 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 r +49 SolutiontoExercise4.320p.233 x 2 +12 x +36 SolutiontoExercise4.321p.233 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 y +64 SolutiontoExercise4.322p.234 11 y 2 +2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(19 SolutiontoExercise4.323p.234 )]TJ/F11 9.9626 Tf 7.749 0 Td [(m )]TJ/F8 9.9626 Tf 9.963 0 Td [(13 n

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267 SolutiontoExercise4.324p.234 5 r 2 +3 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 SolutiontoExercise4.325p.234 3 s +11 SolutiontoExercise4.326p.234 7 x +42 SolutiontoExercise4.328p.234 6 y +24 SolutiontoExercise4.330p.234 5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 SolutiontoExercise4.332p.234 12 x +6 SolutiontoExercise4.334p.234 36 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 SolutiontoExercise4.336p.234 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(63 SolutiontoExercise4.338p.234 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 SolutiontoExercise4.340p.235 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a +18 SolutiontoExercise4.342p.235 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise4.344p.235 )]TJ/F8 9.9626 Tf 7.749 0 Td [(30 y +18 SolutiontoExercise4.346p.235 x 2 +6 x SolutiontoExercise4.348p.235 m 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 m SolutiontoExercise4.350p.235 3 x 2 +6 x SolutiontoExercise4.352p.235 6 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 a SolutiontoExercise4.354p.235 15 x 2 +12 x SolutiontoExercise4.356p.235 2 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b SolutiontoExercise4.358p.235 15 x 4 +12 x 2 SolutiontoExercise4.360p.235 20 a 7 +12 a 6 +8 a 5 SolutiontoExercise4.362p.235 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x 2 SolutiontoExercise4.364p.236 6 x 4 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 x 3 y SolutiontoExercise4.366p.236 2 b 6 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 b 5 x 2 SolutiontoExercise4.368p.236 18 y 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(27 y 6 +72 y 5 +9 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(54 y 3 SolutiontoExercise4.370p.236 a 2 +6 a +8

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268 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.372p.236 y 2 +3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 SolutiontoExercise4.374p.236 i 2 +2 i )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 SolutiontoExercise4.376p.236 6 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 a +6 SolutiontoExercise4.378p.236 18 y 2 +93 y +110 SolutiontoExercise4.380p.236 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x +12 SolutiontoExercise4.382p.236 x 3 + x 2 +2 x +2 SolutiontoExercise4.384p.236 6 x 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise4.386p.237 12 x 5 y 7 +30 x 3 y 5 +12 x 3 y 3 +30 xy SolutiontoExercise4.388p.237 4 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 SolutiontoExercise4.390p.237 x 3 +5 x 2 +4 x SolutiontoExercise4.392p.237 y 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y 4 +6 y 3 SolutiontoExercise4.394p.237 5 y 8 +40 y 7 +35 y 6 SolutiontoExercise4.396p.237 10 x 6 y 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x 5 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x 4 y 3 +3 x 3 y 2 SolutiontoExercise4.398p.237 8 c 3 +40 c +88 SolutiontoExercise4.400p.237 24 a 5 b 9 +42 a 4 b 11 +12 a 3 b 13 +18 a 3 b 3 SolutiontoExercise4.402p.237 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x +21 SolutiontoExercise4.404p.237 21 a 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(22 a 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise4.406p.237 10 a 3 +8 a 3 b +4 a 2 b 2 +5 a 2 b )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 ab SolutiontoExercise4.408p.237 x 2 +2 x +1 SolutiontoExercise4.410p.238 a 2 +4 a +4 SolutiontoExercise4.412p.238 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 x 2 +30 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 SolutiontoExercise4.414p.238 7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 SolutiontoExercise4.416p.238 2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x SolutiontoExercise4.418p.238 6 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(33 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 SolutiontoExercise4.420p.238 9 a 2 b +13 ab 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(ab

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269 SolutiontoExercise4.422p.238 17 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 SolutiontoExercise4.424p.238 12 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 SolutiontoExercise4.426p.238 7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 SolutiontoExercise4.428p.238 26 x +5 SolutiontoExercise4.430p.238 )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 SolutiontoExercise4.432p.239 0 SolutiontoExercise4.434p.239 1 : 98 10 5 SolutiontoExercise4.436p.239 degreeis 6;4 ; 3 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 SolutiontoExercise4.438p.242 x 2 +10 x +25 SolutiontoExercise4.439p.242 x 2 +14 x +49 SolutiontoExercise4.440p.242 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 y +36 SolutiontoExercise4.441p.242 9 a 2 +6 ab + b 2 SolutiontoExercise4.442p.242 81 m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 mn + n 2 SolutiontoExercise4.443p.242 100 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(40 xy +4 y 2 SolutiontoExercise4.444p.242 144 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(168 ab +49 b 2 SolutiontoExercise4.445p.242 25 h 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(150 hk +225 k 2 SolutiontoExercise4.446p.242 x 2 +6 x +9 SolutiontoExercise4.448p.242 x 2 +16 x +64 SolutiontoExercise4.450p.242 y 2 +18 y +81 SolutiontoExercise4.452p.242 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 a +16 SolutiontoExercise4.454p.242 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 a +49 SolutiontoExercise4.456p.242 b 2 +30 b +225 SolutiontoExercise4.458p.243 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 x +144 SolutiontoExercise4.460p.243 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(40 y +400 SolutiontoExercise4.462p.243 16 x 2 +16 x +4

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270 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.464p.243 49 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 x +4 SolutiontoExercise4.466p.243 9 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(54 a +81 SolutiontoExercise4.468p.243 25 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 ab +9 b 2 SolutiontoExercise4.470p.243 4 h 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(32 hk +64 k 2 SolutiontoExercise4.472p.243 a 2 + 2 3 a + 1 9 SolutiontoExercise4.474p.243 x 2 + 4 5 x + 4 25 SolutiontoExercise4.476p.243 y 2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(5 3 y + 25 36 SolutiontoExercise4.478p.243 x 2 +2 : 6 x +1 : 69 SolutiontoExercise4.480p.244 a 2 + a +0 : 25 SolutiontoExercise4.482p.244 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 : 2 x +9 : 61 SolutiontoExercise4.484p.244 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 : 08 b +0 : 0016 SolutiontoExercise4.486p.244 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 SolutiontoExercise4.488p.244 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise4.490p.244 f 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(81 SolutiontoExercise4.492p.244 4 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 SolutiontoExercise4.494p.244 4 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(49 b 2 SolutiontoExercise4.496p.244 25 h 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 k 2 SolutiontoExercise4.498p.244 a 2 )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(4 81 SolutiontoExercise4.500p.244 4 b 2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(36 49 SolutiontoExercise4.502p.245 a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b = a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ab )]TJ/F11 9.9626 Tf 9.963 0 Td [(ab + b 2 = a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 ab + b 2 SolutiontoExercise4.504p.245 rsttermsquared SolutiontoExercise4.506p.245 aSquaretherstterm. bSquarethesecondtermandsubtractitfromtherstterm. SolutiontoExercise4.508p.246 1 80 SolutiontoExercise4.510p.246 10 m 2 +9 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 SolutiontoExercise4.512p.247 allrealnumbers

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271 SolutiontoExercise4.513p.247 allrealnumbersexcept0 SolutiontoExercise4.514p.247 allrealnumbersexcept0 SolutiontoExercise4.515p.247 allrealnumbersexcept6 SolutiontoExercise4.516p.247 allrealnumbersexcept )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise4.517p.247 allrealnumbersgreaterthanorequalto0 SolutiontoExercise4.518p.247 allrealnumbers SolutiontoExercise4.519p.248 x = allrealnumbers SolutiontoExercise4.521p.248 x = allrealnumbers SolutiontoExercise4.523p.248 x = allrealnumbers SolutiontoExercise4.525p.248 x = allrealnumbersexceptzero SolutiontoExercise4.527p.248 h = allrealnumbersexceptzero SolutiontoExercise4.529p.248 s = allrealnumbersexcept6 SolutiontoExercise4.531p.248 commutativepropertyofmultiplication SolutiontoExercise4.533p.248 "howfar" SolutiontoExercise4.535p.248 16 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 SolutiontoExercise4.536p.250 three: 4 x 2 ; 7 x; 12 SolutiontoExercise4.538p.250 two: c; 8 SolutiontoExercise4.540p.251 a 2 SolutiontoExercise4.542p.251 12 y 3 SolutiontoExercise4.544p.251 2 a +2 b SolutiontoExercise4.546p.251 nocommonfactors SolutiontoExercise4.548p.251 12 SolutiontoExercise4.550p.251 2 a 2 b SolutiontoExercise4.552p.251 8 SolutiontoExercise4.554p.251 y +3

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272 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.556p.252 Thevalueof a isequaltothreetimethevalueof b SolutiontoExercise4.558p.252 Thevalueof f isequaltosixtimes g morethenonehalftimethevalueof m squared. SolutiontoExercise4.560p.252 Thevalueof P squaredisequaltothevalueof a cubedtimes k SolutiontoExercise4.562p.252 10 SolutiontoExercise4.564p.252 4.1515 10 12 SolutiontoExercise4.566p.252 7 50 or 0 : 14 SolutiontoExercise4.568p.252 trinomial;cubic;4,3,1 SolutiontoExercise4.570p.252 monomial;zero;147 SolutiontoExercise4.572p.252 binomial;tenth;9,10 SolutiontoExercise4.574p.253 ...becausethereisavariableinthedenominator SolutiontoExercise4.576p.253 linear SolutiontoExercise4.578p.253 linear SolutiontoExercise4.580p.253 quadratic SolutiontoExercise4.582p.253 22 x 2 y 3 +3 xy +6 SolutiontoExercise4.584p.253 11 y 2 +38 y +14 SolutiontoExercise4.586p.253 120 y 3 +86 y 2 +24 y +45 SolutiontoExercise4.588p.253 5 x 2 +2 x +3 SolutiontoExercise4.590p.253 60 c 2 +120 a +60 b SolutiontoExercise4.592p.254 36 m 4 +7 m 3 +4 m 2 +16 m SolutiontoExercise4.594p.254 3 a 2 b 2 c 2 +7 abc 2 + ab 2 c +12 abc SolutiontoExercise4.596p.254 6 a 6 +30 a 4 SolutiontoExercise4.598p.254 10 m 13 +15 m 10 +5 m 8 +5 m 7 +5 m 6 SolutiontoExercise4.600p.254 x 2 +5 x +6 SolutiontoExercise4.602p.254 a 2 +4 a +3 SolutiontoExercise4.604p.254 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 xy

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273 SolutiontoExercise4.606p.254 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 x 4 SolutiontoExercise4.608p.254 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 SolutiontoExercise4.610p.254 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 a 2 +18 a SolutiontoExercise4.612p.254 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 y 3 +15 y 2 +16 y +10 SolutiontoExercise4.614p.255 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise4.616p.255 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 a 2 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 q 2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 b 2 SolutiontoExercise4.618p.255 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a )]TJ/F8 9.9626 Tf 9.962 0 Td [(27 SolutiontoExercise4.620p.255 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise4.622p.255 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(29 SolutiontoExercise4.624p.255 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 x +24 SolutiontoExercise4.626p.255 16 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 bc )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 c 2 SolutiontoExercise4.628p.255 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a +9 SolutiontoExercise4.630p.255 36 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(48 x +16 SolutiontoExercise4.632p.255 x 2 +2 xy + y 2 SolutiontoExercise4.634p.255 m 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 SolutiontoExercise4.636p.255 9 c 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(100 SolutiontoExercise4.638p.256 25 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 b 2 SolutiontoExercise4.640p.256 2 y 2 +7 ay +3 a 2 SolutiontoExercise4.642p.256 x 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 SolutiontoExercise4.644p.256 48 y 2 +32 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(256 SolutiontoExercise4.646p.256 m 4 n +3 m 3 n 2 +2 m 2 n 3 SolutiontoExercise4.648p.256 3 p 5 +21 p 4 +63 p 3 +129 p 2 +84 p SolutiontoExercise4.650p.256 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +4 SolutiontoExercise4.652p.256 x 4 +2 x 2 y + y 2 SolutiontoExercise4.654p.256 9 x 4 y 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(24 x 6 y 4 +16 x 8 y 2

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274 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS SolutiontoExercise4.656p.257 allrealnumbers SolutiontoExercise4.658p.257 allrealnumbersexcept2 SolutiontoExercise4.660p.257 x canequalanyrealnumber; y canequalanynumberexcept )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 SolutiontoExercise4.661p.257 two: 3 a a +1 ; )]TJ/F8 9.9626 Tf 9.409 0 Td [( a +2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise4.662p.257 5 x 3 SolutiontoExercise4.663p.257 8 x b )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 SolutiontoExercise4.664p.257 8 y 3 z SolutiontoExercise4.665p.257 108 SolutiontoExercise4.666p.257 trinomial;5thdegree; numbericalcoecients:3,4,8 SolutiontoExercise4.667p.257 15 x 2 +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise4.668p.257 6 a 2 SolutiontoExercise4.669p.257 x 2 +6 x +8 SolutiontoExercise4.670p.257 6 a 2 +16 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(70 SolutiontoExercise4.671p.257 y 2 +6 y +9 SolutiontoExercise4.672p.258 36 a 2 +84 ay +49 y 2 SolutiontoExercise4.673p.258 16 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(72 xy +81 y 2 SolutiontoExercise4.674p.258 18 x 4 +51 x 3 +15 x 2 SolutiontoExercise4.675p.258 12 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(13 ab +3 b 2 SolutiontoExercise4.676p.258 )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 y 3 +24 y 2 SolutiontoExercise4.677p.258 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 b 7 +8 b 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 b 3 SolutiontoExercise4.678p.258 4 a 6 +12 a 3 b 2 +9 b 4 SolutiontoExercise4.679p.258 4 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(13 a +11 SolutiontoExercise4.680p.258 25 h 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 k 2 SolutiontoExercise4.681p.258 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a 2 +6 a +11 SolutiontoExercise4.682p.258 10 x +7

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275 SolutiontoExercise4.683p.258 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise4.684p.258 392 SolutiontoExercise4.685p.258 allrealnumbersexcept )]TJ/F8 9.9626 Tf 7.749 0 Td [(3

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276 CHAPTER4.ALGEBRAICEXPRESSIONSANDEQUATIONS

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Chapter5 SolvingLinearEquationsand Inequalities 5.1Objectives 1 Aftercompletingthischapter,youshould SolvingEquationsSection5.2 beabletoidentifyvarioustypesofequations understandthemeaningofsolutionsandequivalentequations beabletosolveequationsoftheform x + a = b and x )]TJ/F11 9.9626 Tf 9.963 0 Td [(a = b befamiliarwithandabletosolveliteralequation SolvingEquationsoftheForm ax= b and x a = b Section5.3 understandtheequalitypropertyofadditionandmultiplication beabletosolveequationsoftheform ax= b and x a = b FurtherTechniquesinEquationSolvingSection5.4 becomfortablewithcombiningtechniquesinequationsolving beabletorecognizeidentitiesandcontradictions ApplicationsI-TranslatingfromVerbaltoMathematicalExpressionsSection5.5 beabletotranslatefromverbaltomathematicalexpressions ApplicationsII-SolvingProblemsSection5.6 beabletosolvevariousappliedproblems LinearInequalitiesinOneVariableSection5.7 understandthemeaningofinequalities beabletorecognizelinearinequalities know,andbeabletoworkwith,thealgebraoflinearinequalitiesandwithcompoundinequalities LinearInequalitiesinTwoVariablesSection5.8 beabletoidentifythesolutionofalinearequationintwovariables knowthatsolutionstolinearequationsintwovariablescanbewrittenasorderedpairs 1 Thiscontentisavailableonlineat. 277

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278 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.2SolvingEquations 2 5.2.1Overview TypesofEquations SolutionsandEquivalentEquations LiteralEquations SolvingEquationsoftheForm x + a = b and x )]TJ/F11 9.9626 Tf 9.963 0 Td [(a = b 5.2.2TypesofEquations Identity Someequationsarealwaystrue.Theseequationsarecalledidentities. Identities areequationsthatare trueforallacceptablevaluesofthevariable,thatis,forallvaluesinthedomainoftheequation. 5 x =5 x istrueforallacceptablevaluesof x y +1= y +1 istrueforallacceptablevaluesof y 2+5=7 istrue,andnosubstitutionsarenecessary. Contradiction Someequationsarenevertrue.Theseequationsarecalledcontradictions. Contradictions areequations thatarenevertrueregardlessofthevaluesubstitutedforthevariable. x = x +1 isnevertrueforanyacceptablevalueof x 0 k =14 isnevertrueforanyacceptablevalueof k 2=1 isnevertrue. ConditionalEquation Thetruthofsomeequationsisconditionaluponthevaluechosenforthevariable.Suchequationsarecalled conditionalequations. Conditionalequations areequationsthataretrueforatleastonereplacementof thevariableandfalseforatleastonereplacementofthevariable. x +6=11 istrueonlyontheconditionthat x =5 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 istrueonlyontheconditionthat y =6 5.2.3SolutionsandEquivalentEquations SolutionsandSolvinganEquation Thecollectionofvaluesthatmakeanequationtruearecalled solutions oftheequation.Anequationis solved whenallitssolutionshavebeenfound. EquivalentEquations Someequationshavepreciselythesamecollectionofsolutions.Suchequationsarecalled equivalentequations .Theequations 2 x +1=7 ; 2 x =6 and x =3 areequivalentequationsbecausetheonlyvaluethatmakeseachonetrueis3. 5.2.4SampleSetA Tellwhyeachequationisanidentity,acontradiction,orconditional. Example5.1 Theequation x )]TJ/F8 9.9626 Tf 10.27 0 Td [(4=6 isaconditionalequationsinceitwillbetrueonlyontheconditionthat x =10 2 Thiscontentisavailableonlineat.

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279 Example5.2 Theequation x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2= x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 isanidentitysinceitistrueforallvaluesof x .Forexample, if x =5 ; 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2=5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 istrue x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 istrue Example5.3 Theequation a +5= a +1 isacontradictionsinceeveryvalueof a producesafalsestatement.For example, if a =8 ; 8+5=8+1 isfalse if a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+1 isfalse 5.2.5PracticeSetA Foreachofthefollowingequations,write"identity,""contradiction,"or"conditional."Ifyoucan,ndthe solutionbymakinganeducatedguessbasedonyourknowledgeofarithmetic. Exercise5.1 Solutiononp.338. x +1=10 Exercise5.2 Solutiononp.338. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=7 Exercise5.3 Solutiononp.338. 5 a =25 Exercise5.4 Solutiononp.338. x 4 =9 Exercise5.5 Solutiononp.338. 18 b =6 Exercise5.6 Solutiononp.338. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2= y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise5.7 Solutiononp.338. x +4= x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise5.8 Solutiononp.338. x + x + x =3 x Exercise5.9 Solutiononp.338. 8 x =0 Exercise5.10 Solutiononp.338. m )]TJ/F8 9.9626 Tf 9.963 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 5.2.6LiteralEquations LiteralEquations Someequationsinvolvemorethanonevariable.Suchequationsarecalled literalequations Anequationissolvedforaparticularvariableifthatvariablealoneequalsanexpressionthatdoesnot containthatparticularvariable. Thefollowingequationsareexamplesofliteralequations. 1. y =2 x +7 .Itissolvedfor y 2. d = rt .Itissolvedfor d 3. I = prt .Itissolvedfor I .

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280 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 4. z = x )]TJ/F10 6.9738 Tf 6.226 0 Td [(u s .Itissolvedfor z 5. y +1= x +4 .Thisequationisnotsolvedforanyparticularvariablesincenovariableisisolated. 5.2.7SolvingEquationoftheform x + a = b and x )]TJ/F24 11.9552 Tf 11.955 0 Td [(a = b Recallthattheequalsignofanequationindicatesthatthenumberrepresentedbytheexpressionontheleft sideisthesameasthenumberrepresentedbytheexpressionontherightside. Thisisthethis numbersameasnumber ### x =6 x +2=8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=5 Thissuggeststhefollowingprocedures: 1.Wecanobtainanequivalentequationanequationhavingthesamesolutionsastheoriginalequation by adding the samenumber to bothsides oftheequation. 2.Wecanobtainanequivalentequationby subtracting the samenumber from bothsides ofthe equation. Wecanusetheseresultstoisolate x ,thussolvingfor x Example5.4:Solving x + a = b for x x + a = b The a isassociatedwith x byaddition.Undotheassociation x + a )]TJ/F11 9.9626 Tf 9.963 0 Td [(a = b )]TJ/F11 9.9626 Tf 9.963 0 Td [(a bysubtracting a from both sides. x +0= b )]TJ/F11 9.9626 Tf 9.963 0 Td [(aa )]TJ/F11 9.9626 Tf 9.962 0 Td [(a =0 and 0 istheadditiveidentity. x +0= x: x = b )]TJ/F11 9.9626 Tf 9.963 0 Td [(a Thisequationisequivalenttotherstequation,anditis solvedfor x: Example5.5:Solving x )]TJ/F11 9.9626 Tf 9.962 0 Td [(a = b for x x )]TJ/F11 9.9626 Tf 9.963 0 Td [(a = b The a isassociatedwith x bysubtraction.Undotheassociation x )]TJ/F11 9.9626 Tf 9.962 0 Td [(a + a = b + a byadding a to both sides. x +0= b + a )]TJ/F11 9.9626 Tf 7.749 0 Td [(a + a =0 and 0 istheadditiveidentity. x +0= x: x = b + a Thisequationisequivalenttotherstequation,anditis solvedfor x: Example5.6:MethodforSolving x + a = b and x )]TJ/F11 9.9626 Tf 9.962 0 Td [(a = b for x Tosolvetheequation x + a = b for x subtract a from both sidesoftheequation. Tosolvetheequation x )]TJ/F11 9.9626 Tf 9.963 0 Td [(a = b for x add a to both sidesoftheequation. 5.2.8SampleSetB Example5.7 Solve x +7=10 for x .

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281 x +7=107 isassociatedwith x byaddition.Undotheassociation x +7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=10 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 bysubtracting 7 from both sides. x +0=37 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=0 and 0 istheadditiveidentity :x +0= x: x =3 x isisolated,andtheequation x =3 isequivalenttothe originalequation x +7=10 : Therefore,thesetwo equationhavethesamesolution.Thesolutionto x =3 isclearly 3 : Thus,thesolutionto x +7=10 isalso 3 : Check :Substitute3for x intheoriginalequation. x +7 = 10 3+7 = 10 Isthiscorrect? 10 = 10 Yes,thisiscorrect. Example5.8 Solve m )]TJ/F8 9.9626 Tf 9.962 0 Td [(2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 for m m )]TJ/F8 9.9626 Tf 9.962 0 Td [(2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(92 isassociatedwith m bysubtraction.Undotheassociation m )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(9+2 byadding 2 from both sides. m +0= )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+2=0 and 0 istheadditiveidentity :m +0= m: m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 Check :Substitute )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 for m intheoriginalequation. m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Yes,thisiscorrect. Example5.9 Solve y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : 181= )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 : 915 for y y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : 181= )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 : 915 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : 181+2 : 181= )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 : 915+2 : 181 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 : 734 OntheCalculator Type 16 : 915 Press + = )]TJETq1 0 0 1 197.703 170.552 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQq1 0 0 1 166.664 170.353 cm[]0 d 0 J 0.398 w 0 0 m 31.238 0 l SQBT/F15 9.9626 Tf 96.907 158.438 Td [(Press + Type 2 : 181 Press = Displayreads: )]TJ/F8 9.9626 Tf 7.748 0 Td [(14 : 734 Example5.10 Solve y + m = s for y .

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282 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES y + m = sm isassociatedwith y byaddition.Undotheassociation y + m )]TJ/F11 9.9626 Tf 9.963 0 Td [(m = s )]TJ/F11 9.9626 Tf 9.963 0 Td [(m bysubtracting m from both sides. y +0= s )]TJ/F11 9.9626 Tf 9.963 0 Td [(mm )]TJ/F11 9.9626 Tf 9.962 0 Td [(m =0 and 0 istheadditiveidentity :y +0= y: y = s )]TJ/F11 9.9626 Tf 9.963 0 Td [(m Check :Substitute s )]TJ/F11 9.9626 Tf 9.962 0 Td [(m for y intheoriginalequation. y + m = s s )]TJ/F11 9.9626 Tf 9.962 0 Td [(m + m = s Isthiscorrect? s = s TrueYes,thisiscorrect. Example5.11 Solve k )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 h = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 h +5 for k k )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 h = )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 h +53 h isassociatedwith k bysubtraction.Undotheassociation k )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 h +3 h = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 h +5+3 h byadding 3 h to both sides. k +0= )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 h +5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 h +3 h =0 and 0 istheadditiveidentity :k +0= k: k = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 h +5 5.2.9PracticeSetB Exercise5.11 Solutiononp.338. Solve y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3=8 for y: Exercise5.12 Solutiononp.338. Solve x +9= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 for x: Exercise5.13 Solutiononp.338. Solve m +6=0 for m: Exercise5.14 Solutiononp.338. Solve g )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 : 2=1 : 3 for g: Exercise5.15 Solutiononp.338. solve f +2 d =5 d for f: Exercise5.16 Solutiononp.338. Solve x +8 y =2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 for x: Exercise5.17 Solutiononp.338. Solve y +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=5 x +8 for y: 5.2.10Exercises Forthefollowingproblems,classifyeachoftheequationsasanidentity,contradiction,orconditionalequation. Exercise5.18 Solutiononp.338. m +6=15 Exercise5.19 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(8= )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise5.20 Solutiononp.338. x +1= x +1

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283 Exercise5.21 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(2= k )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise5.22 Solutiononp.338. g + g + g + g =4 g Exercise5.23 x +1=0 Forthefollowingproblems,determinewhichoftheliteralequationshavebeensolvedforavariable.Write "solved"or"notsolved." Exercise5.24 Solutiononp.338. y =3 x +7 Exercise5.25 m =2 k + n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise5.26 Solutiononp.338. 4 a = y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise5.27 hk =2 k + h Exercise5.28 Solutiononp.338. 2 a = a +1 Exercise5.29 5 m =2 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise5.30 Solutiononp.338. m = m Forthefollowingproblems,solveeachoftheconditionalequations. Exercise5.31 h )]TJ/F8 9.9626 Tf 9.963 0 Td [(8=14 Exercise5.32 Solutiononp.338. k +10=1 Exercise5.33 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=5 Exercise5.34 Solutiononp.338. y +6= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Exercise5.35 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(8= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.36 Solutiononp.339. x +14=0 Exercise5.37 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(12=0 Exercise5.38 Solutiononp.339. g +164= )]TJ/F8 9.9626 Tf 7.749 0 Td [(123 Exercise5.39 h )]TJ/F8 9.9626 Tf 9.963 0 Td [(265= )]TJ/F8 9.9626 Tf 7.749 0 Td [(547 Exercise5.40 Solutiononp.339. x +17= )]TJ/F8 9.9626 Tf 7.748 0 Td [(426 Exercise5.41 h )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 : 82= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 : 56

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284 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.42 Solutiononp.339. y +17 : 003= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 056 Exercise5.43 k +1 : 0135= )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 : 0032 Exercise5.44 Solutiononp.339. Solve n + m =4 for n: Exercise5.45 Solve P +3 Q )]TJ/F8 9.9626 Tf 9.963 0 Td [(8=0 for P: Exercise5.46 Solutiononp.339. Solve a + b )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 c = d )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 f for b: Exercise5.47 Solve x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y +5 z +1=2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 z +8 for x: Exercise5.48 Solutiononp.339. Solve 4 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 b + c +11=6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 b for c: 5.2.11ExercisesforReview Exercise5.49 Section2.7 Simplify )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 x 5 y 2 3 Exercise5.50 Solutiononp.339. Section3.7 Write 20 x 3 y 7 5 x 5 y 3 sothatonlypositiveexponentsappear. Exercise5.51 Section4.2 Writethenumberoftermsthatappearintheexpression 5 x 2 +2 x )]TJ/F8 9.9626 Tf 10.221 0 Td [(6+ a + b andthenlistthem. Exercise5.52 Solutiononp.339. Section4.7 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 Exercise5.53 Section4.8 Specifythedomainoftheequation y = 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 5.3SolvingEquationsoftheFormax=bandx/a=b 3 5.3.1Overview EqualityPropertyofDivisionandMultiplication Solving ax = b and x a = b for x 3 Thiscontentisavailableonlineat.

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285 5.3.2EqualityPropertyofDivisionandMultiplication Recallingthattheequalsignofanequationindicatesthatthenumberrepresentedbytheexpressiononthe leftsideisthesameasthenumberrepresentedbytheexpressionontherightsidesuggeststheequality propertyofdivisionandmultiplication,whichstates: 1.Wecanobtainanequivalentequationby dividingbothsides oftheequationbythesamenonzero number,thatis,if c 6 =0 ; then a = b isequivalentto a c = b c 2.Wecanobtainanequivalentequationby multiplyingbothsides oftheequationbythesamenonzero number,thatis,if c 6 =0 ; then a = b isequivalentto ac = bc Wecanusetheseresultstoisolate x; thussolvingtheequationfor x Example5.12 Solving ax = b for x ax = b a isassociatedwith x bymultiplication. Undotheassociationbydividingbothsidesby a ax a = b a ax a = b a 1 x = b a a a =1 and 1 isthemultiplicativeidentity.1 x = x Example5.13 Solving x a = b for x x = b a Thisequationisequivalenttotherstandissolvedby x x a = b a isassociatedwith x bydivision.Undotheassociation bymultiplyingbothsidesby a a x a = a b a x a = ab 1 x = ab a a =1 and 1 isthemultiplicativeidentity.1 x = x x = ab Thisequationisequivalenttotherstandissolvedfor x 5.3.3Solving ax = b and x a = b for x Example5.14 MethodforSolving ax = b and x a = b Tosolve ax = b for x dividebothsides oftheequationby a Tosolve x a = b for x multiplybothsides oftheequationby a 5.3.4SampleSetA Example5.15 Solve 5 x =35 for x .

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286 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5 x =35 5 isassociatedwithxbymultiplication.Undothe associationbydividingbothsidesby5. 5 x 5 = 35 5 x =7 1 x =7 5 5 =1 and1ismultiplicativeidentity.1 x = x: x =7 Check :5=35 Isthiscorrect? 35=35 Yes,thisiscorrect. Example5.16 Solve x 4 =5 for x x 4 =54 isasssociatedwith x bydivision.Undotheassociationby multiplying both sidesby 4 : 4 x 4 =4 5 x =4 5 1 x =20 4 4 =1 and 1 isthemultiplicativeidentity : 1 x = x: x =20 Check : 20 4 =5 Isthiscorrect? 5=5 Yes,thisiscorrect. Example5.17 Solve 2 y 9 =3 for y MethodUseofcancelling: 2 y 9 =39 isassociatedwith y bydivision.Undotheassociationby multiplying both sidesby 9 : 2 y = 2 y =272 isassociatedwith y bymultiplication.Undothe associationbydividing both sidesby 2 : y = 27 2 y = 27 2 Check : 27 9 =3 Isthiscorrect? 27 9 =3 Isthiscorrect? 3=3 Yes,thisiscorrect. MethodUseofreciprocals:

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287 2 y 9 =3 Since 2 y 9 = 2 9 y; 2 9 isassociatedwith y bymultiplication. Then,Since 9 2 2 9 =1 ; themultiplicativeidentity,wecan )]TJ/F7 6.9738 Tf 5.761 -4.147 Td [(9 2 )]TJ/F7 6.9738 Tf 17.035 -3.625 Td [(2 y 9 = )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(9 2 undotheassociativebymultiplying both sidesby 9 2 : )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(9 2 2 9 y = 27 2 1 y = 27 2 y = 27 2 Example5.18 Solvetheliteralequation 4 ax m =3 b for x 4 ax m =3 bm isassociatedwith x bydivision.Undotheassociationby multiplying both sidesby m m 4 ax m = m 3 b 4 ax =3 bm 4 a isassociatedwith x bymultiplication.Undothe associationbymultiplying both sidesby 4 a: ax a = 3 bm 4 a x = 3 bm 4 a Check : 4 a 3 bm 4 a m =3 b Isthiscorrect? a 3 bm a m =3 b Isthiscorrect? 3 b m m =3 b Isthiscorrect? 3 b =3 b Yes,thisiscorrect. 5.3.5PracticeSetA Exercise5.54 Solutiononp.339. Solve 6 a =42 for a Exercise5.55 Solutiononp.339. Solve )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 m =16 for m Exercise5.56 Solutiononp.339. Solve y 8 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 for y Exercise5.57 Solutiononp.339. Solve 6 : 42 x =1 : 09 for x Roundtheresulttotwodecimalplaces. Exercise5.58 Solutiononp.339. Solve 5 k 12 =2 for k Exercise5.59 Solutiononp.339. Solve )]TJ/F10 6.9738 Tf 6.227 0 Td [(ab 2 c =4 d for b Exercise5.60 Solutiononp.339. Solve 3 xy 4 =9 xh for y Exercise5.61 Solutiononp.339. Solve 2 k 2 mn 5 pq = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 n for m .

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288 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.3.6Exercises Inthefollowingproblems,solveeachoftheconditionalequations. Exercise5.62 Solutiononp.339. 3 x =42 Exercise5.63 5 y =75 Exercise5.64 Solutiononp.339. 6 x =48 Exercise5.65 8 x =56 Exercise5.66 Solutiononp.339. 4 x =56 Exercise5.67 3 x =93 Exercise5.68 Solutiononp.339. 5 a = )]TJ/F8 9.9626 Tf 9.963 0 Td [(80 Exercise5.69 9 m = )]TJ/F8 9.9626 Tf 9.962 0 Td [(108 Exercise5.70 Solutiononp.339. 6 p = )]TJ/F8 9.9626 Tf 9.962 0 Td [(108 Exercise5.71 12 q = )]TJ/F8 9.9626 Tf 9.962 0 Td [(180 Exercise5.72 Solutiononp.339. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a =16 Exercise5.73 )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 x =100 Exercise5.74 Solutiononp.339. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x = )]TJ/F8 9.9626 Tf 9.963 0 Td [(42 Exercise5.75 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 m = )]TJ/F8 9.9626 Tf 9.963 0 Td [(40 Exercise5.76 Solutiononp.339. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 k =126 Exercise5.77 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 y =126 Exercise5.78 Solutiononp.340. x 6 =1 Exercise5.79 a 5 =6 Exercise5.80 Solutiononp.340. k 7 =6 Exercise5.81 x 3 =72 Exercise5.82 Solutiononp.340. x 8 =96

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289 Exercise5.83 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise5.84 Solutiononp.340. m 7 = )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise5.85 k 18 =47 Exercise5.86 Solutiononp.340. f )]TJ/F7 6.9738 Tf 6.227 0 Td [(62 =103 Exercise5.87 3 : 06 m =12 : 546 Exercise5.88 Solutiononp.340. 5 : 012 k =0 : 30072 Exercise5.89 x 2 : 19 =5 Exercise5.90 Solutiononp.340. y 4 : 11 =2 : 3 Exercise5.91 4 y 7 =2 Exercise5.92 Solutiononp.340. 3 m 10 = )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise5.93 5 k 6 =8 Exercise5.94 Solutiononp.340. 8 h )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise5.95 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 z 21 = )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise5.96 Solutiononp.340. Solve pq =7 r for p Exercise5.97 Solve m 2 n =2 s for n Exercise5.98 Solutiononp.340. Solve 2 : 8 ab =5 : 6 d for b Exercise5.99 Solve mnp 2 k =4 k for p Exercise5.100 Solutiononp.340. Solve )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 a 2 b 3 c = )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a 2 for b Exercise5.101 Solve 3 pcb 2 m =2 b for pc Exercise5.102 Solutiononp.340. Solve 8 rst 3 p = )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 prs for t Exercise5.103 Solve for Exercise5.104 Solutiononp.340. Solve 3 r 2 r = r for .

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290 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.3.7ExercisesforReview Exercise5.105 Section2.7 Simplify 2 x 0 y 0 z 3 z 2 5 Exercise5.106 Solutiononp.340. Section4.4 Classify 10 x 3 )]TJ/F8 9.9626 Tf 12.09 0 Td [(7 x asamonomial,binomial,ortrinomial.Stateitsdegreeand writethenumericalcoecientofeachitem. Exercise5.107 Section4.5 Simplify 3 a 2 )]TJ/F8 9.9626 Tf 11.655 0 Td [(2 a +4 a a +2 Exercise5.108 Solutiononp.340. Section4.8 Specifythedomainoftheequation y = 3 7+ x Exercise5.109 Section5.2 Solvetheconditionalequation x +6= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 5.4FurtherTechniquesinEquationSolving 4 5.4.1Overview CombiningTechniquesinEquationSolving RecognizingIdentitiesandContrdictions 5.4.2CombiningTechniquesinEquationSolving InSectionsSection5.2andSection5.3weworkedwithtechniquesthatinvolvedtheuseofaddition,subtraction,multiplication,anddivisiontosolveequations.Wecancombinethesetechniquestosolvemore complicatedequations.Todoso,itishelpfultorecallthatanequationissolvedforaparticularvariable whenallothernumbersand/orlettershavebeendisassociatedfromitanditisaloneononesideoftheequal sign.Wewillalsonotethat Toassociatenumbersandlettersweusetheorderofoperations. 1.Multiply/divide 2.Add/subtract Toundoanassociationbetweennumbersandlettersweusetheorderofoperationsinreverse. 1.Add/subtract 2.Multiply/divide 4 Thiscontentisavailableonlineat.

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291 5.4.3SampleSetA Example5.19 Solve 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=9 for x: 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=9 First,undotheassociationbetween x and 7 : The 7 isassociatedwith x bysubtraction. Undotheassociationbyadding 7 to both sides. 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7+7=9+7 4 x =16 Now,undotheassociationbetween x and 4 : The 4 isassociatedwith x bymultiplication : Undotheassociationbydividing both sidesby 4 : 4 x 4 = 16 4 16 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=9 Isthiscorrect? x =4 Check :4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=9 Isthiscorrect? 9=9 Yes,thisiscorrect. Example5.20 Solve 3 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 : 3 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 isassociatedwith y bysubtraction. Undotheassociationbyadding 5 to both sides. 3 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5+5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11+5 3 y 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(64 isassociatedwith y bydivision. Undotheassociationbymultiplying both sidesby 4 : 4 3 y 4 =4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 4 3 y 4 =4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 3 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(243 isassociatedwith y bymultiplication. Undotheassociationbydividing both sidesby 3 : 3 y 3 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(24 3 3 y 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Check : 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Isthiscorrect? )]TJ/F7 6.9738 Tf 6.226 0 Td [(24 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(11= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Yes,thisiscorrect. Example5.21 Solve 8 a 3 b +2 m =6 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 for a:

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292 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 8 a 3 b +2 m =6 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(52 m isassociatedwith a byaddition.Undotheassociation bysubtracting 2 m from both sides : 8 a 3 b +2 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m =6 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m 8 a 3 b =4 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(53 bassociatedwitha bydivision.Undotheassociation bymultiplying both sidesby 3 b: b )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(8 a 3 b =3 b m )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 8 a =12 bm )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 b 8 isassociatedwith a bymultiplication.Undothe multiplicationbydividing both sidesby 8 : 8 a 8 = 12 bm )]TJ/F7 6.9738 Tf 6.226 0 Td [(15 b 8 a = 12 bm )]TJ/F7 6.9738 Tf 6.226 0 Td [(15 b 8 5.4.4PracticeSetA Exercise5.110 Solutiononp.340. Solve 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=11 for y: Exercise5.111 Solutiononp.340. Solve 5 m 2 +6=1 for m: Exercise5.112 Solutiononp.340. Solve 2 n +3 m =4 for n: Exercise5.113 Solutiononp.340. Solve 9 k 2 h +5= p )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 for k: Sometimeswhensolvinganequationitisnecessarytosimplifytheexpressionscomposingit. 5.4.5SampleSetB Example5.22 Solve 4 x +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 for x: 4 x +1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Check :4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(9+1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(9= )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.748 0 Td [(36+1+27= )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(8= )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Yes,thisiscorrect. Example5.23 Solve 3 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4+1 for m: 3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4+1 3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(18= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 m =15

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293 Check :3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4+1 Isthiscorrect? 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Isthiscorrect? 27 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(3= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Yes,thisiscorrect. 5.4.6PracticeSetB Solveandcheckeachequation. Exercise5.114 Solutiononp.340. 16 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 x =8 for x: Exercise5.115 Solutiononp.340. 4 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 for y: Exercise5.116 Solutiononp.340. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 +3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 +2 a 2 +7 a =0 for a: Exercise5.117 Solutiononp.340. 5 m m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 m 2 +2 a m +3=10 for a: Oftenthevariablewewishtosolveforwillappearonbothsidesoftheequalsign.Wecanisolatethevariable oneithertheleftorrightsideoftheequationbyusingthetechniquesofSectionsSection5.2andSection5.3. 5.4.7SampleSetC Example5.24 Solve 6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4=2 x +8 for x: 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=2 x +8 Toisolate x ontheleftside,subtract 2 m frombothsides : 6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x =2 x +8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=8 Add 4 tobothsides : 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4+4=8+4 4 x =12 Dividebothsidesby 4 : 4 x 4 = 12 4 x =3 Check :6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=2+8 Isthiscorrect? 18 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=6+8 Isthiscorrect? 14=14 Yes,thisiscorrect. Example5.25 Solve 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +1=2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [([3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20] for x:

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294 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 x +1=2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [([3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20] )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 x +7=2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [([3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(41] )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 x +7=2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +41 )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 x +7= )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +41 Toisolate x ontherightside,add 18 x tobothsides : )]TJ/F8 9.9626 Tf 7.748 0 Td [(18 x +7+18 x = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +41+18 x 7=17 x +41 Subtract 41 frombothsides : 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(41=17 x +41 )]TJ/F8 9.9626 Tf 9.963 0 Td [(41 )]TJ/F8 9.9626 Tf 7.749 0 Td [(34=17 x Dividebothsidesby 17 : )]TJ/F7 6.9738 Tf 6.227 0 Td [(34 17 = 17 x 17 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2= x Sincetheequation )]TJ/F8 9.9626 Tf 9.962 0 Td [(2= x isequivalenttotheequation x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; wecanwritetheansweras x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 : x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Check :6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2+1=2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 9.962 0 Td [([3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20] Isthiscorrect? 6+6+1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [([3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20] Isthiscorrect? 6+1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [([ )]TJ/F8 9.9626 Tf 7.749 0 Td [(27 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20] Isthiscorrect? 42+1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [([ )]TJ/F8 9.9626 Tf 7.749 0 Td [(47] Isthiscorrect? 43= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4+47 Isthiscorrect? 43=43 Yes,thisiscorrect. 5.4.8PracticeSetC Exercise5.118 Solutiononp.340. Solve 8 a +5=3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 for a: Exercise5.119 Solutiononp.341. Solve 9 y +3 y +6=15 y +21 for y: Exercise5.120 Solutiononp.341. Solve 3 k +2[4 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+3]=63 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 k for k: 5.4.9RecognizingIdentitiesandContradictions AswenotedinSectionSection5.2,someequationsareidentitiesandsomearecontradictions.Asthe problemsofSampleSetDwillsuggest, RecognizinganIdentity 1.If,whensolvinganequation,allthevariablesareeliminatedandatruestatementresults,theequation isan identity. RecognizingaContradiction 2.If,whensolvinganequation,allthevariablesareeliminatedandafalsestatementresults,theequation isa contradiction.

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295 5.4.10SampleSetD Example5.26 Solve 9 x +3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x =12 for x: 9 x +12 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 x =12 12=12 Thevariablehasbeeneliminatedandtheresultisatruestatement.Theoriginalequationis an identity. Example5.27 Solve )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y +1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 for y: )]TJ/F8 9.9626 Tf 7.748 0 Td [(20+4 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 y +1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 )]TJ/F8 9.9626 Tf 7.749 0 Td [(19= )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 Thevariablehasbeeneliminatedandtheresultisafalsestatement.Theoriginalequationisa contradiction. 5.4.11PracticeSetD Classifyeachequationasanidentityoracontradiction. Exercise5.121 Solutiononp.341. 6 x +3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x =3 Exercise5.122 Solutiononp.341. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 m +4 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=28 Exercise5.123 Solutiononp.341. 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x +1+14=0 Exercise5.124 Solutiononp.341. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x +6+8=3[4 )]TJ/F8 9.9626 Tf 9.962 0 Td [( x +2] )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 5.4.12Exercises Forthefollowingproblems,solveeachconditionalequation.Iftheequationisnotconditional,identifyitas anidentityoracontradiction. Exercise5.125 Solutiononp.341. 3 x +1=16 Exercise5.126 6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=20 Exercise5.127 Solutiononp.341. 4 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=27 Exercise5.128 3 x +4=40 Exercise5.129 Solutiononp.341. 2 y +7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3

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296 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.130 8 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(23 Exercise5.131 Solutiononp.341. 5 x +6= )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise5.132 7 a +2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(26 Exercise5.133 Solutiononp.341. 10 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3= )]TJ/F8 9.9626 Tf 7.749 0 Td [(23 Exercise5.134 14 x +1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(55 Exercise5.135 Solutiononp.341. x 9 +2=6 Exercise5.136 m 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8= )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Exercise5.137 Solutiononp.341. y 4 +6=12 Exercise5.138 x 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=5 Exercise5.139 Solutiononp.341. m 11 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15= )]TJ/F8 9.9626 Tf 7.749 0 Td [(19 Exercise5.140 k 15 +20=10 Exercise5.141 Solutiononp.341. 6+ k 5 =5 Exercise5.142 1 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(n 2 =6 Exercise5.143 Solutiononp.341. 7 x 4 +6= )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 Exercise5.144 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 m 5 +11= )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 Exercise5.145 Solutiononp.341. 3 k 14 +25=22 Exercise5.146 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6+5= )]TJ/F8 9.9626 Tf 7.748 0 Td [(25 Exercise5.147 Solutiononp.341. 16 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+11= )]TJ/F8 9.9626 Tf 7.749 0 Td [(85 Exercise5.148 6 x +14=5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise5.149 Solutiononp.341. 23 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(19=22 y +1 Exercise5.150 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 m +1=3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise5.151 Solutiononp.341. 8 k +7=2 k +1 Exercise5.152 12 n +5=5 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(16

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297 Exercise5.153 Solutiononp.341. 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=2 x +5 Exercise5.154 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y +3+5+4 y =0 Exercise5.155 Solutiononp.341. 3 x +7= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [( x +2 Exercise5.156 4 y +2=3 y +2[1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y ] Exercise5.157 Solutiononp.341. 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8+11=2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise5.158 12 )]TJ/F8 9.9626 Tf 9.962 0 Td [( m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=2 m +3 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m +3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m Exercise5.159 Solutiononp.341. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 k )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 k = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 k )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 k +1 Exercise5.160 3[4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y +2]=2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4[1+2+ y ] Exercise5.161 Solutiononp.341. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5[2 m )]TJ/F8 9.9626 Tf 9.962 0 Td [( m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1]=4 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 m +2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 m +1 Forthefollowingproblems,solvetheliteralequationsfortheindicatedvariable.Whendirected,ndthe valueofthatvariableforthegivenvaluesoftheothervariables. Exercise5.162 Solve I = E R for R: Findthevalueof R when I =0 : 005 and E =0 : 0035 : Exercise5.163 Solutiononp.342. Solve P = R )]TJ/F11 9.9626 Tf 9.963 0 Td [(C for R: Findthevalueof R when P =27 and C =85 : Exercise5.164 Solve z = x )]TJETq1 0 0 1 151.547 353.448 cm[]0 d 0 J 0.339 w 0 0 m 4.518 0 l SQBT/F10 6.9738 Tf 151.547 349.26 Td [(x s for x: Findthevalueof x when z =1 : 96 ;s =2 : 5 ; and x =15 : Exercise5.165 Solutiononp.342. Solve F = S 2 x S 2 y for S 2 x S 2 x representsasinglequantity.Findthevalueof S 2 x when F =2 : 21 and S 2 y =3 : 24 : Exercise5.166 Solve p = nRT V for R: Exercise5.167 Solutiononp.342. Solve x =4 y +7 for y: Exercise5.168 Solve y =10 x +16 for x: Exercise5.169 Solutiononp.342. Solve 2 x +5 y =12 for y: Exercise5.170 Solve )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 x +3 y +15=0 for y: Exercise5.171 Solutiononp.342. Solve m = 2 n )]TJ/F10 6.9738 Tf 6.227 0 Td [(h 5 for n: Exercise5.172 Solve t = Q +6 P 8 for P: Exercise5.173 Solutiononp.342. Solve = +9 j for j .

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298 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.174 Solve for 5.4.13ExercisesforReview Exercise5.175 Solutiononp.342. Section2.6 Simplify x +3 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 4 x +3 : Exercise5.176 Section4.7 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +7 : Exercise5.177 Solutiononp.342. Section4.7 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 : Exercise5.178 Section5.2 Solvetheequation y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 : Exercise5.179 Solutiononp.342. Section5.3 Solvetheequation 4 x 5 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 : 5.5ApplicationI-TranslatingfromVerbaltoMatheticalExpressions 5 5.5.1Overview TranslatingfromVerbaltoMathematicalExpressions 5.5.2TranslatingfromVerbaltoMathematicalExpressions Tosolveaproblemusingalgebra,wemustrstexpresstheproblemalgebraically.Toexpressaproblem algebraically,wemustscrutinizethewordingoftheproblemtodeterminethevariablesandconstantsthat arepresentandtherelationshipsamongthem.Thenwemusttranslatetheverbalphrasesandstatements toalgebraicexpressionsandequations. Tohelpustranslateverbalexpressionstomathematics,wecanusethefollowingtableasamathematics dictionary. MathematicsDictionary WordorPhrase MathematicalOperation Sum,sumof,addedto,increasedby,morethan,plus,and + Dierence,minus,subtractedfrom,decreasedby,less,lessthan )]TJETq1 0 0 1 518.602 158.044 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQq1 0 0 1 93.198 157.845 cm[]0 d 0 J 0.398 w 0 0 m 289.784 0 l SQq1 0 0 1 382.983 157.845 cm[]0 d 0 J 0.398 w 0 0 m 135.819 0 l SQq1 0 0 1 93.398 141.307 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQBT/F15 9.9626 Tf 99.575 146.328 Td [(Product,theproductof,of,muitipliedby,times Quotient,dividedby,ratio Equals,isequalto,is,theresultis,becomes = Anumber,anunknownquantity,anunknown,aquantity x oranysymbol 5 Thiscontentisavailableonlineat.

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299 Table5.1 5.5.3SampleSetA Translatethefollowingphrasesorsentencesintomathematicalexpressionsorequations. Example5.28 six |{z} 6 morethan | {z } + anumber | {z } x | {z } 6+ x : Example5.29 Fifteen | {z } 15 minus | {z } )]TJ/F15 9.9626 Tf 17.699 13.151 Td [(anumber | {z } x | {z } 15 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x : Example5.30 Aquantity | {z } y less |{z} )]TJ/F15 9.9626 Tf 13.773 13.15 Td [(eight | {z } 8 | {z } y )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 : Example5.31 Twice | {z } 2 anumber | {z } x is |{z} = ten. |{z} 10 | {z } 2 x =10 Example5.32 Onehalf | {z } 1 2 of |{z} anumber | {z } z is |{z} = twenty. | {z } 20 | {z } 1 2 z =20 Example5.33 Three | {z } 3 times | {z } anumber | {z } y is |{z} = ve |{z} 5 morethan | {z } + twice | {z } 2 thesamenumber. | {z } y | {z } 3 y =5+2 y

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300 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.5.4PracticeSetA Translatethefollowingphrasesorsentencesintomathematicalexpressionsorequations. Exercise5.180 Solutiononp.342. Elevenmorethananumber. Exercise5.181 Solutiononp.342. Nineminusanumber. Exercise5.182 Solutiononp.342. Aquantitylesstwenty. Exercise5.183 Solutiononp.342. Fourtimesanumberisthirtytwo. Exercise5.184 Solutiononp.342. Onethirdofanumberissix. Exercise5.185 Solutiononp.342. Tentimesanumberiseightmorethanvetimesthesamenumber. Sometimesthestructureofthesentenceindicatestheuseofgroupingsymbols. 5.5.5SampleSetB Translatethefollowingphrasesorsentencesintomathematicalexpressionsorequations. Example5.34 Anumberdividedbyve, | {z } x 5 minus | {z } )]TJ/F15 9.9626 Tf 18.366 13.151 Td [(ten, |{z} 10 is |{z} = fteen. | {z } 15 | {z } x 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10=15 Commassetoterms. Example5.35 Eight | {z } 8 dividedby | {z } vemorethananumber | {z } + x is |{z} = ten |{z} 10 | {z } Thewordingindicatesthisistobeconsideredasasinglequantity. 8 5+ x =10 Example5.36 Anumber | {z } x multipliedby | {z } tenmorethanitself | {z } + x is |{z} = twenty. | {z } 20 | {z } x + x =20 Example5.37 Anumberplusoneisdividedbythreetimesthenumberminustwelveandtheresultisfour. x +1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12=4 x +1 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 =4 Noticethatsincethephrase"threetimesthenumberminustwelve"doesnotcontainacomma,we

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301 gettheexpression 3 x )]TJ/F8 9.9626 Tf 9.511 0 Td [(12 .Ifthephrasehadappearedas"threetimesthenumber,minustwelve," theresultwouldhavebeen x +1 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12=4 Example5.38 Somephrasesandsentencesdonottranslatedirectly.Wemustbecarefultoreadthemproperly. Theword from oftenappearsinsuchphrasesandsentences.Theword from means"apointof departureformotion."Thefollowingtranslationwillillustratethisuse. Theword from indicatesthemotionsubtractionistobeginatthepointof"somequantity." Example5.39 Eightlessthansomequantity.Noticethat lessthan couldbereplacedwith from x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 5.5.6PracticeSetB Translatethefollowingphrasesandsentencesintomathematicalexpressionsorequations. Exercise5.186 Solutiononp.342. Anumberdividedbysixteen,plusone,isve. Exercise5.187 Solutiononp.342. Seventimestwomorethananumberistwenty-one. Exercise5.188 Solutiononp.342. Anumberdividedbytwomorethanitselfiszero. Exercise5.189 Solutiononp.342. Anumberminusveisdividedbytwicethenumberplusthreeandtheresultisseventeen. Exercise5.190 Solutiononp.342. Fifty-twoissubtractedfromsomequantity. Exercise5.191 Solutiononp.342. Anunknownquantityissubtractedfromelevenandtheresultisvelessthantheunknown quantity. 5.5.7Exercises Forthefollowingproblems,translatethefollowingphrasesorsentencesintomathematicalexpressionsor equations. Exercise5.192 Solutiononp.342. Aquantitylessfour. Exercise5.193 Eightmorethananumber. Exercise5.194 Solutiononp.342. Anumberplusseven.

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302 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.195 Anumberminusthree. Exercise5.196 Solutiononp.342. Negativeveplusanunknownquantity. Exercise5.197 Negativesixteenminussomequantity. Exercise5.198 Solutiononp.342. Fourteenaddedtotwiceanumber. Exercise5.199 Tenaddedtothreetimessomenumber. Exercise5.200 Solutiononp.343. Onethirdminusanunknownquantity. Exercise5.201 Twiceanumberiseleven. Exercise5.202 Solutiononp.343. Fourninthsofanumberistwenty-one. Exercise5.203 Onethirdofanumberistwofths. Exercise5.204 Solutiononp.343. Threetimesanumberisninemorethantwicethenumber. Exercise5.205 Fivetimesanumberisthatnumberminustwo. Exercise5.206 Solutiononp.343. Twiceanumberaddedtosixresultsinthirty. Exercise5.207 Tentimesanumberlessfourresultsinsixty-six. Exercise5.208 Solutiononp.343. Anumberlesstwenty-veisequalto 3 : 019 Exercise5.209 Sevenmorethansomenumberisvemorethantwicethenumber. Exercise5.210 Solutiononp.343. Whenanumberisdividedbyfour,theresultissixty-eight. Exercise5.211 Elevenfteenthsoftwomorethananumberiseight. Exercise5.212 Solutiononp.343. Onetenthofanumberisthatnumberlessone. Exercise5.213 Twomorethantwiceanumberisonehalfthenumberlessthree. Exercise5.214 Solutiononp.343. Anumberisequaltoitselfplusfourtimesitself. Exercise5.215 Threefthsofaquantityaddedtothequantityitselfisthirty-nine. Exercise5.216 Solutiononp.343. Anumberplussevenisdividedbytwoandtheresultistwenty-two. Exercise5.217 Tentimesanumberminusoneisdividedbyfourteenandtheresultisone.

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303 Exercise5.218 Solutiononp.343. Anumberisaddedtoitselfthendividedbythree.Thisresultisthendividedbythree.Theentire resultisfteen. Exercise5.219 Tendividedbytwomorethananumberistwenty-one. Exercise5.220 Solutiononp.343. Fivedividedbyanumberplussixisfourteen. Exercise5.221 Twelvedividedbytwiceanumberisfty-ve. Exercise5.222 Solutiononp.343. Twentydividedbyeighttimesanumberaddedtooneisnine. Exercise5.223 Anumberdividedbyitself,plusone,resultsinseven. Exercise5.224 Solutiononp.343. Anumberdividedbyten,plusfour,resultsintwenty-four. Exercise5.225 Anumberplussix,dividedbytwo,isseventy-one. Exercise5.226 Solutiononp.343. Anumberplussix,dividedbytwo,plusve,isforty-three. Exercise5.227 Anumbermultipliedbyitselfaddedtoveisthirty-one. Exercise5.228 Solutiononp.343. Aquantitymultipliedbysevenplustwiceitselfisninety. Exercise5.229 Anumberisincreasedbyoneandthenmultipliedbyvetimesitself.Theresultiseighty-four. Exercise5.230 Solutiononp.343. Anumberisaddedtosixandthatresultismultipliedbythirteen.Thisresultisthendividedby sixtimesthenumber.Theentireresultisequaltofty-nine. Exercise5.231 Anumberissubtractedfromtenandthatresultismultipliedbyfour.Thisresultisthendivided bythreemorethanthenumber.Theentireresultisequaltosix. Exercise5.232 Solutiononp.343. Anunknownquantityisdecreasedbyeleven.Thisresultisthendividedbyfteen.Now,oneis subtractedfromthisresultandveisobtained. Exercise5.233 Tenlessthansomenumber. Exercise5.234 Solutiononp.343. Fivelessthansomeunknownnumber. Exercise5.235 Twelvelessthananumber. Exercise5.236 Solutiononp.343. Onelessthananunknownquantity. Exercise5.237 Sixteenlessthansomenumberisforty-two. Exercise5.238 Solutiononp.343. Eightlessthansomeunknownnumberisthree.

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304 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.239 Sevenisaddedtotenlessthansomenumber.Theresultisone. Exercise5.240 Solutiononp.343. Twenty-threeisdividedbytwolessthantwicesomenumberandtheresultisthirty-four. Exercise5.241 Onelessthansomenumberismultipliedbythreelessthanvetimesthenumberandtheentire resultisdividedbysixlessthanthenumber.Theresultistwenty-sevenlessthaneleventimesthe number. 5.5.8ExercisesforReview Exercise5.242 Solutiononp.343. Section2.3 Supplythemissingword.Thepointonalinethatisassociatedwithaparticular numberiscalledthe ofthatnumber. Exercise5.243 Section2.5 Supplythemissingword.Anexponentrecordsthenumberofidentical in amultiplication. Exercise5.244 Solutiononp.343. Section3.3 Writethealgebraicdenitionoftheabsolutevalueofthenumber a Exercise5.245 Section5.4 Solvetheequation 4 y +5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise5.246 Solutiononp.343. Section5.4 Solvetheequation 2 x +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x =4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6+17 5.6ApplicationII-SolvingProblems 6 5.6.1Overview SolvingAppliedProblems 5.6.2SolvingAppliedProblems Let'sstudysomeinterestingproblemsthatinvolvelinearequationsinonevariable.Inordertosolvesuch problems,weapplythefollowingve-stepmethod: Five-StepMethodforSolvingWordProblems 1.Let x orsomeotherletterrepresenttheunknownquantity. 2.Translatethewordstomathematicalsymbolsandformanequation. 3.Solvethisequation. 4.Askyourself"Doesthisresultseemreasonable?"Checkthesolutionbysubstitutingtheresultinto theoriginalstatementoftheproblem.Iftheanswerdoesn'tcheck,youhaveeithersolvedtheequation incorrectly,oryouhavedevelopedthewrongequation.Checkyourmethodofsolutionrst.Ifthe resultdoesnotcheck,reconsideryourequation. 5.Writetheconclusion. 6 Thiscontentisavailableonlineat.

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305 Ifithasbeenyourexperiencethatwordproblemsaredicult,thenfollowtheve-stepmethodcarefully. Mostpeoplehavedicultybecausetheyneglectstep1. AlwaysstartbyINTRODUCINGAVARIABLE! Keepinmindwhatthevariableisrepresentingthroughouttheproblem. 5.6.3SampleSetA Example5.40 Thisyearanitemcosts $44 ,anincreaseof $3 overlastyear'sprice.Whatwaslastyear'sprice? Step 1: Let x = lastyear'sprice. Step 2: x +3=44 :x +3 representsthe$3increaseinprice. Step 3: x +3=44 x +3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3=44 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x =41 Step 4:41+3= 44Yes,thisiscorrect. Step 5: Lastyear'spricewas $41 : 5.6.4PracticeSetA Exercise5.247 Solutiononp.344. Thisyearanitemcosts $23 ,anincreaseof $4 overlastyear'sprice.Whatwaslastyear'sprice? Step1:Let x = Step2: Step3: Step4: Step5:Lastyear'spricewas 5.6.5SampleSetB Example5.41 Theperimeterlengtharoundofasquareis60cmcentimeters.Findthelengthofaside. Step1:Let x =lengthofaside. Step2:Wecandrawapicture. Step 3: x + x + x + x =60 4 x =60 Dividebothsidesby 4 : x =15 : Step 4:4=60 : Yes,thisiscorrect. Step 5: Thelengthofasideis 15 cm.

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306 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.6.6PracticeSetB Exercise5.248 Solutiononp.344. Theperimeterofatriangleis54inches.Ifeachsidehasthesamelength,ndthelengthofaside. Step1:Let x = Step2: Step3: Step4: Step5:Thelengthofasideis inches. 5.6.7SampleSetC Example5.42 Sixpercentofanumberis54.Whatisthenumber? Step 1: Let x = thenumber Step 2: Wemustconvert 6% toadecimal. 6%= : 06 : 06 x =54 : 06 x occursbecausewewant 6% of x Step 3: : 06 x =54 : Dividebothsidesby : 06 : x = 54 : 06 x =900 Step 4: : 06=54 : Yes,thisiscorrect. Step 5: Thenumberis 900 : 5.6.8PracticeSetC Exercise5.249 Solutiononp.344. Eightpercentofanumberis36.Whatisthenumber? Step1:Let x = Step2: Step3: Step4: Step5:Thenumberis 5.6.9SampleSetD Example5.43 Anastronomernoticesthatonestargivesoabout 3 : 6 timesasmuchenergyasanotherstar. Togetherthestarsgiveo 55 : 844 unitsofenergy.Howmanyunitsofenergydoeseachstaremit?

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307 Step1:Inthisproblemwehavetwounknownsand,therefore,wemightthink,twovariables.However, noticethattheenergygivenobyonestarisgivenintermsoftheotherstar.So,rather thanintroducingtwovariables,weintroduceonlyone.Theotherunknownsisexpressedin termsofthisone.Wemightcallthisquantitythebasequantity.Let x = numberofunitsof energygivenobythelessenergeticstar.Then, 3 : 6 x = numberofunitsofenergygiveno bythemoreenergeticstar. Step2: x +3 : 6 x = 55 : 844 : Step3: x +3 : 6 x = 55 : 844 4 : 6 x = 55 : 844 Dividebothsidesby4.6.Acalculatorwouldbeusefulatthis point. x = 55 : 844 4 : 6 x = 12 : 14 Thewordingoftheproblemimplies two numbersareneeded =foracompletesolution.Weneedthenumberofunitsof energyfortheotherstar. 3 : 6 x = 3 : 6 : 14 = 43 : 704 Step4: 12 : 14+43 : 704 = 55 : 844 : Yes,thisiscorrect. Step 5: Onestargiveso 12 : 14 unitsofenergyandtheotherstargiveso 43 : 704 unitsofenergy. 5.6.10PracticeSetD Exercise5.250 Solutiononp.344. GardenAproduces 5 : 8 timesasmanyvegetablesasgardenB.Togetherthegardensproduce102 poundsofvegetables.HowmanypoundsofvegetablesdoesgardenAproduce? Step1:Let x = Step2: Step3: Step4: Step5: 5.6.11SampleSetE Example5.44 Twoconsecutiveevennumberssumto432.Whatarethetwonumbers? Step1:Let x = thesmallerevennumber.Then x +2= thenextconsecutiveevennumber sinceconsecutiveevennumbersdierby2asdoconsecutiveoddnumbers. Step2: x + x +2=432 : Step3: x + x +2=432 2 x +2=432 2 x =430 x =215 : Also,since x =215 ;x +2=217 :

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308 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Step 4:215+217=432 ; but 215 and 217 areoddnumbersandwearelookingforevennumbers. Uponcheckingourmethodofsolutionandreexaminingourequation ; wendno mistakes. Step 5: Wemustconcludethatthisproblemhasnosolution.Therearenotwoconsecutive even numbersthatsumto432. 5.6.12PracticeSetE Exercise5.251 Solutiononp.344. Thesumoftwoconsecutiveevennumbersis498.Whatarethetwonumbers? Step1: Step2: Step3: Step4: Step5: 5.6.13Exercises Solvethefollowingproblems.Notethatsomeoftheproblemsmayseemtohavenopracticalapplications andmaynotseemveryinteresting.They,alongwiththeotherproblems,will,however,helptodevelopyour logicandproblem-solvingability. Exercise5.252 Solutiononp.344. Ifeighteenissubtractedfromsomenumbertheresultisfty-two.Whatisthenumber? Step1:Let x = Step2:Theequationis Step3:Solvetheequation. Step4:Check Step5:Thenumberis Exercise5.253 Ifninemorethantwiceanumberisforty-six,whatisthenumber? Step1:Let x = Step2:Theequationis Step3:Solvetheequation. Step4:Check Step5:Thenumberis Exercise5.254 Solutiononp.344. Ifninelessthanthreeeighthsofanumberistwoandonefourth,whatisthenumber? Step1:Let x = Step2: Step3: Step4: Step5:Thenumberis .

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309 Exercise5.255 Twentypercentofanumberis68.Whatisthenumber? Step1:Let x = Step2: Step3: Step4: Step5:Thenumberis Exercise5.256 Solutiononp.344. Eightmorethanaquantityis37.Whatistheoriginalquantity? Step1:Let x = Step2: Step3: Step4: Step5:Theoriginalquantityis Exercise5.257 Ifaquantityplus 85% moreofthequantityis 62 : 9 ,whatistheoriginalquantity? Step1:Let x = originalquantity. Step2: x |{z} original quantity + : 85 x | {z } 85%more =62 : 9 Step3: Step4: Step5:Theoriginalquantityis Exercise5.258 Solutiononp.344. Acompanymustincreaseproductionby 12% overlastyear'sproduction.Thenewoutputwillbe 56items.Whatwaslastyear'soutput? Step1:Let P = Step2: Step3: Step4: Step5:Lastyear'soutputwas items. Exercise5.259 Acompanyhasdeterminedthatitmustincreaseproductionofacertainlineofgoodsby 1 1 2 times lastyear'sproduction.Thenewoutputwillbe2885items.Whatwaslastyear'soutput? Step1: Step2: Step3: Step4: Step5:Lastyear'soutputwas items. Exercise5.260 Solutiononp.344. Aprotonisabout1837timesasheavyasanelectron.Ifanelectronweighs 2 : 68 units,howmany unitsdoesaprotonweigh?

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310 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Step1: Step2: Step3: Step4: Step5:Aprotonweighs units. Exercise5.261 Neptuneisabout30timesasfarfromthesunasistheEarth.Ifittakeslight8minutestotravel fromthesuntotheEarth,howmanyminutesdoesittaketotraveltoNeptune? Step1: Step2: Step3: Step4: Step5:Lighttakes minutestoreachNeptune. Exercise5.262 Solutiononp.344. Theradiusofthesunisabout695,202kmkilometers.Thatisabout109timesasbigasthe radiusoftheEarth.Whatistheradiusoftheearth? Step1: Step2: Step3: Step4: Step5:Theradiusoftheearthis km. Exercise5.263 Theperimeterofatriangleis105cm.Ifeachofthetwolegsisexactlytwicethelengthofthe base,howlongiseachleg? Step1:Let x = Drawapicture. Step2: Step3: Step4: Step5:Eachlegis cmlong.Thebaseis Exercise5.264 Solutiononp.344. Alumbercompanyhascontractedtocutboardsintotwopiecessothatonepieceisthreetimesthe lengthoftheotherpiece.Ifaboardis12feetlong,whatisthelengthofeachpieceaftercutting? Step1: Step2: Step3: Step4: Step5:Thelengthoftheshorterpieceis feet,andthelengthofthelongerpieceis feet. Exercise5.265 Astudentdoingachemistryexperimenthasabeakerthatcontains84mlmillilitersofanalcohol andwatersolution.Herlabdirectionstellherthatthereis 4 : 6 timesasmuchwaterasalcoholin thesolution.Howmanymillilitersofalcoholareinthesolution?Howmanymillilitersofwater?

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311 Step1: Step2: Step3: Step4: Step5:Thereare mlofalcoholinthesolution.Thereare mlofwaterinthe solution. Exercise5.266 Solutiononp.344. AstatisticianiscollectingdatatohelphimestimatetheaverageincomeofaccountantsinCalifornia. Heneedstocollect390piecesofdataandheis 2 3 done.Howmanypiecesofdatahasthestatistician collected? Step1: Step2: Step3: Step4: Step5:Thestatisticianhascollected piecesofdata. Supposethestatisticianis4piecesofdatashortofbeing 2 3 done.Howmanypiecesofdata hashecollected? Exercise5.267 Atelevisioncommercialadvertisesthatacertaintypeofbatterywilllast,ontheaverage,20hours longerthantwicethelifeofanothertypeofbattery.Ifconsumertestsshowthattheadvertised batterylasts725hours,howmanyhoursmusttheothertypeofbatterylastfortheadvertiser's claimtobevalid? Step1: Step2: Step3: Step4: Step5:Theothertypeofbatterymustlast hoursfortheadvertiser'sclaimtobevalid. Exercise5.268 Solutiononp.344. A1000-mlaskcontainingachloridesolutionwillll3beakersofthesamesizewith210mlof thesolutionleftover.Howmanymillilitersofthechloridesolutionwilleachbeakerhold? Step1: Step2: Step3: Step4: Step5:Eachbeakerwillhold mlofthechloridesolution. Exercise5.269 Astarburns 2 9 ofitsoriginalmassthenblowso 3 7 oftheremainingmassasaplanetarynebula. Ifthenalmassis3unitsofmass,whatwastheoriginalmass? Step1: Step2: Step3: Step4: Step5:Theoriginalmasswas unitsofmass.

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312 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.270 Solutiononp.344. Thesumofanumberandsixteenisforty-two.Whatisthenumber? Exercise5.271 Whenelevenissubtractedfromanumber,theresultis85.Whatisthenumber? Exercise5.272 Solutiononp.344. Threetimesanumberisdividedby6andtheresultis10.5.Whatisthenumber? Exercise5.273 Whenanumberismultipliedbyitself,theresultis144.Whatisthenumber? Exercise5.274 Solutiononp.344. Anumberistripled,thenincreasedbyseven.Theresultis48.Whatisthenumber? Exercise5.275 Eighttimesanumberisdecreasedbythreetimesthenumber,givingadierenceof22.Whatis thenumber? Exercise5.276 Solutiononp.344. Onenumberisfteenmorethananothernumber.Thesumofthetwonumbersis27.Whatare they? Exercise5.277 Thelengthofarectangleis6metersmorethanthreetimesthewidth.Theperimeterofthe rectangleis44metersWhatarethedimensionsoftherectangle? Exercise5.278 Solutiononp.344. Sevenisaddedtotheproductof41andsomenumber.Theresult,whendividedbyfour,is63. Whatisthenumber? Exercise5.279 Thesecondsideofatriangleisvetimesthelengthofthesmallestside.Thethirdistwicethe lengthofthesecondside.Theperimeterofthetriangleis48inches.Findthelengthofeachside. Exercise5.280 Solutiononp.345. PersonAisfourtimesasoldaspersonB,whoissixtimesasoldaspersonC,whoistwiceasold aspersonD.Howoldiseachpersoniftheircombinedagesare189months? Exercise5.281 Twoconsecutiveoddintegerssumto151.Whatarethey? Exercise5.282 Solutiononp.345. Threeconsecutiveintegerssumto36.Whatarethey? Exercise5.283 Threeconsecutiveevenintegersaddupto131.Whatarethey? Exercise5.284 Solutiononp.345. AsaconsequenceofEinstein'stheoryofrelativity,therateoftimepassageisdierentforaperson inastationarypositionandapersoninmotion.Hardtobelieve,buttrue!Tothemovingobserver, therateoftimepassageisslowerthanthatofthestationaryobserver,thatis,themovingperson agesslowerthanthestationaryobserver.Thisfacthasbeenprovenmanytimesbyexperiments withradioactivematerials.Theeectiscalledtimedilationandisreallyonlynoticeablewhen anobjectistravelingatnearthespeedoflight,000milespersecond.Consideringtheseideas, trytosolvethefollowingproblems: Twopeoplehaveidenticalclocks.Oneisstandingontheearthandtheotherismovingina spacecraftat 95% thespeedoflight,176,700milespersecond.Themovingperson'srateoftime passageatthisspeedisabout 0 : 31 timesasfastasthepersonstandingonearth. a.Iftwodaysofearthtimepass,howmanydaysactuallypassonthespacecraft? b.If30yearsofearthtimepass,howmanyyearshaveactuallypassedonthespacecraft?

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313 Stepa: Stepb: Stepc: Stepd: Stepe: yearshavepassedonthespacecraft. c.If30yearshavepassedonthespacecraft,howmanyyearshavepassedontheearth? d.Aspacetravelermakesaround-tripvoyagetothestarCapella.Thetriptakesher120years travelingat176,000milespersecond.Ifitistheyear2000onearthwhensheleaves,what earthyearwillitbewhenshereturns? 5.6.14ExercisesforReview Exercise5.285 Section4.8 Specifythedomainoftheequation y = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +4 Exercise5.286 Solutiononp.345. Section5.2 Classifytheequation x +4=1 asanidentity,acontradiction,oraconditional equation. Exercise5.287 Section5.2 Classifytheequation 2 x +3=2 x +3 asanidentity,acontradictionoraconditional equation. Exercise5.288 Solutiononp.345. Section5.4 Solvetheequation 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+12= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 x +4 Exercise5.289 Section5.5 Translatethefollowingsentencetoamathematicalequation.Threelessthanan unknownnumberismultipliedbynegativefour.Theresultistwomorethantheoriginalunknown number. 5.7LinearinequalitiesinOneVariable 7 5.7.1Overview Inequalities LinearInequalities TheAlgebraofLinearInequalities CompoundInequalities 5.7.2Inequalities RelationshipsofInequality Wehavediscoveredthatanequationisamathematicalwayofexpressingtherelationshipofequalitybetween quantities.Notallrelationshipsneedberelationshipsofequality,however.Certainlythenumberofhuman beingsonearthisgreaterthan20.Also,theaverageAmericanconsumeslessthan10gramsofvitamin Ceveryday.Thesetypesofrelationshipsarenotrelationshipsofequality,butrather,relationshipsof inequality 7 Thiscontentisavailableonlineat.

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314 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.7.3LinearInequalities LinearInequality A linearinequality isamathematicalstatementthatonelinearexpressionisgreaterthanorlessthan anotherlinearexpression. InequalityNotation Thefollowingnotationisusedtoexpressrelationshipsofinequality: > Strictlygreaterthan < Strictlylessthan Greaterthanorequalto Lessthanorequalto Notethattheexpression x> 12 hasinnitelymanysolutions.Anynumberstrictlygreaterthan12will satisfythestatement.Somesolutionsare13,15,90, 12 : 1 ; 16 : 3 and 102 : 51 5.7.4SampleSetA Thefollowing are linearinequalitiesinonevariable. Example5.45 1. x 12 2. x +7 > 4 3. y +3 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 4. P +26 < 10 P )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 5. 2 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 5 > 15 Thefollowing arenot linearinequalitiesinonevariable. Example5.46 1. x 2 < 4 Theterm x 2 isquadratic,notlinear. 2. x 5 y +3 Therearetwovariables.Thisisalinearinequalityintwovariables. 3. y +1 6 =5 Althoughthesymbol 6 = certainlyexpressesaninequality,itiscustomarytouseonlythe symbols <;>; ; 5.7.5PracticeSetA Alinearequation,weknow,mayhaveexactlyonesolution,innitelymanysolutions,ornosolution.Speculateonthenumberofsolutionsofalinearinequality. Hint: Considertheinequalities x
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315 TheAlgebraofLinearInequalities Let a;b; and c representrealnumbersandassumethat ab Then,if abc and a c > b c : Ifbothsidesofaninequalityaremultipliedordividedbythesame negative number, theinequality signmustbereversed changedirectioninorderfortheresultinginequalitytobeequivalentto theoriginalinequality.Seeproblem4inthenextsetofexamples. Forexample,considertheinequality 3 < 7 Example5.47 For 3 < 7 ,if8isaddedtobothsides,weget 3+8 < 7+8 : 11 < 15 True Example5.48 For 3 < 7 ,if8issubtractedfrombothsides,weget 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 < 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 : )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 < )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 True Example5.49 For 3 < 7 ,ifbothsidesaremultipliedby8apositivenumber,weget 8 < 8 24 < 56 True Example5.50 For 3 < 7 ,ifbothsidesaremultipliedby )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 anegativenumber,weget )]TJ/F8 9.9626 Tf 7.748 0 Td [(83 > )]TJ/F8 9.9626 Tf 7.749 0 Td [(87 Noticethechangeindirectionoftheinequalitysign. )]TJ/F8 9.9626 Tf 7.748 0 Td [(24 > )]TJ/F8 9.9626 Tf 9.963 0 Td [(56 True Ifwehadforgottentoreversethedirectionoftheinequalitysignwewouldhaveobtainedthe incorrectstatement )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 < )]TJ/F8 9.9626 Tf 9.962 0 Td [(56 Example5.51 For 3 < 7 ,ifbothsidesaredividedby8apositivenumber,weget 3 8 < 7 8 True Example5.52 For 3 < 7 ,ifbothsidesaredividedby )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 anegativenumber,weget 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 > 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 True since )]TJ/F11 9.9626 Tf 9.963 0 Td [(: 375 )]TJ/F11 9.9626 Tf 9.962 0 Td [(: 875

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316 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.7.7SampleSetB Solvethefollowinglinearinequalities.Drawanumberlineandplaceapointateachsolution. Example5.53 3 x> 15 Dividebothsidesby3.The 3 isapositivenumber ; soweneednotreversethesenseoftheinequality. x> 5 Thus,allnumbersstrictlygreaterthan5aresolutionstotheinequality 3 x> 15 Example5.54 2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 16 Add1tobothsides : 2 y 17 Dividebothsidesby 2 : y 17 2 Example5.55 )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 x +5 < 14 Subtract 5 frombothsides. )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 x< 9 Dividebothsidesby )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 : Wemustreversethesenseoftheinequality sincewearedividingbyanegativenumber. x> )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(9 8 Example5.56 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y +2 < 6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 < 6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 < 6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 y< )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 y> 1 Example5.57 2 z +7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 )]TJ/F8 9.9626 Tf 18.265 0 Td [(6 Multiplyby )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 z +7 24 Noticethechangeinthesenseoftheinequality. 2 z 17 z 17 2

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317 5.7.8PracticeSetB Solvethefollowinglinearinequalities. Exercise5.290 Solutiononp.345. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 5 Exercise5.291 Solutiononp.345. x +4 > 9 Exercise5.292 Solutiononp.345. 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 15 Exercise5.293 Solutiononp.345. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y +16 7 Exercise5.294 Solutiononp.345. 7 s )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 < 2 s +8 Exercise5.295 Solutiononp.345. 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 h +4 < )]TJ/F11 9.9626 Tf 9.963 0 Td [(h 2+6 Exercise5.296 Solutiononp.345. 18 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 x Exercise5.297 Solutiononp.345. )]TJ/F7 6.9738 Tf 9.178 3.923 Td [(3 b 16 4 Exercise5.298 Solutiononp.345. )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 z +10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise5.299 Solutiononp.345. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 3 5 6 5.7.9CompoundInequalities CompoundInequality Anothertypeofinequalityisthe compoundinequality .Acompoundinequalityisoftheform: aa .Surely,ifthenumber a islessthanthenumber x ,thenumber x mustbegreater thanthenumber a .Thus,wecanread a
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318 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 4. 1 4 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 6 7 9 Theterm 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 6 representssomenumberbetweenandincluding 1 4 and 7 9 .Hence, 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 6 representssome numbergreaterthanorequalto 1 4 tobutlessthanorequalto 7 9 Considerproblem3above, 1 x> )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(47 2 Remembertoreversethedirectionoftheinequality signs. )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(47 2
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319 Exercise5.304 Solutiononp.345. 9 < )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 < 14 Exercise5.305 Solutiononp.345. Does 4 16 Exercise5.310 Solutiononp.346. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 8 Exercise5.311 9 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 6 Exercise5.312 Solutiononp.346. 2 z +8 < 7 Exercise5.313 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 > 21 Exercise5.314 Solutiononp.346. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 20 Exercise5.315 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 x< 40 Exercise5.316 Solutiononp.346. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 z< 77 Exercise5.317 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y> 39 Exercise5.318 Solutiononp.346. x 4 12 Exercise5.319 y 7 > 3 Exercise5.320 Solutiononp.346. 2 x 9 4 Exercise5.321 5 y 2 15 Exercise5.322 Solutiononp.346. 10 x 3 4 Exercise5.323 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y 4 < 8

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320 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.324 Solutiononp.346. )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 b 5 < 24 Exercise5.325 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a 7 )]TJ/F8 9.9626 Tf 18.265 0 Td [(24 Exercise5.326 Solutiononp.346. 8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 > 6 Exercise5.327 14 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F8 9.9626 Tf 18.265 0 Td [(18 Exercise5.328 Solutiononp.346. 21 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 < )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise5.329 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +7 )]TJ/F8 9.9626 Tf 18.265 0 Td [(5 Exercise5.330 Solutiononp.346. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 y +10 )]TJ/F8 9.9626 Tf 18.265 0 Td [(4 Exercise5.331 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 < 31 Exercise5.332 Solutiononp.346. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 30 Exercise5.333 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y + 4 3 )]TJ/F7 6.9738 Tf 19.461 3.922 Td [(2 3 Exercise5.334 Solutiononp.346. 5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 15 Exercise5.335 4 x +1 > )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 Exercise5.336 Solutiononp.346. 6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 48 Exercise5.337 3 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +3 > )]TJ/F8 9.9626 Tf 9.962 0 Td [(27 Exercise5.338 Solutiononp.346. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y +3 > 0 Exercise5.339 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(77 0 Exercise5.340 Solutiononp.346. 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise5.344 Solutiononp.346. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 7 x +4 Exercise5.345 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 > 5 x Exercise5.346 Solutiononp.346. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 > )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +12

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321 Exercise5.347 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 4 Exercise5.348 Solutiononp.346. 5 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 14 Exercise5.349 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(3+ x Exercise5.350 Solutiononp.346. 3[4+5 x +1] < )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise5.351 2[6+2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7] 4 Exercise5.352 Solutiononp.346. 7[ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1] 91 Exercise5.353 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 < 3 x +8 Exercise5.354 Solutiononp.346. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 > )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 )]TJ/F11 9.9626 Tf 7.748 0 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15+1 Exercise5.355 )]TJ/F11 9.9626 Tf 7.749 0 Td [(: 0091 x 2 : 885 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 : 014 Exercise5.356 Solutiononp.346. Whatnumberssatisfythecondition:twiceanumberplusoneisgreaterthannegativethree? Exercise5.357 Whatnumberssatisfythecondition:eightmorethanthreetimesanumberislessthanorequal tofourteen? Exercise5.358 Solutiononp.347. Onenumberisvetimeslargerthananothernumber.Thedierencebetweenthesetwonumbers islessthantwenty-four.Whatarethelargestpossiblevaluesforthetwonumbers?Istherea smallestpossiblevalueforeithernumber? Exercise5.359 Theareaofarectangleisfoundbymultiplyingthelengthoftherectanglebythewidthofthe rectangle.Ifthelengthofarectangleis8feet,whatisthelargestpossiblemeasureforthewidth ifitmustbeanintegerpositivewholenumberandtheareamustbelessthan48squarefeet? 5.7.13ExercisesforReview Exercise5.360 Solutiononp.347. Section2.7 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 y 3 z 2 5 Exercise5.361 Section3.3 Simplify )]TJ/F8 9.9626 Tf 9.409 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( j)]TJ/F8 9.9626 Tf 22.692 0 Td [(8 j ] Exercise5.362 Solutiononp.347. Section4.6 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x +4 Exercise5.363 Section5.6 Twenty-vepercentofanumberis 12 : 32 .Whatisthenumber? Exercise5.364 Solutiononp.347. Section5.6 Theperimeterofatriangleis40inches.Ifthelengthofeachofthetwolegsis exactlytwicethelengthofthebase,howlongiseachleg?

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322 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.8LinearEquationsinTwoVariables 8 5.8.1Overview SolutionstoLinearEquationsinTwoVariables OrderedPairsasSolutions 5.8.2SolutionstoLinearEquationsinTwoVariables SolutiontoanEquationinTwoVariables Wehavediscoveredthatanequationisamathematicalwayofexpressingtherelationshipofequalitybetween quantities.Iftherelationshipisbetweentwoquantities,theequationwillcontaintwovariables.Wesaythat anequationintwovariableshasasolutionifanordered pair ofvaluescanbefoundsuchthatwhenthese twovaluesaresubstitutedintotheequationatruestatementresults.Thisisillustratedwhenweobserve somesolutionstotheequation y =2 x +5 1. x =4 ;y =13; since 13=2+5 istrue. 2. x =1 ;y =7; since 7=2+5 istrue. 3. x =0 ;y =5; since 5=2+5 istrue. 4. x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ;y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7; since )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6+5 istrue. 5.8.3OrderedPairsasSolutions Itisimportanttokeepinmindthatasolutiontoalinearequationintwovariablesisanorderedpairof values,onevalueforeachvariable.Asolutionisnotcompletelyknownuntilthevaluesof both variables arespecied. IndependentandDependentVariables Recallthat,inanequation,anyvariablewhosevaluecanbefreelyassignedissaidtobean independent variable. Anyvariablewhosevalueisdeterminedoncetheothervalueorvalueshavebeenassignedissaid tobea dependentvariable. If,inalinearequation,theindependentvariableis x andthedependent variableis y ,andasolutiontotheequationis x = a and y = b ,thesolutioniswrittenasthe ORDEREDPAIR a;b OrderedPair Inan orderedpair a;b ,therstcomponent, a ,givesthevalueoftheindependentvariable,andthe secondcomponent, b ,givesthevalueofthedependentvariable. Wecanuseorderedpairstoshowsomesolutionstotheequation y =6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 Example5.60 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 If x =0 and y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 ,wegetatruestatementuponsubstitutionandcomputataion. 8 Thiscontentisavailableonlineat.

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323 y =6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7=6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 Yes,thisiscorrect. Example5.61 ; 41 If x =8 and y =41 ,wegetatruestatementuponsubstitutionandcomputataion. y =6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 41=6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Isthiscorrect? 41=48 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Isthiscorrect? 41=41 Yes,thisiscorrect. Example5.62 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(31 If x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 and y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(31 ,wegetatruestatementuponsubstitutionandcomputataion. y =6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(31=6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(31= )]TJ/F8 9.9626 Tf 7.748 0 Td [(24 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Isthiscorrect? )]TJ/F8 9.9626 Tf 7.749 0 Td [(31= )]TJ/F8 9.9626 Tf 7.748 0 Td [(31 Yes,thisiscorrect. Theseareonlythreeoftheinntelymanysolutionstothisequation. 5.8.4SampleSetA Findasolutiontoeachofthefollowinglinearequationsintwovariablesandwritethesolutionasanordered pair. Example5.63 y =3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 ; if x =1 Substitute1for x ,compute,andsolvefor y y =3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 =3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Hence,onesolutionis ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Example5.64 y =15 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Substitute )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 for x ,compute,andsolvefor y y =15 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 =15+40 =55 Hence,onesolutionis )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 ; 55 Example5.65 b = )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 a +21 ; if a =2

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324 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Substitute2for a ,compute,andsolvefor b b = )]TJ/F8 9.9626 Tf 7.748 0 Td [(9+21 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(18+21 =3 Hence,onesolutionis ; 3 Example5.66 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y =1 ; if x =0 Substitute0for x ,compute,andsolvefor y 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y =1 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y =1 y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 2 Hence,onesolutionis )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 5.8.5PracticeSetA Findasolutiontoeachofthefollowinglinearequationsintwovariablesandwritethesolutionasanordered pair. Exercise5.365 Solutiononp.347. y =7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 ; if x =3 Exercise5.366 Solutiononp.347. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 n +1 ; if n =2 Exercise5.367 Solutiononp.347. b =3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 ; if a =0 Exercise5.368 Solutiononp.347. 10 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(20=0 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Exercise5.369 Solutiononp.347. 3 a +2 b +6=0 ; if a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 5.8.6Exercises Forthefollowingproblems,solvethelinearequationsintwovariables. Exercise5.370 Solutiononp.347. y =8 x +14 ; if x =1 Exercise5.371 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +1 ; if x =0 Exercise5.372 Solutiononp.347. y =5 x +6 ; if x =4 Exercise5.373 x + y =7 ; if x =8 Exercise5.374 Solutiononp.347. 3 x +4 y =0 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3

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325 Exercise5.375 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x + y =1 ; if x = 1 2 Exercise5.376 Solutiononp.347. 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y +1=0 ; if x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise5.377 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y =4 ; if y =7 Exercise5.378 Solutiononp.347. 2 x +6 y =1 ; if y =0 Exercise5.379 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =0 ; if y = 14 3 Exercise5.380 Solutiononp.347. y = x; if x =1 Exercise5.381 x + y =0 ; if x =0 Exercise5.382 Solutiononp.347. y + 3 4 = x; if x = 9 4 Exercise5.383 y +17= x; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise5.384 Solutiononp.347. )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 y +14 x =1 ; if x =8 Exercise5.385 3 5 y + 1 4 x = 1 2 ; if x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise5.386 Solutiononp.347. 1 5 x + y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 ; if y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.387 y +7 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x =0 ; if x = Exercise5.388 Solutiononp.347. 2 x +31 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3=0 ; if x = a Exercise5.389 436 x +189 y =881 ; if x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4231 Exercise5.390 Solutiononp.347. y =6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 ; if x =2 Exercise5.391 y =2 x +5 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.392 Solutiononp.347. 5 y =9 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 ; if x =2 Exercise5.393 3 y =4 x +1 ; if x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise5.394 Solutiononp.347. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y =3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 ; if x =6 Exercise5.395 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 y =7 x +2 ; if x =0 Exercise5.396 Solutiononp.347. b =4 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 ; if a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7

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326 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.397 b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 a +21 ; if a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 Exercise5.398 Solutiononp.347. 4 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=2 a +1 ; if a =0 Exercise5.399 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 m +11= n +1 ; if n =4 Exercise5.400 Solutiononp.348. 3 t +2=4 s )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 ; if s =1 Exercise5.401 7 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=10 )]TJ/F11 9.9626 Tf 9.963 0 Td [(s ; if s =5 Exercise5.402 Solutiononp.348. y =0 x +5 ; if x =1 Exercise5.403 2 y =0 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 Exercise5.404 Solutiononp.348. )]TJ/F11 9.9626 Tf 7.749 0 Td [(y =0 x +10 ; if x =3 Exercise5.405 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y =0 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ; if x =0 Exercise5.406 Solutiononp.348. y =0 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+6 ; if x =1 Exercise5.407 y =0 x +9 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; if x =12 5.8.6.1CalculatorProblems Exercise5.408 Solutiononp.348. AnexaminationofthewinningspeedsintheIndianapolis500automobileracefrom1961to1970 producestheequation y =1 : 93 x +137 : 60 ,where x isthenumberofyearsfrom1960and y isthe winningspeed.Statisticalmethodswereusedtoobtaintheequation,and,foragivenyear,the equationgivesonlytheapproximatewinningspeed.Usetheequation y =1 : 93 x +137 : 60 tond theapproximatewinningspeedin a.1965 b.1970 c.1986 d.1990 Exercise5.409 Inelectricitytheory,Ohm'slawrelateselectricalcurrenttovoltagebytheequation y =0 : 00082 x where x isthevoltageinvoltsand y isthecurrentinamperes.Thisequationwasfoundby statisticalmethodsandforagivenvoltageyieldsonlyanapproximatevalueforthecurrent.Use theequation y =0 : 00082 x tondtheapproximatecurrentforavoltageof a.6volts b.10volts

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327 Exercise5.410 Solutiononp.348. Statisticalmethodshavebeenusedtoobtainarelationshipbetweentheactualandreported numberofGermansubmarinessunkeachmonthbytheU.S.NavyinWorldWarII.Theequation expressingtheapproximatenumberofactualsinkings, y ,foragivennumberofreportedsinkings, x ,is y =1 : 04 x +0 : 76 .FindtheapproximatenumberofactualsinkingsofGermansubmarinesif thereportednumberofsinkingsis a.4 b.9 c.10 Exercise5.411 Statisticalmethodshavebeenusedtoobtainarelationshipbetweentheheartweightinmilligrams andthebodyweightinmilligramsof10-month-olddiabeticospringofcrossbredmalemice.The equationexpressingtheapproximatebodyweightforagivenheartweightis y =0 : 213 x )]TJ/F8 9.9626 Tf 10.416 0 Td [(4 : 44 Findtheapproximatebodyweightforaheartweightof a.210mg b.245mg Exercise5.412 Solutiononp.348. Statisticalmethodshavebeenusedtoproducetheequation y =0 : 176 x )]TJ/F8 9.9626 Tf 10.546 0 Td [(0 : 64 .Thisequation givestheapproximateredbloodcellcountinmillionsofadog'sblood, y ,foragivenpackedcell volumeinmillimeters, x .Findtheapproximateredbloodcellcountforapackedcellvolumeof a.40mm b.42mm Exercise5.413 Anindustrialmachinecanrunatdierentspeeds.Themachinealsoproducesdefectiveitems, andthenumberofdefectiveitemsitproducesappearstoberelatedtothespeedatwhichthe machineisrunning.Statisticalmethodsfoundthattheequation y =0 : 73 x )]TJ/F8 9.9626 Tf 10.235 0 Td [(0 : 86 isabletogive theapproximatenumberofdefectiveitems, y ,foragivenmachinespeed, x .Usethisequationto ndtheapproximatenumberofdefectiveitemsforamachinespeedof a.9 b.12 Exercise5.414 Solutiononp.348. Acomputercompanyhasfound,usingstatisticaltechniques,thatthereisarelationshipbetween theaptitudetestscoresofassemblylineworkersandtheirproductivity.Usingdataaccumulated overaperiodoftime,theequation y =0 : 89 x )]TJ/F8 9.9626 Tf 10.221 0 Td [(41 : 78 wasderived.The x representsanaptitude testscoreand y theapproximatecorrespondingnumberofitemsassembledperhour.Estimatethe numberofitemsproducedbyaworkerwithanaptitudescoreof a.80 b.95 Exercise5.415 Chemists,makinguseofstatisticaltechniques,havebeenabletoexpresstheapproximateweight ofpotassiumbromide, W ,thatwilldissolvein100gramsofwaterat T degreescentigrade.The equationexpressingthisrelationshipis W =0 : 52 T +54 : 2 .Usethisequationtopredictthepotassium bromideweightthatwilldissolvein100gramsofwaterthatisheatedtoatemperatureof a.70degreescentigrade

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328 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES b.95degreescentigrade Exercise5.416 Solutiononp.348. Themarketingdepartmentatalargecompanyhasbeenabletoexpresstherelationshipbetween thedemandforaproductanditspricebyusingstatisticaltechniques.Thedepartmentfound, byanalyzingstudiesdoneinsixdierentmarketareas,thattheequationgivingtheapproximate demandforaproductinthousandsofunitsforaparticularpriceincentsis y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 : 15 x +257 : 11 Findtheapproximatenumberofunitsdemandedwhenthepriceis a. $0 : 12 b. $0 : 15 Exercise5.417 Themanagementofaspeed-readingprogramclaimsthattheapproximatespeedgaininwords perminute, G ,isrelatedtothenumberofweeksspentinitsprogram, W ,isgivenbytheequation G =26 : 68 W )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 : 44 .Predicttheapproximatespeedgainforastudentwhohasspent a.3weeksintheprogram b.10weeksintheprogram 5.8.7ExercisesforReview Exercise5.418 Solutiononp.348. Section4.6 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x +5 Exercise5.419 Section4.7 Findtheproduct. x +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise5.420 Solutiononp.348. Section5.5 Solvetheequation 6[2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4+1]=3[2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7] Exercise5.421 Section5.7 Solvetheinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [( a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a +10 Exercise5.422 Solutiononp.348. Section5.7 Solvethecompoundinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 < 4 y +11 < 27 5.9SummaryofKeyConcepts 9 5.9.1SummaryofKeyConcepts IdentitySection5.2 Anequationthatistrueforallacceptablevaluesofthevariableiscalled identity x +3= x +3 isan identity. ContradictionSection5.2 Contradictions areequationsthatarenevertrueregardlessofthevaluesubstitutedforthevariable. x +1= x isacontradiction. ConditionalEquationSection5.2 Anequationwhosetruthisconditionaluponthevalueselectedforthevariableiscalleda conditional equation 9 Thiscontentisavailableonlineat.

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329 SolutionsandSolvinganEquationSection5.2 Thecollectionofvaluesthatmakeanequationtruearecalledthe solutions oftheequation.Anequation issaidtobe solved whenallitssolutionshavebeenfound. EquivalentEquationsSection5.2,Section5.3 Equationsthathavepreciselythesamecollectionofsolutionsarecalled equivalentequations Anequivalentequationcanbeobtainedfromaparticularequationbyapplyingthe same binaryoperation to both sidesoftheequation,thatis, 1.addingorsubtractingthe same numbertoorfrom both sidesofthatparticularequation. 2.multiplyingordividing both sidesofthatparticularequationbythe samenon-zero number. LiteralEquationSection5.2 A literalequation isanequationthatiscomposedofmorethanonevariable. RecognizinganIdentitySection5.4 If,whensolvinganequation,allthevariablesareeliminatedandatruestatementresults,theequationis an identity RecognizingaContradictionSection5.4 If,whensolvinganequation,allthevariablesareeliminatedandafalsestatementresults,theequationisa contradiction TranslatingfromVerbaltoMathematicalExpressionsSection5.5 Whensolvingwordproblemsitisabsolutelynecessarytoknowhowcertainwordstranslateintomathematical symbols. Five-StepMethodforSolvingWordProblemsSection5.6 1.Let x orsomeotherletterrepresenttheunknownquantity. 2.Translatethewordstomathematicsandformanequation.Adiagrammaybehelpful. 3.Solvetheequation. 4.Checkthesolutionbysubstitutingtheresultintotheoriginalstatementoftheproblem. 5.Writeaconclusion. LinearInequalitySection5.7 A linearinequality isamathematicalstatementthatonelinearexpressionisgreaterthanorlessthan anotherlinearexpression. InequalityNotationSection5.7 > Strictlygreaterthan < Strictlylessthan Greaterthanorequalto Lessthanequalto CompoundInequalitySection5.7 Aninequalityoftheform a
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330 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES 5.10ExerciseSupplement 10 5.10.1ExerciseSupplement 5.10.1.1SolvingEquationsSection5.2-FurtherTechniquesinEquationSolvingSection5.4 Solvetheequationsforthefollowingproblems. Exercise5.423 Solutiononp.348. y +3=11 Exercise5.424 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=4 Exercise5.425 Solutiononp.348. r )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=16 Exercise5.426 a +2=0 Exercise5.427 Solutiononp.348. x +6= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise5.428 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise5.429 Solutiononp.348. x +8=8 Exercise5.430 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=4 Exercise5.431 Solutiononp.348. 2 x =32 Exercise5.432 4 x =24 Exercise5.433 Solutiononp.348. 3 r = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise5.434 6 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(30 Exercise5.435 Solutiononp.348. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(30 Exercise5.436 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(72 Exercise5.437 Solutiononp.348. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x =6 Exercise5.438 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise5.439 Solutiononp.348. 3 x +7=19 Exercise5.440 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=29 Exercise5.441 Solutiononp.349. 4 x +2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 10 Thiscontentisavailableonlineat.

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331 Exercise5.442 6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(29 Exercise5.443 Solutiononp.349. 8 x +6= )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise5.444 9 a +5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(22 Exercise5.445 Solutiononp.349. m 6 +4=8 Exercise5.446 b 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=5 Exercise5.447 Solutiononp.349. y 9 =54 Exercise5.448 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(17 Exercise5.449 Solutiononp.349. c 6 =15 Exercise5.450 3 a 4 =9 Exercise5.451 Solutiononp.349. 4 y 5 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 Exercise5.452 r 4 =7 Exercise5.453 Solutiononp.349. 6 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 =11 Exercise5.454 9 x 7 =6 Exercise5.455 Solutiononp.349. c 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8=0 Exercise5.456 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 +4= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.457 Solutiononp.349. x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15= )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 Exercise5.458 3 x 4 +2=14 Exercise5.459 Solutiononp.349. 3 r +2 5 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.460 6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 7 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise5.461 Solutiononp.349. 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 6 +2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise5.462 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 8 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise5.463 Solutiononp.349. 4 x +2=20 Exercise5.464 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=16

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332 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.465 Solutiononp.349. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=63 Exercise5.466 3 x +7=5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(21 Exercise5.467 Solutiononp.349. )]TJ/F8 9.9626 Tf 9.409 0 Td [( r +1=33 Exercise5.468 Solve I = prt for t: Findthevalueof t when I =3500 ;P =3000 ; and r =0 : 05 : Exercise5.469 Solutiononp.349. Solve A = LW for W: Findthevalueof W when A =26 and L =2 : Exercise5.470 Solve p = mv for m: Findthevalueof m when p =4240 and v =260 : Exercise5.471 Solutiononp.349. Solve P = R )]TJ/F11 9.9626 Tf 9.963 0 Td [(C for R: Findthevalueof R when P =480 and C =210 : Exercise5.472 Solve P = nRT V for n: Exercise5.473 Solutiononp.349. Solve y =5 x +8 for x: Exercise5.474 Solve 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x =12 for y: Exercise5.475 Solutiononp.349. Solve 4 y +2 x +8=0 for y: Exercise5.476 Solve k = 4 m +6 7 for m: Exercise5.477 Solutiononp.349. Solve t = 10 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b 2 c for b: 5.10.1.2ApplicationI-TranslatingfromVerbaltoMatheticalExpressionsSection5.5 Forthefollowingproblems,translatethephrasesorsentencestomathematicalexpressionsorequations. Exercise5.478 Aquantitylesseight. Exercise5.479 Solutiononp.349. Anumber,timesfourplusseven. Exercise5.480 Negativetenminussomenumber. Exercise5.481 Solutiononp.349. Twofthsofanumberminusve. Exercise5.482 Oneseventhofanumberplustwoninthsofthenumber. Exercise5.483 Solutiononp.349. Threetimesanumberisforty. Exercise5.484 Twiceaquantityplusnineisequaltothequantityplussixty.

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333 Exercise5.485 Solutiononp.349. Fourtimesanumberminusveisdividedbyseven.Theresultistenmorethanthenumber. Exercise5.486 Anumberisaddedtoitselfvetimes,andthatresultismultipliedbyeight.Theentireresultis twelve. Exercise5.487 Solutiononp.349. Anumbermultipliedbyelevenmorethanitselfissix. Exercise5.488 Aquantitylessthreeisdividedbytwomorethanthequantityitself.Theresultisonelessthan theoriginalquantity. Exercise5.489 Solutiononp.349. Anumberisdividedbytwicethenumber,andeighttimesthenumberisaddedtothatresult.The resultisnegativeone. Exercise5.490 Anunknownquantityisdecreasedbysix.Thisresultisthendividedbytwenty.Tenissubtracted fromthisresultandnegativetwoisobtained. Exercise5.491 Solutiononp.350. Onelessthansomenumberisdividedbyvetimesthenumber.Theresultisthecubeofthe number. Exercise5.492 Ninelessthansomenumberismultipliedbythenumberlessnine.Theresultisthesquareofsix timesthenumber. 5.10.1.3ApplicationII-SolvingProblemsSection5.6 Forthefollowingproblems,ndthesolution. Exercise5.493 Solutiononp.350. Thisyearanitemcosts$106,anincreaseof$10overlastyear'sprice.Whatwaslastyear'sprice? Exercise5.494 Theperimeterofasquareis44inches.Findthelengthofaside. Exercise5.495 Solutiononp.350. Ninepercentofanumberis77.4.Whatisthenumber? Exercise5.496 Twoconsecutiveintegerssumto63.Whatarethey? Exercise5.497 Solutiononp.350. Fourconsecutiveoddintegersaddto56.Whatarethey? Exercise5.498 Iftwenty-oneissubtractedfromsomenumberandthatresultismultipliedbytwo,theresultis thirty-eight.Whatisthenumber? Exercise5.499 Solutiononp.350. If37%moreofaquantityis159.1,whatisthequantit? Exercise5.500 Astatisticianiscollectingdatatohelpherestimatethenumberofpickpocketsinacertaincity. Sheneeds108piecesofdataandis 3 4 done.Howmanypiecesofdatahasshecollected?

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334 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.501 Solutiononp.350. Thestatisticianinproblem78iseightpiecesofdatashortofbeing 5 6 done.Howmanypiecesof datahasshecollected? Exercise5.502 Atelevisioncommercialadvertisesthatacertaintypeoflightbulbwilllast,ontheaverage,200 hourslongerthanthreetimesthelifeofanothertypeofbulb.Ifconsumertestsshowthatthe advertisedbulblasts4700hours,howmanyhoursmusttheothertypeofbulblastfortheadvertiser's claimtobevalid? 5.10.1.4LinearinequalitiesinOneVariableSection5.7 Solvetheinequalitiesforthefollowingproblems. Exercise5.503 Solutiononp.350. y +3 < 15 Exercise5.504 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 12 Exercise5.505 Solutiononp.350. 4 x +3 > 23 Exercise5.506 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 < 1 Exercise5.507 Solutiononp.350. 6 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 )]TJ/F8 9.9626 Tf 18.265 0 Td [(27 Exercise5.508 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y 14 Exercise5.509 Solutiononp.350. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 a )]TJ/F8 9.9626 Tf 18.265 0 Td [(88 Exercise5.510 x 7 > )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise5.511 Solutiononp.350. b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 4 Exercise5.512 2 a 7 < 6 Exercise5.513 Solutiononp.350. 16 c 3 )]TJ/F8 9.9626 Tf 18.264 0 Td [(48 Exercise5.514 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 c +3 5 Exercise5.515 Solutiononp.350. )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 y +4 > 15 Exercise5.516 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 > )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise5.517 Solutiononp.350. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 x +10+2 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 Exercise5.518 5 x +4 7 x +16 Exercise5.519 Solutiononp.350. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 < 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11

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335 Exercise5.520 4 x +1+2 )]TJ/F8 9.9626 Tf 18.265 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+4 Exercise5.521 Solutiononp.350. )]TJ/F8 9.9626 Tf 9.409 0 Td [( x +6+2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 < 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x +11 Exercise5.522 Whatnumberssatisfythecondition:ninelessthannegativefourtimesanumberisstrictlygreater thannegativeone? 5.10.1.5LinearEquationsinTwoVariablesSection5.8 Solvetheequationsforthefollowingproblems. Exercise5.523 Solutiononp.350. y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 x +4 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise5.524 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 x +11 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.525 Solutiononp.350. 3 a +2 b =14 ; if b =4 Exercise5.526 4 m +2 k =30 ; if m =8 Exercise5.527 Solutiononp.350. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 r +5 s = )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 ; if s =0 Exercise5.528 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 ; if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.529 Solutiononp.350. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a +19=2 b +6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 ; if b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise5.530 6 t +8= )]TJ/F8 9.9626 Tf 9.409 0 Td [( a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 ; if a =10 Exercise5.531 Solutiononp.350. )]TJ/F8 9.9626 Tf 9.41 0 Td [( a + b =5 ; if a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise5.532 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a a +1=2 b +1 ; if a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 5.11ProciencyExam 11 5.11.1ProciencyExam Solvetheequationsandinequalitiesforthefollowingproblems. Exercise5.533 Solutiononp.350. Section5.2 x +8=14 Exercise5.534 Solutiononp.350. Section5.2 6 a +3= )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Exercise5.535 Solutiononp.351. Section5.3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a 8 =6 11 Thiscontentisavailableonlineat.

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336 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES Exercise5.536 Solutiononp.351. Section5.4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 +16=11 Exercise5.537 Solutiononp.351. Section5.3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 4 +6=3 Exercise5.538 Solutiononp.351. Section5.4 5 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(8=7 b +12 Exercise5.539 Solutiononp.351. Section5.4 3 a +4=2 a +3 Exercise5.540 Solutiononp.351. Section5.4 5 y +3 )]TJ/F8 9.9626 Tf 9.963 0 Td [( y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise5.541 Solutiononp.351. Section5.3 )]TJ/F7 6.9738 Tf 6.226 0 Td [( x +3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x 3 =2 Exercise5.542 Solutiononp.351. Section5.4 Solve 2 p )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 q +1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 for p Exercise5.543 Solutiononp.351. Section5.3 Solve p = nRT V for T Exercise5.544 Solutiononp.351. Section5.4 Solve for 4 Exercise5.545 Solutiononp.351. Section5.7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 4 Exercise5.546 Solutiononp.351. Section5.7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a +1 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise5.547 Solutiononp.351. Section5.7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a +6 )]TJ/F11 9.9626 Tf 18.264 0 Td [(a +11 Exercise5.548 Solutiononp.351. Section5.7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 3 > )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Translatethephrasesorsentencesintomathematicalexpressionsorequationsforthefollowingproblems. Exercise5.549 Solutiononp.351. Section5.5 Threeaddedtotwiceanumber. Exercise5.550 Solutiononp.351. Section5.5 Eightlessthantwothirdsofanumber. Exercise5.551 Solutiononp.351. Section5.5 Twomorethanfourtimesanumber. Exercise5.552 Solutiononp.351. Section5.5 Anumberisaddedtoitselfandthisresultismultipliedbytheoriginalnumber cubed.Theresultistwelve. Exercise5.553 Solutiononp.351. Section5.5 Anumberisdecreasedbyveandthatresultisdividedbytenmorethanthe originalnumber.Theresultissixtimestheoriginalnumber. Solvethefollowingproblems. Exercise5.554 Solutiononp.351. Section5.6 Eightpercentofanumberis 1 : 2 .Whatisthenumber? Exercise5.555 Solutiononp.351. Section5.6 Threeconsecutiveoddintegerssumto38.Whatarethey?

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337 Exercise5.556 Solutiononp.351. Section5.6 Fivemorethanthreetimesanumberisstrictlylessthanseventeen.Whatisthe number? Exercise5.557 Solutiononp.351. Section5.8 Solve y =8 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 for y if x =3 ,andwritethesolutionasanorderedpair.

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338 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutionstoExercisesinChapter5 SolutiontoExercise5.1p.279 conditional, x =9 SolutiontoExercise5.2p.279 conditional, y =11 SolutiontoExercise5.3p.279 conditional, a =5 SolutiontoExercise5.4p.279 conditional, x =36 SolutiontoExercise5.5p.279 conditional, b =3 SolutiontoExercise5.6p.279 identity SolutiontoExercise5.7p.279 contradiction SolutiontoExercise5.8p.279 identity SolutiontoExercise5.9p.279 conditional, x =0 SolutiontoExercise5.10p.279 conditional, m =2 SolutiontoExercise5.11p.282 y =11 SolutiontoExercise5.12p.282 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 SolutiontoExercise5.13p.282 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise5.14p.282 g =8 : 5 SolutiontoExercise5.15p.282 f =3 d SolutiontoExercise5.16p.282 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise5.17p.282 y = x +9 SolutiontoExercise5.18p.282 conditional SolutiontoExercise5.20p.282 identity SolutiontoExercise5.22p.283 identity SolutiontoExercise5.24p.283 solved SolutiontoExercise5.26p.283 notsolved SolutiontoExercise5.28p.283 notsolved SolutiontoExercise5.30p.283 notsolved SolutiontoExercise5.32p.283 k = )]TJ/F8 9.9626 Tf 7.748 0 Td [(9

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339 SolutiontoExercise5.34p.283 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(17 SolutiontoExercise5.36p.283 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 SolutiontoExercise5.38p.283 g = )]TJ/F8 9.9626 Tf 7.748 0 Td [(287 SolutiontoExercise5.40p.283 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(443 SolutiontoExercise5.42p.283 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 : 059 SolutiontoExercise5.44p.284 n =4 )]TJ/F11 9.9626 Tf 9.962 0 Td [(m SolutiontoExercise5.46p.284 b = )]TJ/F11 9.9626 Tf 7.749 0 Td [(a +3 c + d )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 f SolutiontoExercise5.48p.284 c =2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 SolutiontoExercise5.50p.284 4 y 4 x 2 SolutiontoExercise5.52p.284 9 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x +1 SolutiontoExercise5.54p.287 a =7 SolutiontoExercise5.55p.287 m = )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(4 3 SolutiontoExercise5.56p.287 y = )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 SolutiontoExercise5.57p.287 x =0 : 17 roundedtotwodecimalplaces SolutiontoExercise5.58p.287 k = 24 5 SolutiontoExercise5.59p.287 b = )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 cd a SolutiontoExercise5.60p.287 y =12 h SolutiontoExercise5.61p.287 m = )]TJ/F7 6.9738 Tf 6.226 0 Td [(15 pq k 2 SolutiontoExercise5.62p.288 x =14 SolutiontoExercise5.64p.288 x =8 SolutiontoExercise5.66p.288 x =14 SolutiontoExercise5.68p.288 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(16 SolutiontoExercise5.70p.288 p = )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 SolutiontoExercise5.72p.288 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 SolutiontoExercise5.74p.288 x =7

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340 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutiontoExercise5.76p.288 k = )]TJ/F8 9.9626 Tf 7.748 0 Td [(42 SolutiontoExercise5.78p.288 x =6 SolutiontoExercise5.80p.288 k =42 SolutiontoExercise5.82p.288 x =768 SolutiontoExercise5.84p.289 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(56 SolutiontoExercise5.86p.289 f = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6386 SolutiontoExercise5.88p.289 k =0 : 06 SolutiontoExercise5.90p.289 y =9 : 453 SolutiontoExercise5.92p.289 m = )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 3 SolutiontoExercise5.94p.289 h = 21 8 SolutiontoExercise5.96p.289 p = 7 r q SolutiontoExercise5.98p.289 b = 2 d a SolutiontoExercise5.100p.289 b = 15 c 8 SolutiontoExercise5.102p.289 t = )]TJ/F7 6.9738 Tf 8.944 4.444 Td [(3 p 2 4 SolutiontoExercise5.104p.289 = 2 r 3 SolutiontoExercise5.106p.290 binomial;3rddegree; 10 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise5.108p.290 allrealnumbersexcept )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise5.110p.292 y =4 SolutiontoExercise5.111p.292 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise5.112p.292 n = 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 m 2 SolutiontoExercise5.113p.292 k = 2 hp )]TJ/F7 6.9738 Tf 6.226 0 Td [(14 h 9 SolutiontoExercise5.114p.293 x =11 SolutiontoExercise5.115p.293 y =19 SolutiontoExercise5.116p.293 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 SolutiontoExercise5.117p.293 a = 10+5 m 6

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341 SolutiontoExercise5.118p.294 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 SolutiontoExercise5.119p.294 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise5.120p.294 k =5 SolutiontoExercise5.121p.295 identity, 3=3 SolutiontoExercise5.122p.295 contradiction, )]TJ/F8 9.9626 Tf 7.748 0 Td [(28=28 SolutiontoExercise5.123p.295 identity, 0=0 SolutiontoExercise5.124p.295 contradiction, )]TJ/F8 9.9626 Tf 7.748 0 Td [(22=6 SolutiontoExercise5.125p.295 x =5 SolutiontoExercise5.127p.295 a =7 SolutiontoExercise5.129p.295 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise5.131p.296 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise5.133p.296 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise5.135p.296 x =36 SolutiontoExercise5.137p.296 y =24 SolutiontoExercise5.139p.296 m = )]TJ/F8 9.9626 Tf 7.748 0 Td [(44 SolutiontoExercise5.141p.296 k = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 SolutiontoExercise5.143p.296 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 SolutiontoExercise5.145p.296 k = )]TJ/F8 9.9626 Tf 7.748 0 Td [(14 SolutiontoExercise5.147p.296 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise5.149p.296 y =20 SolutiontoExercise5.151p.296 k = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 SolutiontoExercise5.153p.296 contradiction SolutiontoExercise5.155p.297 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise5.157p.297 x = 19 14 SolutiontoExercise5.159p.297 k =3

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342 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutiontoExercise5.161p.297 m =2 SolutiontoExercise5.163p.297 R =112 SolutiontoExercise5.165p.297 S x 2 = F S y 2 ; S x 2 =7 : 1604 SolutiontoExercise5.167p.297 y = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 4 SolutiontoExercise5.169p.297 y = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +12 5 SolutiontoExercise5.171p.297 n = 5 m + h 2 SolutiontoExercise5.173p.297 SolutiontoExercise5.175p.298 x +3 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 7 SolutiontoExercise5.177p.298 4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +1 SolutiontoExercise5.179p.298 x = )]TJ/F7 6.9738 Tf 6.226 0 Td [(15 4 SolutiontoExercise5.180p.300 11+ x SolutiontoExercise5.181p.300 9 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x SolutiontoExercise5.182p.300 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 SolutiontoExercise5.183p.300 4 x =32 SolutiontoExercise5.184p.300 x 3 =6 SolutiontoExercise5.185p.300 10 x =8+5 x SolutiontoExercise5.186p.301 x 16 +1=5 SolutiontoExercise5.187p.301 7+ x =21 SolutiontoExercise5.188p.301 x 2+ x =0 SolutiontoExercise5.189p.301 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 2 x +3 =17 SolutiontoExercise5.190p.301 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(52 SolutiontoExercise5.191p.301 11 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise5.192p.301 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise5.194p.301 b +7 SolutiontoExercise5.196p.302 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5+ c

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343 SolutiontoExercise5.198p.302 2 d +14 SolutiontoExercise5.200p.302 1 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(e SolutiontoExercise5.202p.302 4 9 f =21 SolutiontoExercise5.204p.302 3 g =2 g +9 SolutiontoExercise5.206p.302 2 h +6=30 SolutiontoExercise5.208p.302 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(25=3 : 019 SolutiontoExercise5.210p.302 m 4 =68 SolutiontoExercise5.212p.302 n 10 = n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 SolutiontoExercise5.214p.302 x = x +4 x SolutiontoExercise5.216p.302 Q +7 2 =22 SolutiontoExercise5.218p.303 r + r 3 3 =15 SolutiontoExercise5.220p.303 5 s +6 =14 SolutiontoExercise5.222p.303 20 8 x +1=9 SolutiontoExercise5.224p.303 v 10 +4=24 SolutiontoExercise5.226p.303 w +6 2 +5=43 SolutiontoExercise5.228p.303 7 y +2 y =90 SolutiontoExercise5.230p.303 z +16 6 z =59 SolutiontoExercise5.232p.303 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 15 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1=5 SolutiontoExercise5.234p.303 n )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise5.236p.303 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise5.238p.303 p )]TJ/F8 9.9626 Tf 9.963 0 Td [(8=3 SolutiontoExercise5.240p.304 23 2 n )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 =34 SolutiontoExercise5.242p.304 graph SolutiontoExercise5.244p.304 j a j = f a; if a 0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a; if a< 0

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344 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutiontoExercise5.246p.304 x =3 SolutiontoExercise5.247p.305 Lastyear'spricewas $19 SolutiontoExercise5.248p.306 Thelengthofasideis18inches. SolutiontoExercise5.249p.306 Thenumberis450. SolutiontoExercise5.250p.307 GardenAproduces87poundsofvegetables. SolutiontoExercise5.251p.308 Thetwonumbersare248and250. SolutiontoExercise5.252p.308 Step1:Let x =theunknownquantity. Step2:Theequationis x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18=52 : Step3:Solvetheequation.Add18toeachside. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18+18=52+18 x =70 Step4:Check 70 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18=52; True. Step5:Thenumberis70. SolutiontoExercise5.254p.308 Step5:Thenumberis30. SolutiontoExercise5.256p.309 Step5:Theoriginalquantityis29. SolutiontoExercise5.258p.309 Step5:Lastyear'soutputwas50items. SolutiontoExercise5.260p.309 Step5:Aprotonweighs4923.16units. SolutiontoExercise5.262p.310 Step5:Theradiusoftheearthis6378km. SolutiontoExercise5.264p.310 Step5:Thelengthoftheshorterpieceis3feet,andthelengthofthelongerpieceis9feet. SolutiontoExercise5.266p.311 Step5:Thestatisticianhascollected260piecesofdata. SolutiontoExercise5.268p.311 Step5:Eachbeakerwillhold263 1 3 mlofchloridesolution. SolutiontoExercise5.270p.312 Step5:Theunknownnumberis26. SolutiontoExercise5.272p.312 Step5:Theunknownnumberis21. SolutiontoExercise5.274p.312 Step5:Theunknownnumberis 41 3 : SolutiontoExercise5.276p.312 Step5:Oneunknownnumberis6;theotheris21.

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345 SolutiontoExercise5.278p.312 Step5:Theunknownnumberis 245 41 : SolutiontoExercise5.280p.312 Step5:TheageofDis3months;Cis6months;Bis36months;Ais144months. SolutiontoExercise5.282p.312 Step5:Therstintegeris11;secondis12;thirdis13. SolutiontoExercise5.284p.312 a Step5:Thetimepassedinspaceis 0 : 62 days. b Step5: 9 : 3 yearshavepassedonthespacecraft. c Step5: 96 : 77 yearshavepassedontheearth. d Step5:Earthyearwhenshereturnswillbe2387. SolutiontoExercise5.286p.313 conditional SolutiontoExercise5.288p.313 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise5.290p.317 y 11 SolutiontoExercise5.291p.317 x> 5 SolutiontoExercise5.292p.317 x 4 SolutiontoExercise5.293p.317 y 9 5 SolutiontoExercise5.294p.317 s< 29 2 SolutiontoExercise5.295p.317 h> 1 18 SolutiontoExercise5.296p.317 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(30 SolutiontoExercise5.297p.317 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(64 3 SolutiontoExercise5.298p.317 z< )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 7 SolutiontoExercise5.299p.317 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 SolutiontoExercise5.300p.318 9
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346 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutiontoExercise5.306p.319 x< 5 SolutiontoExercise5.308p.319 y )]TJ/F8 9.9626 Tf 18.265 0 Td [(17 SolutiontoExercise5.310p.319 x 3 SolutiontoExercise5.312p.319 z< )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 SolutiontoExercise5.314p.319 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(4 SolutiontoExercise5.316p.319 z> )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 SolutiontoExercise5.318p.319 x 48 SolutiontoExercise5.320p.319 x 18 SolutiontoExercise5.322p.319 x 6 5 SolutiontoExercise5.324p.319 b> )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 SolutiontoExercise5.326p.320 x< )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(15 4 SolutiontoExercise5.328p.320 y> 16 21 SolutiontoExercise5.330p.320 y 2 SolutiontoExercise5.332p.320 x 15 SolutiontoExercise5.334p.320 x 4 SolutiontoExercise5.336p.320 x 5 SolutiontoExercise5.338p.320 y< )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise5.340p.320 x< 6 SolutiontoExercise5.342p.320 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(7 SolutiontoExercise5.344p.320 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(4 SolutiontoExercise5.346p.320 x> 8 SolutiontoExercise5.348p.321 y )]TJ/F8 9.9626 Tf 18.265 0 Td [(9 SolutiontoExercise5.350p.321 x< )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise5.352p.321 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(3 SolutiontoExercise5.354p.321 x< )]TJ/F8 9.9626 Tf 9.963 0 Td [(2

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347 SolutiontoExercise5.356p.321 x> )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise5.358p.321 Firstnumber:anynumberstrictlysmallerthat6. Secondnumber:anynumberstrictlysmallerthan30. Nosmallestpossiblevalueforeithernumber. Nolargestpossiblevalueforeithernumber. SolutiontoExercise5.360p.321 x 10 y 15 z 10 SolutiontoExercise5.362p.321 2 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 SolutiontoExercise5.364p.321 16inches SolutiontoExercise5.365p.324 ; 1 SolutiontoExercise5.366p.324 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 SolutiontoExercise5.367p.324 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 SolutiontoExercise5.368p.324 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 SolutiontoExercise5.369p.324 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 SolutiontoExercise5.370p.324 ; 22 SolutiontoExercise5.372p.324 ; 26 SolutiontoExercise5.374p.324 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 9 4 SolutiontoExercise5.376p.325 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(29 3 SolutiontoExercise5.378p.325 )]TJ/F7 6.9738 Tf 5.762 -4.148 Td [(1 2 ; 0 SolutiontoExercise5.380p.325 ; 1 SolutiontoExercise5.382p.325 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(9 4 ; 3 2 SolutiontoExercise5.384p.325 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 ; 111 20 SolutiontoExercise5.386p.325 )]TJ/F8 9.9626 Tf 7.749 0 Td [(40 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 SolutiontoExercise5.388p.325 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a; 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 31 SolutiontoExercise5.390p.325 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(30 SolutiontoExercise5.392p.325 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 ; )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(9 5 SolutiontoExercise5.394p.325 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6 ; )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(21 2 SolutiontoExercise5.396p.325 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(40

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348 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutiontoExercise5.398p.326 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 7 4 SolutiontoExercise5.400p.326 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 ; )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(38 3 SolutiontoExercise5.402p.326 ; 5 SolutiontoExercise5.404p.326 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 SolutiontoExercise5.406p.326 ; 6 SolutiontoExercise5.408p.326 a Approximately147mphusing ; 147 : 25 b Approximately157mphusing ; 156 : 9 c Approximately188mphusing ; 187 : 78 d Approximately196mphusing ; 195 : 5 SolutiontoExercise5.410p.327 a Approximately5sinkingsusing ; 4 : 92 b Approximately10sinkingsusing ; 10 : 12 c Approximately11sinkingsusing ; 11 : 16 SolutiontoExercise5.412p.327 a Approximately 6 : 4 using ; 6 : 4 b Approximately 4 : 752 using ; 7 : 752 SolutiontoExercise5.414p.327 a Approximately29itemsusing ; 29 : 42 b Approximately43itemsusing ; 42 : 77 SolutiontoExercise5.416p.328 a Approximately87unitsusing ; 87 : 31 b Approximately45unitsusing ; 44 : 86 SolutiontoExercise5.418p.328 12 x 2 +17 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise5.420p.328 x =0 SolutiontoExercise5.422p.328 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3
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349 SolutiontoExercise5.439p.330 x =4 SolutiontoExercise5.441p.330 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise5.443p.331 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise5.445p.331 m =24 SolutiontoExercise5.447p.331 y =486 SolutiontoExercise5.449p.331 c =90 SolutiontoExercise5.451p.331 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 SolutiontoExercise5.453p.331 a = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(55 6 SolutiontoExercise5.455p.331 c =16 SolutiontoExercise5.457p.331 x =28 SolutiontoExercise5.459p.331 r = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(7 3 SolutiontoExercise5.461p.331 x = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(45 4 SolutiontoExercise5.463p.331 x =3 SolutiontoExercise5.465p.332 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 SolutiontoExercise5.467p.332 r = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(17 4 SolutiontoExercise5.469p.332 W =13 SolutiontoExercise5.471p.332 R =690 SolutiontoExercise5.473p.332 x = y )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 5 SolutiontoExercise5.475p.332 y = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(1 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise5.477p.332 b = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(2 ct )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 a 3 SolutiontoExercise5.479p.332 x +7 SolutiontoExercise5.481p.332 2 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise5.483p.332 3 x =40 SolutiontoExercise5.485p.333 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 7 = x +10 SolutiontoExercise5.487p.333 x x +11=6

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350 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES SolutiontoExercise5.489p.333 x 2 x +8 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise5.491p.333 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 5 x = x 3 SolutiontoExercise5.493p.333 lastyear'sprice =$96 SolutiontoExercise5.495p.333 x =860 SolutiontoExercise5.497p.333 x =11 x +2=13 x +4=15 x +6=17 SolutiontoExercise5.499p.333 x =116 : 13139 SolutiontoExercise5.501p.334 82 piecesofdata SolutiontoExercise5.503p.334 y< 12 SolutiontoExercise5.505p.334 x> 5 SolutiontoExercise5.507p.334 a )]TJ/F7 6.9738 Tf 19.461 3.923 Td [(7 2 SolutiontoExercise5.509p.334 a 11 SolutiontoExercise5.511p.334 b )]TJ/F8 9.9626 Tf 18.265 0 Td [(12 SolutiontoExercise5.513p.334 c )]TJ/F8 9.9626 Tf 18.265 0 Td [(9 SolutiontoExercise5.515p.334 y< )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise5.517p.334 x> )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(9 14 SolutiontoExercise5.519p.334 x> 3 2 SolutiontoExercise5.521p.335 x< 7 3 SolutiontoExercise5.523p.335 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 19 SolutiontoExercise5.525p.335 ; 4 SolutiontoExercise5.527p.335 ; 0 SolutiontoExercise5.529p.335 )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(7 2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise5.531p.335 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; 0 SolutiontoExercise5.533p.335 x =6

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351 SolutiontoExercise5.534p.335 a = )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 6 SolutiontoExercise5.535p.335 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(16 SolutiontoExercise5.536p.335 x =10 SolutiontoExercise5.537p.336 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise5.538p.336 b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 SolutiontoExercise5.539p.336 a = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(3 2 SolutiontoExercise5.540p.336 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 SolutiontoExercise5.541p.336 x =9 SolutiontoExercise5.542p.336 p = 6 q )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 SolutiontoExercise5.543p.336 T = Vp nR SolutiontoExercise5.544p.336 SolutiontoExercise5.545p.336 a 12 SolutiontoExercise5.546p.336 a> 2 SolutiontoExercise5.547p.336 a )]TJ/F8 9.9626 Tf 18.265 0 Td [(23 SolutiontoExercise5.548p.336 x< 6 SolutiontoExercise5.549p.336 3+2 a SolutiontoExercise5.550p.336 2 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 SolutiontoExercise5.551p.336 2+4 x SolutiontoExercise5.552p.336 2 x )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 3 =12 SolutiontoExercise5.553p.336 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x +10 =6 x SolutiontoExercise5.554p.336 x =15 SolutiontoExercise5.555p.336 Therearenothreeconsecutiveoddintegersthataddto38. SolutiontoExercise5.556p.337 x< 4 SolutiontoExercise5.557p.337 ; 13

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352 CHAPTER5.SOLVINGLINEAREQUATIONSANDINEQUALITIES

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Chapter6 FactoringPolynomials 6.1Objectives 1 Aftercompletingthischapter,youshould FindingtheFactorsofaMonomialSection6.2 beremindedofproductsofpolynomials beabletodetermineasecondfactorofapolynomialgivenarstfactor FactoringaMonomialfromaPolynomialSection6.3 beabletofactoramonomialfromapolynomial TheGreatestCommonFactorSection6.4 understandmoreclearlythefactorizationprocess beabletodeterminethegreatestcommonfactoroftwoormoreterms FactoringbyGroupingSection6.5 knowhowtofactorapolynomialusingthegroupingmethodandwhentotrythegroupingmethod FactoringTwoSpecialProductsSection6.6 knowthefundamentalrulesoffactoring beabletofactorthedierenceoftwosquaresandperfectsquaretrinomials FactoringTrinomialswithLeadingCoecient1Section6.7 beabletofactortrinomialswithleadingcoecient1 becomefamiliarwithsomefactoringhints FactoringTrinomialswithLeadingCoecientOtherThan1Section6.8 beabletofactortrinomialswithleadingcoecientotherthan1 1 Thiscontentisavailableonlineat. 353

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354 CHAPTER6.FACTORINGPOLYNOMIALS 6.2FindingthefactorsofaMonomial 2 6.2.1Overview ProductsofPolynomials Factoring 6.2.2ProductsofPolynomials Previously,westudiedmultiplicationofpolynomialsSectionSection4.6.Weweregiven factors andasked tondtheir product ,asshownbelow. Example6.1 Giventhefactors4and8,ndtheproduct. 4 8=32 .Theproductis32. Example6.2 Giventhefactors 6 x 2 and 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 ,ndtheproduct. 6 x 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=12 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(42 x 2 Theproductis 12 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(42 x 2 Example6.3 Giventhefactors x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y and 3 x + y ,ndtheproduct. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y x + y =3 x 2 + xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 =3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 Theproductis 3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 Example6.4 Giventhefactors a +8 and a +8 ,ndtheproduct. a +8 2 = a 2 +16 a +64 Theproductis a 2 +16 a +64 6.2.3Factoring Now,let'sreversethesituation.Wewillbegiventheproduct,andwewilltrytondthefactors.This process,whichisthereverseofmultiplication,iscalled factoring Factoring Factoring istheprocessofdeterminingthefactorsofagivenproduct. 6.2.4SampleSetA Example6.5 Thenumber24istheproduct,andonefactoris6.Whatistheotherfactor? We'relookingforanumber suchthat 6 =24 .Weknowfromexperience that =4 .Asproblemsbecomeprogressivelymorecomplex,ourexperiencemaynotgive usthesolutiondirectly.Weneedamethodforndingfactors.Todevelopthismethodwecanuse therelativelysimpleproblem 6 =24 asaguide. Tondthenumber ,wewould divide 24by6. 2 Thiscontentisavailableonlineat.

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355 24 6 =4 Theotherfactoris4. Example6.6 Theproductis 18 x 3 y 4 z 2 andonefactoris 9 xy 2 z .Whatistheotherfactor? Weknowthatsince 9 xy 2 z isafactorof 18 x 3 y 4 z 2 ,theremustbesomequantity suchthat 9 xy 2 z =18 x 3 y 4 z 2 .Dividing 18 x 3 y 4 z 2 by 9 xy 2 z ,weget 18 x 3 y 4 z 2 9 xy 2 z =2 x 2 y 2 z Thus,theotherfactoris 2 x 2 y 2 z Checkingwillconvinceusthat 2 x 2 y 2 z isindeedtheproperfactor. )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 x 2 y 2 z )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(9 xy 2 z =18 x 2+1 y 2+2 z 1+1 =18 x 3 y 4 z 2 Weshouldtrytondthequotientmentallyandavoidactuallywritingthedivisionproblem. Example6.7 Theproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 a 5 b n and 3 ab 4 isafactor.Findtheotherfactor. Mentallydividing )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 a 5 b n by 3 ab 4 ,weget )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 a 5 b n 3 ab 4 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 a 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 a 4 b n )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Thus,theotherfactoris )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 a 4 b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 6.2.5PracticeSetA Exercise6.1 Solutiononp.400. Theproductis84andonefactoris6.Whatistheotherfactor? Exercise6.2 Solutiononp.400. Theproductis 14 x 3 y 2 z 5 andonefactoris 7 xyz .Whatistheotherfactor? 6.2.6Exercises Inthefollowingproblems,therstquantityrepresentstheproductandthesecondquantityrepresentsa factorofthatproduct.Findtheotherfactor. Exercise6.3 Solutiononp.400. 30 ; 6 Exercise6.4 45 ; 9 Exercise6.5 Solutiononp.400. 10 a; 5 Exercise6.6 16 a; 8 Exercise6.7 Solutiononp.400. 21 b; 7 b Exercise6.8 15 a; 5 a

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356 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.9 Solutiononp.400. 20 x 3 ; 4 Exercise6.10 30 y 4 ; 6 Exercise6.11 Solutiononp.400. 8 x 4 ; 4 x Exercise6.12 16 y 5 ; 2 y Exercise6.13 Solutiononp.400. 6 x 2 y; 3 x Exercise6.14 9 a 4 b 5 ; 9 a 4 Exercise6.15 Solutiononp.400. 15 x 2 b 4 c 7 ; 5 x 2 bc 6 Exercise6.16 25 a 3 b 2 c; 5 ac Exercise6.17 Solutiononp.400. 18 x 2 b 5 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 xb 4 Exercise6.18 22 b 8 c 6 d 3 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 b 8 c 4 Exercise6.19 Solutiononp.400. )]TJ/F8 9.9626 Tf 7.749 0 Td [(60 x 5 b 3 f 9 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 x 2 b 2 f 2 Exercise6.20 39 x 4 y 5 z 11 ; 3 xy 3 z 10 Exercise6.21 Solutiononp.400. 147 a 20 b 6 c 18 d 2 ; 21 a 3 bd Exercise6.22 )]TJ/F8 9.9626 Tf 7.749 0 Td [(121 a 6 b 8 c 10 ; 11 b 2 c 5 Exercise6.23 Solutiononp.400. 1 8 x 4 y 3 ; 1 2 xy 3 Exercise6.24 7 x 2 y 3 z 2 ; 7 x 2 y 3 z Exercise6.25 Solutiononp.400. 5 a 4 b 7 c 3 d 2 ; 5 a 4 b 7 c 3 d Exercise6.26 14 x 4 y 3 z 7 ; 14 x 4 y 3 z 7 Exercise6.27 Solutiononp.400. 12 a 3 b 2 c 8 ; 12 a 3 b 2 c 8 Exercise6.28 6 a +1 2 a +5 ; 3 a +1 2 Exercise6.29 Solutiononp.400. 8 x + y 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y ; 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y Exercise6.30 14 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 6 a +4 2 ; 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 a +4 Exercise6.31 Solutiononp.400. 26 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y 10 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y 12 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y 7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y 7

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357 Exercise6.32 34 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 4 + a 8 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(17 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 4 + a 2 Exercise6.33 Solutiononp.400. x + y x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y ;x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y Exercise6.34 a +3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 ;a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise6.35 Solutiononp.400. 48 x n +3 y 2 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; 8 x 3 y n +5 Exercise6.36 0 : 0024 x 4 n y 3 n +5 z 2 ; 0 : 03 x 3 n y 5 6.2.7ExercisesforReview Exercise6.37 Solutiononp.400. Section2.6 Simplify )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 4 y 0 z 2 3 Exercise6.38 Section3.3 Simplify f)]TJ/F8 9.9626 Tf 22.14 0 Td [([ )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 6 j ] g Exercise6.39 Solutiononp.400. Section4.7 Findtheproduct. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 6.3FactoringaMonomialfromaPolynomial 3 6.3.1Overview TheFactorizationProcess 6.3.2TheFactorizationProcess Weintroducetheprocessoffactoringamonomialfromapolynomialbyexaminingaproblem:Supposethat 12 x 2 +20 x istheproductandoneofthefactorsis 4 x .Tondtheotherfactorwecouldsetuptheproblem thisway: 4 x =12 x 2 +20 x Sincetheproduct 12 x 2 +20 x consistsoftwoterms,theexpressionmultiplying 4 x mustconsistoftwo terms,since,bythedistributiveproperty Nowweseethatthisproblemissimplyanextensionofndingthefactorsofamonomial. 1 stterm :4 x =12 x 2 2 ndterm :4 x =20 x = 12 x 2 4 x = 20 x 4 x =3 x =5 Thus, 4 x x +5=12 x 2 +20 x Usually,thesedivisionscanbedonementallyandthetermsofthefactorlledindirectly. 3 Thiscontentisavailableonlineat.

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358 CHAPTER6.FACTORINGPOLYNOMIALS 6.3.3SampleSetA Example6.8 Theproductis 3 x 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x 4 andonefactoris x 4 .Findtheotherfactor. Wehavetheproblem: x 4 times"whatexpression"yields 3 x 7 )]TJ/F8 9.9626 Tf 8.385 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 8.385 0 Td [(3 x 4 ?Mathematically, x 4 =3 x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 4 Sincetherearefourtermsintheproduct,theremustbefourtermsinsidetheparentheses.To ndeachofthefourterms,we'lldividementallyeachtermoftheproductby x 4 .Theresulting quotientwillbethenecessarytermofthefactor. 1 stterm : 3 x 7 x 4 =3 x 7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 =3 x 3 Place 3 x 3 intothe 1 stpositioninthe : 2 ndterm : )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x 6 x 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x 2 Place )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 intothe 2 ndpositioninthe : 3 rdterm : 4 x 5 x 4 =4 x Place 4 x intothe 3 rdpositioninthe : 4 thterm : )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x 4 x 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Place )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 intothe 4 thpositioninthe : Therefore,theotherfactoris 3 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Thisresultcanbecheckedbyapplyingthedistributiveproperty. x 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 2 +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 =3 x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 4 Isthiscorrect? 3 x 4+3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 4+2 +4 x 4+1 =3 x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 4 Isthiscorrect? 3 x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x 4 =3 x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x 4 Yes,thisiscorrect. Thus, x 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 2 +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 =3 x 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 6 +4 x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 4 Again,ifthedivisionscanbeperformedmentally,theprocesscanproceedveryquickly. Example6.9 Theproductis 10 x 3 y 6 +15 x 3 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x 2 y 4 andafactoris 5 x 2 y 4 .Findtheotherfactor. 5 x 2 y 4 =10 x 3 y 6 +15 x 3 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x 2 y 4 Sincetherearethreetermsintheproduct,theremustbethreetermsinsidetheparentheses. Tondeachofthesethreeterms,we'lldivideeachtermoftheproductby 5 x 2 y 4 1 stterm : 10 x 3 y 6 5 x 2 y 4 =2 xy 2 Placethe 2 xy 2 intothe1stpositioninthe : 2 ndterm : 15 x 3 y 4 5 x 2 y 4 =3 x Placethe 3 x intothe2ndpositioninthe : 3 rdterm : )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 y 4 5 x 2 y 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Placethe )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 intothe3rdpositioninthe : Theotherfactoris 2 xy 2 +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ,and 5 x 2 y 4 )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(2 xy 2 +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 =10 x 3 y 6 +15 x 3 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x 2 y 4 Example6.10 Theproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 3 +2 c andafactoris )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 .Findtheotherfactor. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 3 +2 c Sincetherearethreetermsintheproduct,theremustbethreetermsinsidetheparentheses. Wewilldividementallyeachtermoftheproductby )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 stterm : )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 =4 a 2 Place 4 a 2 intothe1stpositioninsidethe : 2 ndterm : )]TJ/F10 6.9738 Tf 6.226 0 Td [(b 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = b 3 Place b 3 intothe2ndpositioninsidethe : 3 rdterm : 2 c )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 c Place )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 c intothe3rdpositioninsidethe : Theotherfactoris 4 a 2 + b 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 c ,and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 a 2 + b 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 c = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 3 +2 c Withoutwritingthe )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ,weget

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359 )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(4 a 2 + b 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 c = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 3 +2 c Example6.11 Theproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 b 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 a 3 b 2 +9 a 2 b 2 andafactoris )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 b 2 .Findtheotherfactor. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 b 2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 a 2 b 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 a 3 b 2 +9 a 2 b 2 Mentallydividingeachtermoftheoriginaltrinomialby )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 b 2 ,weget b 3 +5 a )]TJ/F8 9.9626 Tf 9.342 0 Td [(3 astheother factor,and )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 b 2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(b 3 +5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 b 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 a 3 b 2 +9 a 2 b 2 6.3.4PracticeSetA Exercise6.40 Solutiononp.400. Theproductis 3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x andafactoris 3 x .Findtheotherfactor. Exercise6.41 Solutiononp.400. Theproductis 5 y 4 +10 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 y 2 andafactoris 5 y 2 .Findtheotherfactor. Exercise6.42 Solutiononp.400. Theproductis 4 x 5 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x 4 y 4 +16 x 3 y 5 +24 xy 7 andafactoris 4 xy 3 .Findtheotherfactor. Exercise6.43 Solutiononp.400. Theproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(25 a 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 a 2 +5 andafactoris )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 .Findtheotherfactor. Exercise6.44 Solutiononp.401. Theproductis )]TJ/F11 9.9626 Tf 7.748 0 Td [(a 2 + b 2 andafactoris )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 .Findtheotherfactor. 6.3.5Exercises Forthefollowingproblems,therstquantityrepresentstheproductandthesecondquantityafactor.Find theotherfactor. Exercise6.45 Solutiononp.401. 4 x +10 ; 2 Exercise6.46 6 y +18 ; 3 Exercise6.47 Solutiononp.401. 5 x +25 ; 5 Exercise6.48 16 a +64 ; 8 Exercise6.49 Solutiononp.401. 3 a 2 +9 a; 3 a Exercise6.50 14 b 2 +16 b; 2 b Exercise6.51 Solutiononp.401. 21 x 2 +28 x; 7 x Exercise6.52 45 y 2 +50 y; 5 y Exercise6.53 Solutiononp.401. 18 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a; 2 a Exercise6.54 20 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a; 4 a

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360 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.55 Solutiononp.401. 7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 x; 7 x Exercise6.56 6 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 y; 6 y Exercise6.57 Solutiononp.401. 8 a 2 +4 a; 4 a Exercise6.58 26 b 2 +13 b; 13 b Exercise6.59 Solutiononp.401. 9 x 2 +6 x +18 ; 6 Exercise6.60 12 b 2 +16 b +20 ; 4 Exercise6.61 Solutiononp.401. 21 x 2 +7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 ; 7 Exercise6.62 35 x 2 +40 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 ; 5 Exercise6.63 Solutiononp.401. 14 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(28 y +14 ; 14 Exercise6.64 36 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 a +12 ; 4 Exercise6.65 Solutiononp.401. 4 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 ; 2 Exercise6.66 6 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 ; 3 Exercise6.67 Solutiononp.401. 18 x 3 +20 x; 2 x Exercise6.68 40 y 3 +24 y; 4 y Exercise6.69 Solutiononp.401. 16 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x 2 ; 4 x 2 Exercise6.70 11 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 x +11 ; 11 Exercise6.71 Solutiononp.401. 10 a 3 +12 a 2 +16 a +8 ; 2 Exercise6.72 14 b 3 +16 b 2 +26 b +30 ; 2 Exercise6.73 Solutiononp.401. 8 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a +16 ; 4 Exercise6.74 25 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 x 2 +15 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 ; 5 Exercise6.75 Solutiononp.401. 4 x 6 +16 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 x; 4 x Exercise6.76 9 a 5 +6 a 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 a 4 +24 a 2 ; 3 a 2 Exercise6.77 Solutiononp.401. 10 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 x 2 ; 5 x 2

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361 Exercise6.78 12 x 3 y 5 +20 x 3 y 2 ; 4 x 3 y 2 Exercise6.79 Solutiononp.401. 10 a 4 b 3 +4 a 3 b 4 ; 2 a 3 b 3 Exercise6.80 8 a 3 b 6 c 8 +12 a 2 b 5 c 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 a 2 b 7 c 5 ; 4 a 2 b 5 c 5 Exercise6.81 Solutiononp.401. 4 x 5 y 4 + x 2 + x;x Exercise6.82 14 a 5 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a 4 b 4 +7 a 3 ;a 3 Exercise6.83 Solutiononp.401. 64 a 5 b 3 c 11 +56 a 4 b 4 c 10 )]TJ/F8 9.9626 Tf 9.963 0 Td [(48 a 3 b 5 c 9 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 a 3 b 2 c 5 ; 8 a 3 b 2 c 5 Exercise6.84 3 h 3 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 h 6 b 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 h 2 b + hb;hb Exercise6.85 Solutiononp.401. 5 a +10 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Exercise6.86 6 b +8 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise6.87 Solutiononp.401. 8 x 2 +12 x; )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x Exercise6.88 20 a 2 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a 2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a 2 Exercise6.89 Solutiononp.401. a + b; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise6.90 x + y; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise6.91 Solutiononp.401. a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b + c; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise6.92 2 x +4 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z; )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise6.93 Solutiononp.402. )]TJ/F11 9.9626 Tf 7.749 0 Td [(a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b )]TJ/F11 9.9626 Tf 9.963 0 Td [(c; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise6.94 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x +1 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 6.3.6ExercisesforReview Exercise6.95 Solutiononp.402. Section4.2 Howmany 4 y 2 'saretherein 24 x 2 y 3 ? Exercise6.96 Section4.7 Findtheproduct. y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 Exercise6.97 Solutiononp.402. Section5.4 Solve 2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a =7 Exercise6.98 Section6.2 Giventhat 3 m 2 n isafactorof 12 m 3 n 4 ,ndtheotherfactor.

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362 CHAPTER6.FACTORINGPOLYNOMIALS 6.4TheGreatestCommonFactor 4 6.4.1Overview FactoringMethod GreatestCommonFactor 6.4.2FactoringMethod InthelasttwotypesofproblemsSectionsSection6.2andSection6.3,weknewoneofthefactorsand wereabletodeterminetheotherfactorthroughdivision.Suppose,now,we'regiventheproductwithout anyfactors.Ourproblemistondthefactors,ifpossible.Thisprocedureandtheprevioustwoprocedures arebasedonthedistributiveproperty. Wewillusethedistributivepropertyinreverse. ab + ac | {z } product = a b + c | {z } factors Wenoticethatintheproduct, a iscommontobothterms.Infact, a isacommonfactorofbothterms. Since a iscommontobothterms,wewill factoritout andwrite a Nowweneedtodeterminewhattoplaceinsidetheparentheses.Thisistheprocedureoftheprevious section.Divideeachtermoftheproductbytheknownfactor a: ab a = b and ac a = c Thus, b and c aretherequiredtermsoftheotherfactor.Hence, ab + ac = a b + c Whenfactoringamonomialfromapolynomial,weseekoutfactorsthatarenotonlycommontoeach termofthepolynomial,butfactorsthathavetheseproperties: 1.Thenumericalcoecientsarethelargestcommonnumericalcoecients. 2.Thevariablespossessthelargestexponentscommontoallthevariables. 6.4.3GreatestCommonFactor Amonomialfactorthatmeetstheabovetworequirementsiscalledthe greatestcommonfactor ofthe polynomial. 6.4.4SampleSetA Example6.12 Factor 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 : Thegreatestcommonfactoris3. 4 Thiscontentisavailableonlineat.

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363 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18=3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 6 Factorout 3 : 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18=3 Divideeachtermoftheproductby 3 : 3 x 3 = x and )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 3 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Trytoperformthisdivisionmentally. 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18=3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Example6.13 Factor 9 x 3 +18 x 2 +27 x: Noticethat 9 x isthegreatestcommonfactor. 9 x 3 +18 x 2 +27 x =9 x x 2 +9 x 2 x +9 x 3 : Factorout 9 x: 9 x 3 +18 x 2 +27 x =9 x Mentallydivide 9 x intoeachtermoftheproduct. 9 x 3 +18 x 2 +27 x =9 x )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 +2 x +3 Example6.14 Factor 10 x 2 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 xy 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 y 5 : Noticethat 5 y 3 isthegreatestcommonfactor.Factorout 5 y 3 : 10 x 2 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 xy 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 y 5 =5 y 3 Mentallydivide 5 y 3 intoeachtermoftheproductandplacetheresultingquotientsinsidethe : 10 x 2 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 xy 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 y 5 =5 y 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 y 2 Example6.15 Factor )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 x 5 +8 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 2 : Weseethatthegreatestcommonfactoris )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x 2 : )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 x 5 +8 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x 2 Mentallydividing )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x 2 intoeachtermoftheproduct,weget )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 x 5 +8 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +1 6.4.5PracticeSetA Exercise6.99 Solutiononp.402. Factor 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(48 : Exercise6.100 Solutiononp.402. Factor 6 y 3 +24 y 2 +36 y: Exercise6.101 Solutiononp.402. Factor 10 a 5 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 a 4 b 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 b 6 : Exercise6.102 Solutiononp.402. Factor )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 m 4 +28 m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 m:

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364 CHAPTER6.FACTORINGPOLYNOMIALS 6.4.6 Considerthisproblem:factor Ax + Ay: Surely, Ax + Ay = A x + y : Weknowfromtheverybeginningof ourstudyofalgebrathatlettersrepresentsinglequantities.Wealsoknowthataquantityoccurringwithin asetofparenthesesistobeconsideredasasinglequantity.Supposethattheletter A isrepresentingthe quantity a + b : Thenwehave Ax + Ay = A x + y a + b x + a + b y = a + b x + y Whenweobservetheexpression a + b x + a + b y wenoticethat a + b iscommontobothterms.Sinceitiscommon,wefactoritout. a + b Asusual,wedeterminewhattoplaceinsidetheparenthesesbydividingeachtermoftheproductby a + b : a + b x a + b = x and a + b y a + b = y Thus,weget a + b x + a + b y = a + b x + y ThisisaforerunnerofthefactoringthatwillbedoneinSection 5 : 4 : 6.4.7SampleSetB Example6.16 Factor x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 b: Noticethat x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 isthegreatestcommonfactor.Factorout x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 : x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 b = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Then ; x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 a x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = a and x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 b x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 = b: x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 b = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 a + b Example6.17 Factor 3 x 2 x +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x x +1 Noticethat x and x +1 arecommontobothterms.Factorthemout.We'llperformthis factorizationbyletting A = x x +1 : Thenwehave 3 xA )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 A = A x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 But A = x x +1 ; so 3 x 2 x +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x x +1= x x +1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 6.4.8PracticeSetB Exercise6.103 Solutiononp.402. Factor y +4 a + y +4 b: Exercise6.104 Solutiononp.402. Factor 8 m 3 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 m 2 n )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 :

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365 6.4.9Exercises Forthefollowingproblems,factorthepolynomials. Exercise6.105 Solutiononp.402. 9 a +18 Exercise6.106 6 a +24 Exercise6.107 Solutiononp.402. 8 b +12 Exercise6.108 16 x +12 Exercise6.109 Solutiononp.402. 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise6.110 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Exercise6.111 Solutiononp.402. 21 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 Exercise6.112 16 f )]TJ/F8 9.9626 Tf 9.962 0 Td [(36 Exercise6.113 Solutiononp.402. 12 x 2 +18 x Exercise6.114 10 y 2 +15 y Exercise6.115 Solutiononp.402. 8 y 2 +18 Exercise6.116 7 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(21 Exercise6.117 Solutiononp.402. 3 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 Exercise6.118 2 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise6.119 Solutiononp.402. 6 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 y Exercise6.120 ax 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a Exercise6.121 Solutiononp.402. by 2 + b Exercise6.122 7 by 2 +14 b Exercise6.123 Solutiononp.402. 5 a 2 x 2 +10 x Exercise6.124 24 ax 2 +28 a Exercise6.125 Solutiononp.402. 10 x 2 +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(15

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366 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.126 12 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 Exercise6.127 Solutiononp.402. 15 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 y +9 Exercise6.128 ax 2 + ax + a Exercise6.129 Solutiononp.402. by 3 + by 2 + by + b Exercise6.130 2 y 2 +6 y +4 xy Exercise6.131 Solutiononp.402. 9 x 2 +6 xy +4 x Exercise6.132 30 a 2 b 2 +40 a 2 b 2 +50 a 2 b 2 Exercise6.133 Solutiononp.402. 13 x 2 y 5 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(26 x 2 y 5 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(39 x 2 y 5 Exercise6.134 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise6.135 Solutiononp.402. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 y +10 Exercise6.136 Ab + Ac Exercise6.137 Solutiononp.403. Nx + Ny Exercise6.138 Qx + Qy Exercise6.139 Solutiononp.403. Ax )]TJ/F11 9.9626 Tf 9.963 0 Td [(Ay Exercise6.140 x +4 b + x +4 c Exercise6.141 Solutiononp.403. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 a + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 b Exercise6.142 x +7 a + x +7 b Exercise6.143 Solutiononp.403. a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b w )]TJ/F8 9.9626 Tf 9.962 0 Td [( a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b x Exercise6.144 )]TJ/F11 9.9626 Tf 9.963 0 Td [(v X + )]TJ/F11 9.9626 Tf 9.962 0 Td [(v Y Exercise6.145 Solutiononp.403. 3 x 5 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x 3 y 4 +27 x 5 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x 2 y 6 Exercise6.146 8 a 3 b 15 +24 a 2 b 14 +48 a 3 b 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 a 3 b 7 +80 a 4 b 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a 3 b 7 +4 a 2 b Exercise6.147 Solutiononp.403. )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 x 3 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 3 y 2 +16 x 4 y 3 +2 x 2 y

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367 6.4.10ExercisesforReview Exercise6.148 Section5.6 Aquantityplus 21% moreofthatquantityis 26 : 25 : Whatistheoriginalquantity? Exercise6.149 Solutiononp.403. Section5.8 Solvetheequation 6 t )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=4 )]TJ/F11 9.9626 Tf 9.962 0 Td [(s if s =2 : Exercise6.150 Section6.3 Giventhat 4 a 3 isafactorof 8 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a 2 ; ndtheotherfactor. 6.5FactoringbyGrouping 5 6.5.1Overview UsingGroupingtoFactoraPolynomial KnowingwhentoTrytheGroupingMethod 6.5.2UsingGroupingtoFactoraPolynomial Sometimesapolynomialwillnothaveaparticularfactorcommontoeveryterm.However,wemaystillbe abletoproduceafactoredformforthepolynomial. Thepolynomial x 3 +3 x 2 )]TJ/F8 9.9626 Tf 9.136 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.135 0 Td [(18 hasnosinglefactorthatiscommontoeveryterm.However,wenotice thatifwe group togetherthersttwotermsandthesecondtwoterms,weseethateachresultingbinomial hasaparticularfactorcommontobothterms. Factor x 2 outofthersttwoterms,andfactor )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 outofthesecondtwoterms. x 2 x +3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x +3 Nowlookcloselyatthisbinomial.Eachofthetwotermscontainsthefactor x +3 Factorout x +3 x +3 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 isthenalfactorization. x 3 +3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(18= x +3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 6.5.3KnowingwhentoTrytheGroupingMethod Wearealertedtotheideaofgroupingwhenthepolynomialweareconsideringhas either ofthesequalities: 1.nofactorcommonto all terms 2.an even numberofterms Whenfactoringbygrouping,thesign + or )]TJ/F8 9.9626 Tf 7.748 0 Td [( ofthefactorwearetakingoutwill usually butnotalways bethesameasthesignoftherstterminthatgroup. 5 Thiscontentisavailableonlineat.

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368 CHAPTER6.FACTORINGPOLYNOMIALS 6.5.4SampleSetA Example6.18 Factor 8 a 2 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 b 4 +14 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 1.Wenoticethereisnofactorcommontoallterms. 2.Weseetherearefourterms,anevennumber. 3.Weseethatterms1and2have +4 b 4 incommonsincethe1stterminthegroupis +8 a 2 b 4 4.Wenoticethatthe3rdand4thtermshave +7 incommonsincethe1stterminthegroupis +14 a 2 8 a 2 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 b 4 +14 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7= a 2 -1b 4 +7 6.5.5PracticeSetA Usethegroupingmethodtofactorthefollowingpolynomials. Exercise6.151 Solutiononp.403. ax + ay + bx + by Exercise6.152 Solutiononp.403. 2 am +8 m +5 an +20 n Exercise6.153 Solutiononp.403. a 2 x 3 +4 a 2 y 3 +3 bx 3 +12 by 3 Exercise6.154 Solutiononp.403. 15 mx +10 nx )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 my )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ny Exercise6.155 Solutiononp.403. 40 abx )]TJ/F8 9.9626 Tf 9.962 0 Td [(24 abxy )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 c 2 x +21 c 2 xy Exercise6.156 Solutiononp.403. Whenfactoringthepolynomial 8 a 2 b 4 )]TJ/F8 9.9626 Tf 8.729 0 Td [(4 b 4 +14 a 2 )]TJ/F8 9.9626 Tf 8.729 0 Td [(7 inSampleSetA,wegroupedtogetherterms1 and2and3and4.Couldwehavegroupedtogetherterms1and3and2and4?Trythis. 8 a 2 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 b 4 +14 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7= Dowegetthesameresult?Iftheresultsdonotlookpreciselythesame,recallthecommutativepropertyof multiplication. 6.5.6Exercises Forthefollowingproblems,usethegroupingmethodtofactorthepolynomials.Somepolynomialsmaynot befactorableusingthegroupingmethod. Exercise6.157 Solutiononp.403. 2 ab +3 a +18 b +27 Exercise6.158 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +4 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 Exercise6.159 Solutiononp.403. xy + x +3 y +3

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369 Exercise6.160 mp +3 mq + np +3 nq Exercise6.161 Solutiononp.403. ar +4 as +5 br +20 bs Exercise6.162 14 ax )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 bx +21 ay )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 by Exercise6.163 Solutiononp.403. 12 mx )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 bx +21 ay )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 by Exercise6.164 36 ak )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 ah )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 bk +6 bh Exercise6.165 Solutiononp.403. a 2 b 2 +2 a 2 +3 b 2 +6 Exercise6.166 3 n 2 +6 n +9 m 3 +12 m Exercise6.167 Solutiononp.403. 8 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y 3 +12 z 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 z Exercise6.168 x 2 +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y 2 + y Exercise6.169 Solutiononp.403. x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x + xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y Exercise6.170 2 n 2 +12 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 mn )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 m Exercise6.171 Solutiononp.403. 4 pq )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 p +3 q 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(21 Exercise6.172 8 x 2 +16 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 y Exercise6.173 Solutiononp.403. 12 s 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(27 s )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 st +18 t Exercise6.174 15 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 xy +8 y Exercise6.175 Solutiononp.403. a 4 b 4 +3 a 5 b 5 +2 a 2 b 2 +6 a 3 b 3 Exercise6.176 4 a 3 bc )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 a 2 bc 3 +10 abc 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 bc 4 Exercise6.177 Solutiononp.403. 5 x 2 y 3 z +3 x 3 yw )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 y 3 z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 wxyz Exercise6.178 a 3 b 2 cd + abc 2 dx )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 2 bxy )]TJ/F11 9.9626 Tf 9.963 0 Td [(cx 2 y Exercise6.179 Solutiononp.403. 5 m 10 n 17 p 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(m 6 n 7 p 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(40 m 4 n 10 qt 2 +8 pqt 2

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370 CHAPTER6.FACTORINGPOLYNOMIALS 6.5.7ExercisesforReview Exercise6.180 Section2.6 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 5 y 3 )]TJ/F11 9.9626 Tf 14.147 -8.07 Td [(x 2 y Exercise6.181 Solutiononp.404. Section3.8 Usescienticnotationtondtheproductof )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 10 2 Exercise6.182 Section4.8 Findthedomainoftheequation y = 6 x +5 Exercise6.183 Solutiononp.404. Section5.8 Constructthegraphoftheinequality y )]TJ/F8 9.9626 Tf 18.265 0 Td [(2 Exercise6.184 Section6.4 Factor 8 a 4 b 4 +12 a 3 b 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 a 2 b 3 6.6FactoringTwoSpecialProducts 6 6.6.1Overview TheDierenceofTwoSquares FundamentalRulesofFactoring PerfectSquareTrinomials 6.6.2TheDierenceofTwoSquares Recallthatwhenwemultipliedtogetherthetwobinomials a + b and a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b ,weobtainedtheproduct a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b = a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 PerfectSquare Noticethattheterms a 2 and b 2 intheproductcanbeproducedbysquaring a and b ,respectively.Aterm thatisthesquareofanothertermiscalleda perfectsquare .Thus,both a 2 and b 2 areperfectsquares. Theminussignbetween a 2 and b 2 meansthatwearetakingthe dierence ofthetwosquares. Sinceweknowthat a + b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b = a 2 )]TJ/F11 9.9626 Tf 9.092 0 Td [(b 2 ,weneedonlyturntheequationaroundtondthefactorization form. a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 = a + b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b Thefactorizationformsaysthatwecanfactor a 2 )]TJ/F11 9.9626 Tf 10.691 0 Td [(b 2 ,thedierenceoftwosquares,byndingthe termsthatproducetheperfectsquaresandsubstitutingthesequantitiesintothefactorizationform. Whenusingrealnumbersasweare,thereisnofactoredformforthesumoftwosquares.Thatis,using realnumbers, a 2 + b 2 cannot befactored 6 Thiscontentisavailableonlineat.

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371 6.6.3SampleSetA Example6.19 Factor x 2 )]TJ/F8 9.9626 Tf 10.324 0 Td [(16 .Both x 2 and16areperfectsquares.Thetermsthat,whensquared,produce x 2 and16are x and4,respectively.Thus, x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16= x +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Wecancheckourfactorizationsimplybymultiplying. x +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4= x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 = x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 : Example6.20 49 a 2 b 4 )]TJ/F8 9.9626 Tf 10.295 0 Td [(121 .Both 49 a 2 b 4 and121areperfectsquares.Thetermsthat,whensquared,produce 49 a 2 b 4 and121are 7 ab 2 and11,respectively.Substitutingthesetermsintothefactorizationform weget 49 a 2 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(121= )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(7ab 2 +11ab 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Wecancheckourfactorizationbymultiplying. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 ab 2 +11 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(7 ab 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 =49 a 2 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 ab 2 +11 ab 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(121 =49 a 2 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(121 Example6.21 3 x 2 )]TJ/F8 9.9626 Tf 9.309 0 Td [(27 .Thisdoesn'tlooklikethedierenceoftwosquaressincewedon'treadilyknowtheterms thatproduce 3 x 2 and27.However,noticethat3iscommontoboththeterms.Factorout3. 3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Nowweseethat x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 isthedierenceoftwosquares.Factoringthe x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 weget 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(27=3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 =3 x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Becarefulnottodropthefactor3. 6.6.4PracticeSetA Ifpossible,factorthefollowingbinomialscompletely. Exercise6.185 Solutiononp.404. m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 Exercise6.186 Solutiononp.404. 36 p 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(81 q 2 Exercise6.187 Solutiononp.404. 49 a 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 c 2 Exercise6.188 Solutiononp.404. x 8 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(100 w 12 Exercise6.189 Solutiononp.404. 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(75 Exercise6.190 Solutiononp.404. a 3 b 4 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(am 3 n 2

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372 CHAPTER6.FACTORINGPOLYNOMIALS 6.6.5FundamentalRulesofFactoring Therearetwofundamentalrulesthatwefollowwhenfactoring: FundamentalRulesofFactoring 1.Factoroutallcommonmonomialsrst. 2.Factorcompletely. 6.6.6SampleSetB Factoreachbinomialcompletely. Example6.22 4 a 8 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 b 5 : Factoroutthecommonfactor4b. 4 b )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 b 4 Nowwecanseeadierenceoftwosquares,whereasintheoriginalpolynomialwecouldnot. We'llcompleteourfactorizationbyfactoringthedierenceoftwosquares. 4 a 8 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 b 5 =4 b )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 b 4 =4 b )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 4 +3 b 2 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(a 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b 2 Example6.23 x 16 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 8 : Factorthisdierenceoftwosquares. x 16 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 8 = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 8 + y 4 Sumoftwosquares Doesnotfactor )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 8 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 4 Dierenceoftwosquares Factorit! = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 8 + y 4 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 4 + y 2 )]TJ/F11 9.9626 Tf 13.054 -8.07 Td [(x 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 Factoragain! = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 8 + y 4 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 4 + y 2 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 2 + y )]TJ/F11 9.9626 Tf 10.792 -8.07 Td [(x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y Finally,thefactorizationiscomplete. Thesetypesofproductsappearfromtimetotime,sobeawarethatyoumayhavetofactor morethanonce. 6.6.7PracticeSetB Factoreachbinomialcompletely. Exercise6.191 Solutiononp.404. m 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(n 4 Exercise6.192 Solutiononp.404. 16 y 8 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 6.6.8PerfectSquareTrinomials Recalltheprocessofsquaringabinomial. ab 2 = a 2 +2 ab + b 2 ab 2 = a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 ab + b 2

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373 OurMethodIs WeNotice Squaretherstterm. Thersttermoftheproductshouldbeaperfect square. Taketheproductofthetwotermsanddoubleit. Themiddletermoftheproductshouldbedivisible by2sinceit'smultipliedby2. Squarethelastterm. Thelasttermoftheproductshouldbeaperfect square. Table6.1 Perfectsquaretrinomials always factorasthesquareofabinomial. Torecognizeaperfectsquaretrinomial,lookforthefollowingfeatures: 1.Therstandlasttermsareperfectsquares. 2.Themiddletermisdivisibleby2,andifwedividethemiddleterminhalftheoppositeofdoubling it,wewillgettheproductofthetermsthatwhensquaredproducetherstandlastterms. Inotherwords,factoringaperfectsquaretrinomialamountstondingthetermsthat,whensquared,produce therstandlasttermsofthetrinomial,andsubstitutingintooneoftheformula a 2 +2 ab + b 2 = a + b 2 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ab + b 2 = a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 6.6.9SampleSetC Factoreachperfectsquaretrinomial. Example6.24 x 2 +6 x +9 .Thisexpressionisaperfectsquaretrinomial.The x 2 and9areperfectsquares. Thetermsthatwhensquaredproduce x 2 and9are x and3,respectively. Themiddletermisdivisibleby2,and 6 x 2 =3 x .The 3 x istheproductof x and3,whichare thetermsthatproducetheperfectsquares. x 2 +6 x +9= x +3 2 Example6.25 x 4 )]TJ/F8 9.9626 Tf 8.981 0 Td [(10 x 2 y 3 +25 y 6 .Thisexpressionisaperfectsquaretrinomial.The x 4 and 25 y 6 arebothperfect squares.Thetermsthatwhensquaredproduce x 4 and 25 y 6 are x 2 and 5 y 3 ,respectively. Themiddleterm )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 x 2 y 3 isdivisibleby 2 .Infact, )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 x 2 y 3 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 2 y 3 .Thus, x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x 2 y 3 +25 y 6 = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y 3 2 Example6.26 x 2 +10 x +16 .Thisexpressionis not aperfectsquaretrinomial.Althoughthemiddletermis divisibleby 2 ; ? 10 x 2 =5 x ,the5and x arenotthetermsthatwhensquaredproducetherstand lastterms.Thisexpressionwouldbeaperfectsquaretrinomialifthemiddletermwere 8 x Example6.27 4 a 4 +32 a 2 b )]TJ/F8 9.9626 Tf 10.027 0 Td [(64 b 2 .Thisexpressionis not aperfectsquaretrinomialsincethelastterm )]TJ/F8 9.9626 Tf 7.748 0 Td [(64 b 2 is notaperfectsquaresinceanyquantitysquaredisalwayspositiveorzeroandnevernegative. Thus, 4 a 4 +32 a 2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(64 b 2 cannotbefactoredusingthismethod.

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374 CHAPTER6.FACTORINGPOLYNOMIALS 6.6.10PracticeSetC Factor,ifpossible,thefollowingtrinomials. Exercise6.193 Solutiononp.404. m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 m +16 Exercise6.194 Solutiononp.404. k 2 +10 k +25 Exercise6.195 Solutiononp.404. 4 a 2 +12 a +9 Exercise6.196 Solutiononp.404. 9 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 xy +16 y 2 Exercise6.197 Solutiononp.404. 2 w 3 z +16 w 2 z 2 +32 wz 3 Exercise6.198 Solutiononp.404. x 2 +12 x +49 6.6.11Exercises Forthefollowingproblems,factorthebinomials. Exercise6.199 Solutiononp.404. a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 Exercise6.200 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(25 Exercise6.201 Solutiononp.404. x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 Exercise6.202 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(49 Exercise6.203 Solutiononp.404. a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(100 Exercise6.204 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 Exercise6.205 Solutiononp.404. 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(64 Exercise6.206 2 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(32 Exercise6.207 Solutiononp.404. 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 Exercise6.208 5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(125 Exercise6.209 Solutiononp.404. 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 Exercise6.210 9 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(100 Exercise6.211 Solutiononp.404. 36 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25

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375 Exercise6.212 121 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 Exercise6.213 Solutiononp.404. 12 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(75 Exercise6.214 10 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(320 Exercise6.215 Solutiononp.404. 8 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(50 Exercise6.216 a 2 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 Exercise6.217 Solutiononp.405. x 2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(25 Exercise6.218 x 4 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(36 Exercise6.219 Solutiononp.405. x 4 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 a 2 Exercise6.220 a 2 b 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 y 4 Exercise6.221 Solutiononp.405. 4 a 2 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 b 2 Exercise6.222 16 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 y 2 Exercise6.223 Solutiononp.405. a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 Exercise6.224 a 4 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 4 Exercise6.225 Solutiononp.405. x 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 4 Exercise6.226 x 8 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 Exercise6.227 Solutiononp.405. a 8 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 2 Exercise6.228 b 6 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 Exercise6.229 Solutiononp.405. b 6 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 4 Exercise6.230 9 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 2 Exercise6.231 Solutiononp.405. 25 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 2 Exercise6.232 49 )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 a 2 Exercise6.233 Solutiononp.405. 100 )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 b 4 Exercise6.234 128 )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 x 2

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376 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.235 Solutiononp.405. x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 Exercise6.236 2 ab 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 3 b Exercise6.237 Solutiononp.405. a 4 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 4 Exercise6.238 a 16 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 4 Exercise6.239 Solutiononp.405. x 12 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 12 Exercise6.240 a 2 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 c Exercise6.241 Solutiononp.405. a 3 c 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 ac 2 Exercise6.242 a 4 b 4 c 2 d 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 x 2 y 2 Exercise6.243 Solutiononp.405. 49 x 2 y 4 z 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(64 a 4 b 2 c 8 d 10 Forthefollowingproblems,factor,ifpossible,thetrinomials. Exercise6.244 x 2 +8 x +16 Exercise6.245 Solutiononp.405. x 2 +10 x +25 Exercise6.246 a 2 +4 a +4 Exercise6.247 Solutiononp.405. a 2 +12 a +36 Exercise6.248 b 2 +18 b +81 Exercise6.249 Solutiononp.405. y 2 +20 y +100 Exercise6.250 c 2 +6 c +9 Exercise6.251 Solutiononp.405. a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 a +4 Exercise6.252 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 b +9 Exercise6.253 Solutiononp.405. x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x +25 Exercise6.254 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(22 b +121 Exercise6.255 Solutiononp.405. a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 a +144 Exercise6.256 a 2 +2 a +1

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377 Exercise6.257 Solutiononp.405. x 2 +2 x +1 Exercise6.258 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +1 Exercise6.259 Solutiononp.405. b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b +1 Exercise6.260 4 a 2 +12 a +9 Exercise6.261 Solutiononp.405. 9 x 2 +6 x +1 Exercise6.262 4 x 2 +28 x +49 Exercise6.263 Solutiononp.405. 16 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 a +9 Exercise6.264 25 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 a +4 Exercise6.265 Solutiononp.405. 9 x 2 +6 xy + y 2 Exercise6.266 16 x 2 +24 xy +9 y 2 Exercise6.267 Solutiononp.406. 36 a 2 +60 ab +25 b 2 Exercise6.268 4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 xy +9 y 2 Exercise6.269 Solutiononp.406. 12 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(60 a +75 Exercise6.270 16 x 2 +8 x +1 Exercise6.271 Solutiononp.406. 32 x 2 +16 x +2 Exercise6.272 x 2 + x +1 Exercise6.273 Solutiononp.406. 4 a 2 + a +9 Exercise6.274 9 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 a +49 Exercise6.275 Solutiononp.406. x 5 +8 x 4 +16 x 3 Exercise6.276 12 a 3 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(48 a 2 b 2 +48 ab 3

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378 CHAPTER6.FACTORINGPOLYNOMIALS 6.6.12EXERCISESFORREVIEW Exercise6.277 Solutiononp.406. Section6.4 Factor m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [( m )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y Exercise6.278 Section6.5 Factor 8 xm +16 xn +3 ym +6 yn bygrouping. 6.7FactoringTrinomialswithLeadingCoecient1 7 6.7.1Overview Method FactoringHints 6.7.2Method Let'sconsidertheproductofthetwobinomials x +4 and x +7 Noticethatthe rstterm intheresultingtrinomialcomesfromtheproductofthersttermsin thebinomials: x x = x 2 .The lastterm inthetrinomialcomesfromtheproductofthelasttermsin thebinomials: 4 7=28 .The middleterm comesfromtheadditionoftheouterandinnerproducts: 7 x +4 x =11 x .Also,noticethatthecoecientofthemiddletermisexactlythe sum ofthelasttermsin thebinomials: 4+7=11 Theproblemwe'reinterestedinisthatgivenatrinomial,howcanwendthefactors?Whentheleading coecientthecoecientofthequadratictermis1,theobservationswemadeaboveleadustothefollowing methodoffactoring. MethodofFactoring 1.Writetwosetsofparentheses: 2.Placeabinomialintoeachsetofparentheses.Thersttermofeachbinomialisafactoroftherst termofthetrinomial. 3.Determinethesecondtermsofthebinomialsbydeterminingthefactorsofthethirdtermthatwhen addedtogetheryieldthecoecientofthemiddleterm. 6.7.3SampleSetA Factorthefollowingtrinomials. Example6.28 x 2 +5 x +6 1.Writetwosetsofparentheses: 2.Placethefactorsof x 2 intotherstpositionofeachsetofparentheses: x x 7 Thiscontentisavailableonlineat.

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379 3.Thethirdtermofthetrinomialis6.Weseektwonumberswhose aproductis6and bsumis5. Therequirednumbersare3and2.Place +3 and +2 intotheparentheses. x 2 +5 x +6= x +3 x +2 Thefactorizationiscomplete.We'llchecktobesure. x +3 x +2= x 2 +2 x +3 x +6 = x 2 +5 x +6 Example6.29 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 1.Writetwosetsofparentheses: 2.Placethefactorsof y 2 intotherstpositionofeachsetofparentheses: y y 3.Thethirdtermofthetrinomialis )]TJ/F8 9.9626 Tf 7.748 0 Td [(24 .Weseektwonumberswhose aproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 and bsumis )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Therequirednumbersare )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 and 4 .Place )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 and +4 intotheparentheses. y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(24= y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 y +4 Thefactorizationiscomplete.We'llchecktobesure. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 y +4= y 2 +4 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 = y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 Noticethattheothercombinationsofthefactorsof )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 someofwhichare )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 12;3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(8; and )]TJ/F8 9.9626 Tf 8.97 0 Td [(4 ; 6 donotwork.Forexample, y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y +12= y 2 + 10 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 y +3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(8= y 2 )]TJETq1 0 0 1 230.842 338.238 cm[]0 d 0 J 0.398 w 0 0 m 20.983 0 l SQq1 0 0 1 231.041 321.301 cm[]0 d 0 J 0.398 w 0 0 m 0 16.737 l SQBT/F8 9.9626 Tf 236.222 326.323 Td [(5 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(24 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y +6= y 2 + 2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(24 Inalloftheseequations,themiddletermsareincorrect. Example6.30 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 a +30 1.Writetwosetsofparentheses: 2.Placethefactorsof a 2 intotherstpositionofeachsetofparentheses: a a 3.Thethirdtermofthetrinomialis +30 .Weseektwonumberswhose aproductis30and bsumis )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 Therequirednumbersare )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 and )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 .Place )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 and )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 intotheparentheses. a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 a +30= a )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Thefactorizationiscomplete.We'llchecktobesure. a )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6= a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a +30 = a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 a +30 Example6.31 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(42

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380 CHAPTER6.FACTORINGPOLYNOMIALS Beforewebegin,let'srecallthemostbasicruleoffactoring: factoroutcommonmonomial factorsrst .Noticethat3isthegreatestcommonmonomialfactorof every term.Factorout3. 3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(42=3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Nowwecancontinue. 1.Writetwosetsofparentheses: 3 2.Placethefactorsof x 2 intotherstpositionofeachsetofparentheses: 3 x x 3.Thethirdtermofthetrinomialis )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 .Weseektwonumberswhose aproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 and bsumis )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Therequirednumbersare )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 and 2 .Place )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 and +2 intotheparentheses. 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(42=3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +2 Thefactorizationiscomplete.We'llchecktobesure. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +2=3 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 =3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 =3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(42 6.7.4PracticeSetA Factor,ifpossible,thefollowingtrinomials. Exercise6.279 Solutiononp.406. k 2 +8 k +15 Exercise6.280 Solutiononp.406. y 2 +7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 Exercise6.281 Solutiononp.406. m 2 +10 m +24 Exercise6.282 Solutiononp.406. m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 m +16 6.7.5FactoringHints Factoringtrinomialsmaytakesomepractice,butwithtimeandexperience,youwillbeabletofactormuch morequickly. Therearesomecluesthatarehelpfulindeterminingthefactorsofthethirdtermthatwhenaddedyield thecoecientofthemiddleterm. FactoringHints Lookatthe sign ofthelastterm : a.Ifthesignispositive,weknowthatthetwofactorsmusthavethe same sign,since ++= + and )]TJ/F8 9.9626 Tf 7.749 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(=+ .Thetwofactorswillhavethesamesignasthesignofthemiddleterm. b.Ifthesignisnegative,weknowthattwofactorsmusthave opposite signs,since + )]TJ/F8 9.9626 Tf 7.749 0 Td [(= )]TJ/F8 9.9626 Tf 7.749 0 Td [( and )]TJ/F8 9.9626 Tf 7.749 0 Td [(+= )]TJ/F8 9.9626 Tf 7.748 0 Td [( .

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381 6.7.6SampleSetB Example6.32 Factor x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x +12 1.Writetwosetsofparentheses: 2.Thethirdtermofthetrinomialis +12 .Thesignispositive,sothetwofactorsof12weare lookingformusthavethesamesign.Theywillhavethesignofthemiddleterm.Thesign ofthemiddletermisnegative,sobothfactorsof12arenegative.Theyare )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 and )]TJ/F8 9.9626 Tf 9.992 0 Td [(1 ; )]TJ/F8 9.9626 Tf -395.494 -11.955 Td [(6 and )]TJ/F8 9.9626 Tf 10.378 0 Td [(2 ; or )]TJ/F8 9.9626 Tf 10.377 0 Td [(4 and )]TJ/F8 9.9626 Tf 10.378 0 Td [(3 .Onlythefactors )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 and )]TJ/F8 9.9626 Tf 10.378 0 Td [(3 addto )]TJ/F8 9.9626 Tf 10.377 0 Td [(7 ; so )]TJ/F8 9.9626 Tf 10.377 0 Td [(4 and )]TJ/F8 9.9626 Tf 10.377 0 Td [(3 arethe properfactorsof12tobeused. x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +12= x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 6.7.7PracticeSetB Factor,ifpossible,thefollowingtrinomials. Exercise6.283 Solutiononp.406. 4 k 2 +32 k +28 Exercise6.284 Solutiononp.406. 3 y 4 +24 y 3 +36 y 2 Exercise6.285 Solutiononp.406. x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 y 2 Exercise6.286 Solutiononp.406. )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 a 5 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 a 4 b 2 +15 a 3 b 3 6.7.8Exercises Forthefollowingproblems,factorthetrinomialswhenpossible. Exercise6.287 Solutiononp.406. x 2 +4 x +3 Exercise6.288 x 2 +6 x +8 Exercise6.289 Solutiononp.406. x 2 +7 x +12 Exercise6.290 x 2 +6 x +5 Exercise6.291 Solutiononp.406. y 2 +8 y +12 Exercise6.292 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y +6 Exercise6.293 Solutiononp.406. y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y +4 Exercise6.294 a 2 + a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise6.295 Solutiononp.406. a 2 +3a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4

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382 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.296 x 2 +4x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Exercise6.297 Solutiononp.406. x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Exercise6.298 x 2 +7 x +12 Exercise6.299 Solutiononp.406. y 2 +10 y +16 Exercise6.300 x 2 +6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 Exercise6.301 Solutiononp.406. y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 y +7 Exercise6.302 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 Exercise6.303 Solutiononp.406. a 2 + a )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 Exercise6.304 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a +2 Exercise6.305 Solutiononp.406. a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a +20 Exercise6.306 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 Exercise6.307 Solutiononp.406. x 2 +13 x +42 Exercise6.308 x 2 +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(35 Exercise6.309 Solutiononp.407. x 2 +13 x +40 Exercise6.310 y 2 +6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 Exercise6.311 Solutiononp.407. b 2 +15 b +56 Exercise6.312 3 a 2 +24 a +36 Hint: Alwayssearchforacommonfactor. Exercise6.313 Solutiononp.407. 4 x 2 +12 x +8 Exercise6.314 2 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(18 a +40 Exercise6.315 Solutiononp.407. 5 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(70 y +440 Exercise6.316 6 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(54 x +48 Exercise6.317 Solutiononp.407. x 3 +6 x 2 +8 x

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383 Exercise6.318 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x 2 +15 x Exercise6.319 Solutiononp.407. x 4 +9 x 3 +14 x 2 Exercise6.320 2 a 3 +12 a 2 +10 a Exercise6.321 Solutiononp.407. 4 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(40 a 2 +84 a Exercise6.322 3 xm 2 +33 xm +54 x Exercise6.323 Solutiononp.407. 2 y 2 n 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 y 2 n )]TJ/F8 9.9626 Tf 9.962 0 Td [(48 y 2 Exercise6.324 4 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(42 x 3 +144 x 2 Exercise6.325 Solutiononp.407. y 5 +13 y 4 +42 y 3 Exercise6.326 4 x 2 a 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(48 x 2 a 5 +252 x 2 a 4 6.7.9ExercisesforReview Exercise6.327 Solutiononp.407. Section6.5 Factor 6 xy +2 ax )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 ay )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 2 Exercise6.328 Section6.6 Factor 8 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(50 Exercise6.329 Solutiononp.407. Section6.6 Factor 4 x 2 +17 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 6.8FactoringTrinomialswithLeadingCoecientOtherThan1 8 6.8.1Overview TheMethodofFactorization 6.8.2TheMethodofFactorization Inthelastsectionwesawthatwecouldeasilyfactortrinomialsoftheform x 2 + bx + c byndingthefactors oftheconstant c thataddtothecoecientofthelinearterm b ,asshowninthefollowingexample: Factor x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Thethirdtermofthetrinomialis )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 .Weseektwonumberswhose aproductis )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 and bsumis )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Therequirednumbersare )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 and +3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21= x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 x +3 8 Thiscontentisavailableonlineat.

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384 CHAPTER6.FACTORINGPOLYNOMIALS Theproblemoffactoringthepolynomial ax 2 + bx + c a 6 =1 ,becomesmoreinvolved.Wewillstudytwo methodsoffactoringsuchpolynomials.Eachmethodproducesthesameresult,andyoushouldselectthe methodyouaremostcomfortablewith.Therstmethodiscalledthe trialanderrormethod andrequires someeducatedguesses.WewillexaminetwoexamplesSampleSetsAandB.Then,wewillstudyasecond methodoffactoring.Thesecondmethodiscalledthe collectanddiscardmethod ,anditrequiresless guessingthanthetrialanderrormethod.SampleSetCillustratestheuseofthecollectanddiscardmethod. 6.8.2.1TheTrialandErrorMethodofFactoring ax 2 + bx + c TrialandErrorMethod Considertheproduct Examiningthetrinomial 20 x 2 +23 x +6 ,wecanimmediatelyseesomefactorsoftherstandlastterms. 20 x 2 6 20 x;x 6 ; 1 10 x; 2 x 3 ; 2 5 x; 4 x Table6.2 Ourgoalistochoosethepropercombinationoffactorsoftherstandlasttermsthatyieldthemiddle term 23 x Noticethatthemiddletermcomesfromthe sum ofthe outer and inner productsinthemultiplicationof thetwobinomials. Thisfactprovidesusawaytondthepropercombination. Lookforthecombinationthatwhen multiplied andthen added yieldsthemiddleterm. Thepropercombinationwe'relookingforis

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385 6.8.3SampleSetA Example6.33 Factor 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 Factortherstandlastterms. Thus, 3 x and3aretobemultiplied, 2 x and )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 aretobemultiplied. 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12= Putthefactorsoftheleadingterminimmediately. = 3 x 2 x Since3 x and3aretobemultiplied,theymustbelocatedindierentbinomials. = 3 x 2 x +3 Placethe )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 intheremainingsetofparentheses. = 3 x )]TJ/F8 9.9626 Tf 17.711 0 Td [(4 2 x +3 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12= 3 x )]TJ/F8 9.9626 Tf 17.711 0 Td [(4 2 x +3 Check : x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x +3=6 x 2 +9 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 =6 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Example6.34 Factor 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 Findthefactorsoftherstandlastterms. Thus,the 4 x and )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 aretobemultiplied,and 2 x and3aretobemultiplied.

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386 CHAPTER6.FACTORINGPOLYNOMIALS 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(27= 4 x 2 x = 4 x 2 x )]TJ/F8 9.9626 Tf 17.711 0 Td [(9 = 4 x +3 2 x )]TJ/F8 9.9626 Tf 17.711 0 Td [(9 check : x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9=8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 x +6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 =8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 Example6.35 Factor 15 x 2 +44 x +32 Beforewestartndingthefactorsoftherstandlastterms,noticethattheconstantterm is +32 .Sincetheproductis positive ,thetwofactorswearelookingfor must havethesamesign. Theymustbothbepositiveorbothbenegative.Nowthemiddleterm, +44 x ,isprecededbya positivesign.Weknowthatthemiddletermcomesfromthe sum oftheouterandinnerproducts. Ifthesetwonumbersaretosumtoapositivenumber,theymustbothbepositivethemselves.If theywerenegative,theirsumwouldbenegative.Thus,wecanconcludethatthetwofactorsof +32 thatwearelookingforarebothpositivenumbers.Thiseliminatesseveralfactorsof32and lessensouramountofwork. Factortherstandlastterms. Afterafewtrialsweseethat 5 x and4aretobemultiplied,and 3 x and8aretobemultiplied. 15 x 2 +44 x +32= x +8 x +4 Example6.36 Factor 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(56 x +6 Weseethateachtermiseven,sowecanfactorout2. 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(9 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 x +3 Noticethattheconstanttermispositive.Thus,weknowthatthefactorsof3thatweare lookingformusthavethesamesign.Sincethesignofthemiddletermisnegative,bothfactors mustbenegative. Factortherstandlastterms.

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387 Therearenotmanycombinationstotry,andwendthat 9 x and )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 aretobemultipliedand x and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 aretobemultiplied. 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(56 x +6=2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(9 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 x +3 =2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Ifwehadnotfactoredthe2outrst,wewouldhavegottenthefactorization Thefactorizationisnotcompletesinceoneofthefactorsmaybefactoredfurther. 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(56 x +6= x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 =2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Bythecommutativepropertyofmultiplication. Theresultsarethesame,butitismucheasiertofactorapolynomialafterallcommonfactors havebeenfactoredoutrst. Example6.37 Factor 3 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Therearenocommonfactors.Weseethattheconstanttermisnegative.Thus,thefactorsof )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 musthavedierentsigns. Factortherstandlastterms. Afterafewtrials,weseethat 3 x and )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 aretobemultipliedand x and7aretobemultiplied. 3 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(14=3 x +7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Example6.38 Factor 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(26 xy +15 y 2 Weseethattheconstanttermispositiveandthatthemiddletermisprecededbyaminus sign. Hence,thefactorsof 15 y 2 thatwearelookingformustbothbenegative. Factortherstandlastterms.

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388 CHAPTER6.FACTORINGPOLYNOMIALS Afterafewtrials,weseethat 4 x and )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y aretobemultipliedand 2 x and )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 y aretobe multiplied. 8 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(26 xy +15 y 2 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y 6.8.4PracticeSetA Factorthefollowing,ifpossible. Exercise6.330 Solutiononp.407. 2 x 2 +13 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise6.331 Solutiononp.407. 3 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise6.332 Solutiononp.407. 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Exercise6.333 Solutiononp.407. 16 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(22 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise6.334 Solutiononp.407. 10 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(19 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 Exercise6.335 Solutiononp.407. 6 m 3 +40 m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 m Exercise6.336 Solutiononp.407. 14 p 2 +31 pq )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 q 2 Exercise6.337 Solutiononp.407. )]TJ/F8 9.9626 Tf 7.749 0 Td [(24 w 2 z 2 +14 wz 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 z 4 Exercise6.338 Solutiononp.407. 3 x 2 +6 xy +2 y 2 Asyougetmorepracticefactoringthesetypesofpolynomialsyoubecomefasteratpickingtheproper combinations.Ittakesalotofpractice! Thereisashortcutthatmayhelpinpickingthepropercombinations.Thisprocessdoesnotalwayswork, butitseemstoholdtrueinmanycases.Afteryouhavefactoredtherstandlasttermsandarebeginning tolookforthepropercombinations,startwiththe intermediate factorsandnottheextremeones. 6.8.5SampleSetB Example6.39 Factor 24 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(41 x +12 Factortherstandlastterms.

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389 24 x 2 12 24 x;x )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 12 x; 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 8 x; 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 6 x; 4 x Table6.3 Ratherthanstartingwiththe 24 x;x and )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,picksomeintermediatevalues, 8 x and 3 x the 6 x and 4 x ,orthe )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 and )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ,orthe )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 and )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 24 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(41 x +12= x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 6.8.6PracticeSetB Exercise6.339 Solutiononp.407. Factor 48 x 2 +22 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 Exercise6.340 Solutiononp.407. Factor 54 y 2 +39 yw )]TJ/F8 9.9626 Tf 9.963 0 Td [(28 w 2 6.8.6.1TheCollectandDiscardMethodofFactoring ax 2 + bx + c CollectandDiscardMethod Considerthepolynomial 6 x 2 + x )]TJ/F8 9.9626 Tf 10.157 0 Td [(12 .Webeginbyidentifying a and c .Inthiscase, a =6 and c = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Westartoutaswewouldwith a =1 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12: 6 x 6 x Now,compute a c a c = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12= )]TJ/F8 9.9626 Tf 7.749 0 Td [(72 Findthefactorsof )]TJ/F8 9.9626 Tf 7.749 0 Td [(72 thataddto1,thecoecientof x ,thelinearterm.Thefactorsare9and )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Includethesefactorsintheparentheses. 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12: x +9 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Butwehaveincludedtoomuch.Wemusteliminatethesurplus.Factoreachparentheses. 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12:3 x +3 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Discardthefactorsthatmultiplyto a =6 .Inthiscase,3and2.Weareleftwiththeproperfactorization. 6 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12= x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 6.8.7SampleSetC Example6.40 Factor 10 x 2 +23 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Identify a =10 and b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 .

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390 CHAPTER6.FACTORINGPOLYNOMIALS 10 x 2 +23 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5; 10 x 10 x Compute a c = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(50 Findthefactorsof )]TJ/F8 9.9626 Tf 7.749 0 Td [(50 thataddto +23 ,thecoecientof x ,thelinearterm.Thefactorsare 25and )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 .Placethesenumbersintotheparentheses. 10 x 2 +23 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5: x +25 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Wehavecollectedtoomuch.Factoreachsetofparenthesesandeliminatethesurplus. 10 x 2 +23 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5: x +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Discardthefactorsthatmultiplyto a =10 .Inthiscase,5and2. 10 x 2 +23 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= x +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Example6.41 Factor 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 Identify a =8 and c = )]TJ/F8 9.9626 Tf 7.749 0 Td [(27 8 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(27: 8 x 8 x Compute a c = )]TJ/F8 9.9626 Tf 7.749 0 Td [(27= )]TJ/F8 9.9626 Tf 7.749 0 Td [(216 Findthefactorsof )]TJ/F8 9.9626 Tf 7.749 0 Td [(216 thataddto )]TJ/F8 9.9626 Tf 7.749 0 Td [(30 ,thecoecientof x ,thelinearterm.Thisrequires somethought.Thefactorsare )]TJ/F8 9.9626 Tf 7.749 0 Td [(36 and6.Placethesenumbersintotheparentheses. 8 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(27: x )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 x +6 Wehavecollectedtoomuch.Factoreachsetofparenthesesandeliminatethesurplus. 8 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(27: x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 x +3 Discardthefactorsthatmultiplyto a =8 .Inthiscase,4and2. 8 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(27= x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 x +3 Example6.42 Factor 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 Identify a =18 and c = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y 2 : 18 x 18 x Compute a c = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2= )]TJ/F8 9.9626 Tf 7.749 0 Td [(36 Findthefactorsof )]TJ/F8 9.9626 Tf 7.749 0 Td [(36 thataddto )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ,thecoecientof xy .Inthiscase, )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 and4.Place thesenumbersintotheparentheses,axing y toeach. 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y 2 : x )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 y x +4 y Wehavecollectedtoomuch.Factoreachsetofparenthesesandeliminatethesurplus. 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y 2 : x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y x +2 y Discardthefactorsthatmultiplyto a =18 .Inthiscase,9and4. 18 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y 2 = x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y x +2 y 6.8.8PracticeSetC Exercise6.341 Solutiononp.407. Factor 6 x 2 +7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise6.342 Solutiononp.407. Factor 14 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(31 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 Exercise6.343 Solutiononp.407. Factor 48 x 2 +22 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 .

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391 Exercise6.344 Solutiononp.408. Factor 10 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(23 xw +12 w 2 6.8.9Exercises Factorthefollowingproblems,ifpossible. Exercise6.345 Solutiononp.408. x 2 +3 x +2 Exercise6.346 x 2 +7 x +12 Exercise6.347 Solutiononp.408. 2 x 2 +7 x +5 Exercise6.348 3 x 2 +4 x +1 Exercise6.349 Solutiononp.408. 2 x 2 +11 x +12 Exercise6.350 10 x 2 +33 x +20 Exercise6.351 Solutiononp.408. 3 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise6.352 3 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise6.353 Solutiononp.408. 4 x 2 +8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 Exercise6.354 2 a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise6.355 Solutiononp.408. 9 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a +2 Exercise6.356 16 a 2 +16 a +3 Exercise6.357 Solutiononp.408. 16 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(26 y +3 Exercise6.358 3 y 2 +14 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Exercise6.359 Solutiononp.408. 10 x 2 +29 x +10 Exercise6.360 14 y 2 +29 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 Exercise6.361 Solutiononp.408. 81 a 2 +19 a +2 Exercise6.362 24 x 2 +34 x +5 Exercise6.363 Solutiononp.408. 24 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(34 x +5

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392 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.364 24 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(26 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise6.365 Solutiononp.408. 24 x 2 +26 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise6.366 6 a 2 +13 a +6 Exercise6.367 Solutiononp.408. 6 x 2 +5 xy + y 2 Exercise6.368 6 a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ay )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 2 Forthefollowingproblems,thegiventrinomialoccurswhensolvingthecorrespondingappliedproblem. Factoreachtrinomial.Youdonotneedtosolvetheproblem. Exercise6.369 Solutiononp.408. 5 r 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Ittakes5hourstopaddleaboat12milesdownstreamandthenback.Thecurrentows attherateof1mileperhour.Atwhatratewastheboatpaddled? Exercise6.370 x 2 +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(84 Thelengthofarectangleis5inchesmorethanthewidthoftherectangle.Iftheareaof therectangleis84squareinches,whatarethelengthandwidthoftherectangle? Exercise6.371 Solutiononp.408. x 2 +24 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(145 Asquaremeasures12inchesoneachside.Anothersquareistobedrawnaroundthis squareinsuchawaythatthetotalareais289squareinches.Whatisthedistancefromtheedge ofthesmallersquaretotheedgeofthelargersquare?Thetwosquareshavethesamecenter. Exercise6.372 x 2 +8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 Awomanwishestoconstructarectangularboxthatisopenatthetop.Shewishesitto be4incheshighandhavearectangularbasewhoselengthisthreetimesthewidth.Thematerial usedforthebasecosts$2persquareinch,andthematerialusedforthesidescosts$1.50per squareinch.Thewomanwillspendexactly$120formaterials.Findthedimensionofthebox lengthofthebase,widthofthebase,andheight. Forthefollowingproblems,factorthetrinomialsifpossible. Exercise6.373 Solutiononp.408. 16 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y 2 Exercise6.374 6 a 2 +7 ab +2 b 2 Exercise6.375 Solutiononp.408. 12 a 2 +7 ab +12 b 2 Exercise6.376 9 x 2 +18 xy +8 y 2 Exercise6.377 Solutiononp.408. 8 a 2 +10 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 b 2

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393 Exercise6.378 12 a 2 +54 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(90 Exercise6.379 Solutiononp.408. 12 b 4 +30 b 2 a +12 a 2 Exercise6.380 30 a 4 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a 2 b 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 c 2 Exercise6.381 Solutiononp.408. 3 a 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a 3 b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 b 4 Exercise6.382 20 a 2 b 2 +2 abc 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 a 2 c 4 Exercise6.383 Solutiononp.408. 14 a 2 z 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(40 a 3 z 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(46 a 4 z 2 6.8.10ExercisesforReview Exercise6.384 Section2.7 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 3 b 6 4 Exercise6.385 Solutiononp.408. Section4.6 Findtheproduct. x 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +4 Exercise6.386 Section4.7 Findtheproduct. m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 n 2 Exercise6.387 Solutiononp.408. Section5.3 Solvetheequation 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +7=0 Exercise6.388 Section6.7 Factor x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x 4 +7 x 3 6.9SummaryofKeyConcepts 9 6.9.1SummaryofKeyConcepts FactoringSection6.2 Factoringistheprocessofdeterminingthefactorsofsomeproduct.Factoringisthereverseofmultiplication. GreatestCommonFactorSection6.4 The greatestcommonfactor ofapolynomialisthefactorthatiscommontoeverytermofthepolynomial andalsoissuchthat 1.Thenumericalcoecientisthelargestnumbercommontoeachterm. 2.Thevariablespossessthelargestexponentsthatarecommontoallthevariables. FactoringaMonomialfromaPolynomialSection6.4 If A isthegreatestcommonfactorof Ax + Ay ,then Ax + Ay = A x + y FactoringbyGroupingSection6.5 Wearealertedtotheideaof factoringbygrouping whenthepolynomialweareconsidering 9 Thiscontentisavailableonlineat.

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394 CHAPTER6.FACTORINGPOLYNOMIALS 1.Hasnofactorcommontoallterms. 2.Hasanevennumberofterms. Ax + Ay [U+23B5] A iscommon + Bx + By [U+23B5] B iscommon = A x + y + B x + y [U+23B5] x + y iscommon = x + y A + B SpecialproductsSection6.6 a 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 = a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b a 2 +2 ab + b 2 = a + b 2 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ab + b 2 = a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 FundamentalRuleofFactoringSection6.6 1.Factoroutallcommonmonomialsrst. 2.Factorcompletely. FactoringTrinomialsSection6.7,Section6.8 Onemethodoffactoringatrinomialistolistallthefactorpairsofbothoftherstandlasttermsandthen choosethecombinationthatwhenmultipliedandthenaddedproducesthemiddleterm. 6.10ExerciseSupplement 10 6.10.1ExerciseSupplement 6.10.1.1FindingthefactorsofaMonomialSection6.2 Forthefollowingproblems,therstquantityrepresentstheproductandthesecondquantityrepresentsa factor.Findtheotherfactor. Exercise6.389 Solutiononp.408. 32 a 4 b; 2 b Exercise6.390 35 x 3 y 2 ; 7 x 3 Exercise6.391 Solutiononp.408. 44 a 2 b 2 c; 11 b 2 Exercise6.392 50 m 3 n 5 p 4 q; 10 m 3 q Exercise6.393 Solutiononp.409. 51 a +1 2 b +3 4 ; 3 a +1 Exercise6.394 )]TJ/F8 9.9626 Tf 7.748 0 Td [(26 x +2 y 3 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y Exercise6.395 Solutiononp.409. )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 x 5 y 4 x + y 4 x +3 y 3 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 x x + y x +3 y Exercise6.396 )]TJ/F8 9.9626 Tf 7.749 0 Td [( a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 b 10 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 8 a +3 b 7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [( a )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 b 7 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 7 a +3 b 7 10 Thiscontentisavailableonlineat.

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395 Exercise6.397 Solutiononp.409. 12 x n +6 y 2 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x n +1 y n +3 Exercise6.398 )]TJ/F8 9.9626 Tf 7.748 0 Td [(400 a 3 n +10 b n )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 c 4 n +7 ; 20 a 2 n +8 c 2 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise6.399 Solutiononp.409. 16 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 ; 16 Exercise6.400 35 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(45 ; 513 Exercise6.401 Solutiononp.409. 24 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a; 6 a Exercise6.402 88 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(33 x 3 +44 x 2 +55 x; 11 x Exercise6.403 Solutiononp.409. 9 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 y 2 +36 y; )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 y Exercise6.404 4 m 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 m 4 +16 m 2 ; 4 m Exercise6.405 Solutiononp.409. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x 4 y 3 +10 x 3 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 x 2 y 2 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 x 2 y 2 Exercise6.406 )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 a 5 b 6 c 4 a +2 3 +35 a 5 bc 5 a +2 4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 a 4 b a +2 2 Exercise6.407 Solutiononp.409. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise6.408 a +3 b; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 6.10.1.2FactoringaMonomialfromaPolynomialSection6.3-TheGreatestCommonFactor Section6.4 Forthefollowingproblems,factorthepolynomials. Exercise6.409 Solutiononp.409. 8 a +4 Exercise6.410 10 x +10 Exercise6.411 Solutiononp.409. 3 y 2 +27 y Exercise6.412 6 a 2 b 2 +18 a 2 Exercise6.413 Solutiononp.409. 21 x +5+9 Exercise6.414 14 a +1+35 Exercise6.415 Solutiononp.409. ma 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(m

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396 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.416 15 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(24 y +24 Exercise6.417 Solutiononp.409. r 2 r +1 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 r r +1 2 + r +1 Exercise6.418 Pa + Pb + Pc Exercise6.419 Solutiononp.409. )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x + x +3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x + x 6.10.1.3FactoringbyGroupingSection6.5 Forthefollowingproblems,usethegroupingmethodtofactorthepolynomials.Somemaynotbefactorable. Exercise6.420 4 ax + x +4 ay + y Exercise6.421 Solutiononp.409. xy +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise6.422 2 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 a Exercise6.423 Solutiononp.409. a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a + ab )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 b Exercise6.424 m 2 +5 m + nm +5 n Exercise6.425 Solutiononp.409. r 2 + rs )]TJ/F11 9.9626 Tf 9.963 0 Td [(r )]TJ/F11 9.9626 Tf 9.963 0 Td [(s Exercise6.426 8 a 2 bc +20 a 2 bc +10 a 3 b 3 c +25 a 3 b 3 Exercise6.427 Solutiononp.409. a a +6 )]TJ/F8 9.9626 Tf 9.963 0 Td [( a +6+ a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [( a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise6.428 a x +7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x +7+ a x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 6.10.1.4FactoringTwoSpecialProductsSection6.6-FactoringTrinomialswithLeading CoecientOtherThan1Section6.8 Forthefollowingproblems,factorthepolynomials,ifpossible. Exercise6.429 Solutiononp.409. m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(36 Exercise6.430 r 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(81 Exercise6.431 Solutiononp.409. a 2 +8 a +16 Exercise6.432 c 2 +10 c +25 Exercise6.433 Solutiononp.409. m 2 + m +1

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397 Exercise6.434 r 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(r )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise6.435 Solutiononp.409. a 2 +9 a +20 Exercise6.436 s 2 +9 s +18 Exercise6.437 Solutiononp.409. x 2 +14 x +40 Exercise6.438 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 a +36 Exercise6.439 Solutiononp.409. n 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 n +49 Exercise6.440 a 2 +6 a +5 Exercise6.441 Solutiononp.409. a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 a +20 Exercise6.442 6 x 2 +5 x +1 Exercise6.443 Solutiononp.410. 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Exercise6.444 4 x 2 +7 x +3 Exercise6.445 Solutiononp.410. 42 a 2 +5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise6.446 30 y 2 +7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 Exercise6.447 Solutiononp.410. 56 m 2 +26 m +6 Exercise6.448 27 r 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(33 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise6.449 Solutiononp.410. 4 x 2 +4 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y 2 Exercise6.450 25 a 2 +25 ab +6 b 2 Exercise6.451 Solutiononp.410. 2 x 2 +6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 Exercise6.452 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y 2 +4 y +48 Exercise6.453 Solutiononp.410. x 3 +3 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x Exercise6.454 3 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 y 3 +24 y 2 Exercise6.455 Solutiononp.410. 15 a 2 b 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ab )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 b Exercise6.456 4 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 x 2 +16 x

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398 CHAPTER6.FACTORINGPOLYNOMIALS Exercise6.457 Solutiononp.410. 18 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 a + 1 2 Exercise6.458 a 4 +16 a 2 b +16 b 2 Exercise6.459 Solutiononp.410. 4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 xy +9 y 2 Exercise6.460 49 b 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(84 b 2 +36 Exercise6.461 Solutiononp.410. r 6 s 8 +6 r 3 s 4 p 2 q 6 +9 p 4 q 12 Exercise6.462 a 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a 2 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 b 2 Exercise6.463 Solutiononp.410. 81 a 8 b 12 c 10 )]TJ/F8 9.9626 Tf 9.962 0 Td [(25 x 20 y 18 6.11ProciencyExam 11 6.11.1ProciencyExam Exercise6.464 Solutiononp.410. Section6.2 Theproductis 27 a 3 +9 a 2 +9 a andafactoris 3 a .Findtheotherfactor. Exercise6.465 Solutiononp.410. Section6.2 Theproductis 15 x n +5 y 3 n )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 .Findtheotherfactor. Forthefollowingproblems,factor,ifpossible,thepolynomials. Exercise6.466 Solutiononp.410. Section6.4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 x 2 y 4 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(28 x 2 y 3 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(42 x 2 y 2 Exercise6.467 Solutiononp.410. Section6.4 y +2 a + y +2 c Exercise6.468 Solutiononp.410. Section6.5 6 x 2 y 2 z +5 x 2 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 xyz )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 xy 2 Exercise6.469 Solutiononp.410. Section6.6 4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 c 2 Exercise6.470 Solutiononp.410. Section6.6 m 4 )]TJ/F11 9.9626 Tf 9.962 0 Td [(n 4 Exercise6.471 Solutiononp.410. Section6.6 b 2 +8 b +16 Exercise6.472 Solutiononp.410. Section6.6 9 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 y +25 Exercise6.473 Solutiononp.410. Section6.7 x 2 +5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 Exercise6.474 Solutiononp.410. Section6.7 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 Exercise6.475 Solutiononp.410. Section6.8 4 x 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(36 x 4 +80 x 2 11 Thiscontentisavailableonlineat.

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399 Exercise6.476 Solutiononp.410. Section6.8 9 x 2 +25 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6

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400 CHAPTER6.FACTORINGPOLYNOMIALS SolutionstoExercisesinChapter6 SolutiontoExercise6.1p.355 14 SolutiontoExercise6.2p.355 2 x 2 yz 4 SolutiontoExercise6.3p.355 5 SolutiontoExercise6.5p.355 2 a SolutiontoExercise6.7p.355 3 SolutiontoExercise6.9p.355 5 x 3 SolutiontoExercise6.11p.356 2 x 3 SolutiontoExercise6.13p.356 2 xy SolutiontoExercise6.15p.356 3 b 3 c SolutiontoExercise6.17p.356 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 xb SolutiontoExercise6.19p.356 4 x 3 bf 7 SolutiontoExercise6.21p.356 7 a 17 b 5 c 18 d SolutiontoExercise6.23p.356 1 4 x 3 SolutiontoExercise6.25p.356 d SolutiontoExercise6.27p.356 1 SolutiontoExercise6.29p.356 4 x + y 3 SolutiontoExercise6.31p.356 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y 5 SolutiontoExercise6.33p.357 x + y SolutiontoExercise6.35p.357 6 x n y n )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise6.37p.357 x 12 z 6 SolutiontoExercise6.39p.357 4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 x +16 SolutiontoExercise6.40p.359 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise6.41p.359 y 2 +2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.42p.359 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 3 y +4 x 2 y 2 +6 y 4

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401 SolutiontoExercise6.43p.359 5 a 4 +7 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.44p.359 a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 SolutiontoExercise6.45p.359 2 x +5 SolutiontoExercise6.47p.359 x +5 SolutiontoExercise6.49p.359 a +3 SolutiontoExercise6.51p.359 3 x +4 SolutiontoExercise6.53p.359 9 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise6.55p.359 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise6.57p.360 2 a +1 SolutiontoExercise6.59p.360 3 2 x 2 + x +3 SolutiontoExercise6.61p.360 3 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise6.63p.360 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y +1 SolutiontoExercise6.65p.360 2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 SolutiontoExercise6.67p.360 9 x 2 +10 SolutiontoExercise6.69p.360 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.71p.360 5 a 3 +6 a 2 +8 a +4 SolutiontoExercise6.73p.360 2 a 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 a +4 SolutiontoExercise6.75p.360 x 5 +4 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise6.77p.360 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 SolutiontoExercise6.79p.361 5 a +2 b SolutiontoExercise6.81p.361 4 x 4 y 4 + x +1 SolutiontoExercise6.83p.361 8 a 2 bc 6 +7 ab 2 c 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 b 3 c 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.85p.361 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise6.87p.361 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.89p.361 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b

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402 CHAPTER6.FACTORINGPOLYNOMIALS SolutiontoExercise6.91p.361 )]TJ/F11 9.9626 Tf 7.749 0 Td [(a + b )]TJ/F11 9.9626 Tf 9.962 0 Td [(c SolutiontoExercise6.93p.361 a + b + c SolutiontoExercise6.95p.361 6 x 2 y SolutiontoExercise6.97p.361 a =3 SolutiontoExercise6.99p.363 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 SolutiontoExercise6.100p.363 6 y )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 2 +4 y +6 SolutiontoExercise6.101p.363 2 b 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 a 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 a 4 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 b 2 SolutiontoExercise6.102p.363 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 m )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 m 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 m +1 SolutiontoExercise6.103p.364 y +4 a + b SolutiontoExercise6.104p.364 2 m 2 n )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.105p.365 9 a +2 SolutiontoExercise6.107p.365 4 b +3 SolutiontoExercise6.109p.365 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.111p.365 7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise6.113p.365 6 x x +3 SolutiontoExercise6.115p.365 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 y 2 +9 SolutiontoExercise6.117p.365 3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise6.119p.365 6 y y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.121p.365 b )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 2 +1 SolutiontoExercise6.123p.365 5 x )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 x +2 SolutiontoExercise6.125p.365 5 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.127p.366 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 y +3 SolutiontoExercise6.129p.366 b )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(y 3 + y 2 + y +1 SolutiontoExercise6.131p.366 x x +6 y +4 SolutiontoExercise6.133p.366 13 x 2 y 5 )]TJ/F11 9.9626 Tf 7.749 0 Td [(c )]TJ/F8 9.9626 Tf 9.963 0 Td [(3

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403 SolutiontoExercise6.135p.366 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 y 3 +4 y 2 +7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.137p.366 N x + y SolutiontoExercise6.139p.366 A x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y SolutiontoExercise6.141p.366 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 a + b SolutiontoExercise6.143p.366 a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b w )]TJ/F11 9.9626 Tf 9.963 0 Td [(x SolutiontoExercise6.145p.366 3 x 2 y 3 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 xy +9 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 3 SolutiontoExercise6.147p.366 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 2 y )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(11 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 x 2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise6.149p.367 t =3 SolutiontoExercise6.151p.368 a + b x + y SolutiontoExercise6.152p.368 m +5 n a +4 SolutiontoExercise6.153p.368 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(a 2 +3 b )]TJ/F11 9.9626 Tf 14.147 -8.069 Td [(x 3 +4 y 3 SolutiontoExercise6.154p.368 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y m +2 n SolutiontoExercise6.155p.368 x )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 ab )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 c 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y SolutiontoExercise6.156p.368 yes SolutiontoExercise6.157p.368 b +3 a +9 SolutiontoExercise6.159p.368 y +1 x +3 SolutiontoExercise6.161p.369 a +5 b r +4 s SolutiontoExercise6.163p.369 3 mx )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 bx +7 ay )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 by Notfactorablebygrouping SolutiontoExercise6.165p.369 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 +3 )]TJ/F11 9.9626 Tf 10.792 -8.07 Td [(b 2 +2 SolutiontoExercise6.167p.369 Notfactorablebygrouping SolutiontoExercise6.169p.369 x + y x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.171p.369 Notfactorablebygrouping SolutiontoExercise6.173p.369 s )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 s )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 t SolutiontoExercise6.175p.369 a 2 b 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 b 2 +2 +3 ab SolutiontoExercise6.177p.369 y )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(5 y 2 z +3 xw )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 z

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404 CHAPTER6.FACTORINGPOLYNOMIALS SolutiontoExercise6.179p.369 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m 6 n 7 p 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 qt 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(5 m 4 n 10 )]TJ/F11 9.9626 Tf 9.963 0 Td [(p SolutiontoExercise6.181p.370 6 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 SolutiontoExercise6.183p.370 SolutiontoExercise6.185p.371 m +5 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.186p.371 9 p )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 q p +3 q SolutiontoExercise6.187p.371 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(7 a 2 + bc )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(7 a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(bc SolutiontoExercise6.188p.371 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 4 y 2 +10 w 6 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 4 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 w 6 SolutiontoExercise6.189p.371 3 x +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise6.190p.371 am )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(ab 2 + mn )]TJ/F11 9.9626 Tf 10.793 -8.069 Td [(ab 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(mn SolutiontoExercise6.191p.372 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m 2 + n 2 m )]TJ/F11 9.9626 Tf 9.963 0 Td [(n m + n SolutiontoExercise6.192p.372 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 y 4 +1 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 y 2 +1 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 SolutiontoExercise6.193p.374 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 SolutiontoExercise6.194p.374 k +5 2 SolutiontoExercise6.195p.374 a +3 2 SolutiontoExercise6.196p.374 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 y 2 SolutiontoExercise6.197p.374 2 wz w +4 z 2 SolutiontoExercise6.198p.374 notpossible SolutiontoExercise6.199p.374 a +3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.201p.374 x +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise6.203p.374 a +10 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 SolutiontoExercise6.205p.374 4 a +4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise6.207p.374 3 x +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.209p.374 a +5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.211p.374 y +5 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise6.213p.375 3 a +5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5

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405 SolutiontoExercise6.215p.375 2 y +5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.217p.375 xy +5 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise6.219p.375 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 y 2 +3 a )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 2 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a SolutiontoExercise6.221p.375 b 2 a +3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.223p.375 a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b SolutiontoExercise6.225p.375 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 + y 2 x + y x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y SolutiontoExercise6.227p.375 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 4 + y )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(a 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y SolutiontoExercise6.229p.375 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(b 3 + x 2 )]TJ/F11 9.9626 Tf 10.793 -8.069 Td [(b 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 SolutiontoExercise6.231p.375 + a )]TJ/F11 9.9626 Tf 9.962 0 Td [(a SolutiontoExercise6.233p.375 4 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5+3 b 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b 2 SolutiontoExercise6.235p.375 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 2 +4 x +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise6.237p.376 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 + b 2 a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b SolutiontoExercise6.239p.376 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 6 + x 6 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 3 + y 3 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(x 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 3 SolutiontoExercise6.241p.376 ac 2 a +5 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise6.243p.376 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(7 xy 2 z 3 +8 a 2 bc 4 d 5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(7 xy 2 z 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 a 2 bc 4 d 5 SolutiontoExercise6.245p.376 x +5 2 SolutiontoExercise6.247p.376 a +6 2 SolutiontoExercise6.249p.376 y +10 2 SolutiontoExercise6.251p.376 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 SolutiontoExercise6.253p.376 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 2 SolutiontoExercise6.255p.376 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 2 SolutiontoExercise6.257p.376 x +1 2 SolutiontoExercise6.259p.377 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 SolutiontoExercise6.261p.377 x +1 2 SolutiontoExercise6.263p.377 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2

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406 CHAPTER6.FACTORINGPOLYNOMIALS SolutiontoExercise6.265p.377 x + y 2 SolutiontoExercise6.267p.377 a +5 b 2 SolutiontoExercise6.269p.377 3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 2 SolutiontoExercise6.271p.377 2 x +1 2 SolutiontoExercise6.273p.377 notfactorable SolutiontoExercise6.275p.377 x 3 x +4 2 SolutiontoExercise6.277p.378 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y SolutiontoExercise6.279p.380 k +3 k +5 SolutiontoExercise6.280p.380 y +10 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.281p.380 m +6 m +4 SolutiontoExercise6.282p.380 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise6.283p.381 4 k +7 k +1 SolutiontoExercise6.284p.381 3 y 2 y +2 y +6 SolutiontoExercise6.285p.381 x +2 y x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y SolutiontoExercise6.286p.381 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 a 3 b a +3 b a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b SolutiontoExercise6.287p.381 x +3 x +1 SolutiontoExercise6.289p.381 x +3 x +4 SolutiontoExercise6.291p.381 y +6 y +2 SolutiontoExercise6.293p.381 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.295p.381 a +4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.297p.382 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +3 SolutiontoExercise6.299p.382 y +8 y +2 SolutiontoExercise6.301p.382 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.303p.382 a +6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.305p.382 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2

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407 SolutiontoExercise6.307p.382 x +6 x +7 SolutiontoExercise6.309p.382 x +5 x +8 SolutiontoExercise6.311p.382 b +8 b +7 SolutiontoExercise6.313p.382 4 x +2 x +1 SolutiontoExercise6.315p.382 5 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 y +88 SolutiontoExercise6.317p.382 x x +4 x +2 SolutiontoExercise6.319p.383 x 2 x +7 x +2 SolutiontoExercise6.321p.383 4 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.323p.383 2 y 2 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 n +3 SolutiontoExercise6.325p.383 y 3 y +6 y +7 SolutiontoExercise6.327p.383 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(a y + a SolutiontoExercise6.329p.383 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +5 SolutiontoExercise6.330p.388 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x +7 SolutiontoExercise6.331p.388 x +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.332p.388 a +3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 SolutiontoExercise6.333p.388 b +1 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise6.334p.388 y +3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise6.335p.388 2 m m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 m +7 SolutiontoExercise6.336p.388 p )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 q p +5 q SolutiontoExercise6.337p.388 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 z 2 w )]TJ/F11 9.9626 Tf 9.963 0 Td [(z w )]TJ/F11 9.9626 Tf 9.963 0 Td [(z SolutiontoExercise6.338p.388 notfactorable SolutiontoExercise6.339p.389 x +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.340p.389 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 w y +7 w SolutiontoExercise6.341p.390 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x +3 SolutiontoExercise6.342p.390 x +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5

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408 CHAPTER6.FACTORINGPOLYNOMIALS SolutiontoExercise6.343p.390 x +5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.344p.391 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 w x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 w SolutiontoExercise6.345p.391 x +2 x +1 SolutiontoExercise6.347p.391 x +5 x +1 SolutiontoExercise6.349p.391 x +3 x +4 SolutiontoExercise6.351p.391 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +1 SolutiontoExercise6.353p.391 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +7 SolutiontoExercise6.355p.391 notfactorable SolutiontoExercise6.357p.391 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.359p.391 x +2 x +5 SolutiontoExercise6.361p.391 notfactorable SolutiontoExercise6.363p.391 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise6.365p.392 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x +5 SolutiontoExercise6.367p.392 x + y x + y SolutiontoExercise6.369p.392 r +1 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.371p.392 x +29 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise6.373p.392 x + y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y SolutiontoExercise6.375p.392 notfactorable SolutiontoExercise6.377p.392 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 a 2 +5 ab )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b 2 SolutiontoExercise6.379p.393 6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 b 2 + a )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(b 2 +2 a SolutiontoExercise6.381p.393 3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 3 +2 b 2 )]TJ/F11 9.9626 Tf 10.793 -8.07 Td [(a 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 b 2 SolutiontoExercise6.383p.393 2 a 2 z 2 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(23 a 2 or )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a 2 z 2 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(23 a 2 +20 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 SolutiontoExercise6.385p.393 x 4 + x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x 2 SolutiontoExercise6.387p.393 x = 11 2 SolutiontoExercise6.389p.394 16 a 4

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409 SolutiontoExercise6.391p.394 4 a 2 c SolutiontoExercise6.393p.394 17 a +1 b +3 4 SolutiontoExercise6.395p.394 4 x 4 y 4 x + y 3 x +3 y 2 SolutiontoExercise6.397p.395 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x 5 y n )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 SolutiontoExercise6.399p.395 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise6.401p.395 4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.403p.395 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y 2 +9 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 SolutiontoExercise6.405p.395 x 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +3 SolutiontoExercise6.407p.395 x +2 y + c 2 SolutiontoExercise6.409p.395 4 a +1 SolutiontoExercise6.411p.395 3 y y +9 SolutiontoExercise6.413p.395 3 x +38 SolutiontoExercise6.415p.395 m )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.417p.396 r +1 h r 2 r +1 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 r r +1+1 i SolutiontoExercise6.419p.396 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +4 x SolutiontoExercise6.421p.396 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y +4 SolutiontoExercise6.423p.396 a + b a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 SolutiontoExercise6.425p.396 r )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 r + s SolutiontoExercise6.427p.396 2 a +1 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.429p.396 m +6 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 SolutiontoExercise6.431p.396 a +4 2 SolutiontoExercise6.433p.396 notfactorable SolutiontoExercise6.435p.397 a +5 a +4 SolutiontoExercise6.437p.397 x +10 x +4 SolutiontoExercise6.439p.397 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 2

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410 CHAPTER6.FACTORINGPOLYNOMIALS SolutiontoExercise6.441p.397 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise6.443p.397 a +3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise6.445p.397 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 a +2 SolutiontoExercise6.447p.397 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(28 m 2 +13 m +3 SolutiontoExercise6.449p.397 x +3 y x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y SolutiontoExercise6.451p.397 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +5 SolutiontoExercise6.453p.397 x x +4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise6.455p.397 b )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(15 a 2 b )]TJ/F11 9.9626 Tf 9.963 0 Td [(a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise6.457p.397 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 a )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(1 2 SolutiontoExercise6.459p.398 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y 2 SolutiontoExercise6.461p.398 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(r 3 s 4 +3 p 2 q 6 2 SolutiontoExercise6.463p.398 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(9 a 4 b 6 c 5 +5 x 10 y 9 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(9 a 4 b 6 c 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x 10 y 9 SolutiontoExercise6.464p.398 9 a 2 +3 a +3 SolutiontoExercise6.465p.398 5 x n y 2 n )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise6.466p.398 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14 x 2 y 2 )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(y 2 b +2 yb +3 SolutiontoExercise6.467p.398 a + c y +2 SolutiontoExercise6.468p.398 xy xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z +5 y SolutiontoExercise6.469p.398 4 a +2 c a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 c SolutiontoExercise6.470p.398 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(m 2 + n 2 m + n m )]TJ/F11 9.9626 Tf 9.962 0 Td [(n SolutiontoExercise6.471p.398 b +4 2 SolutiontoExercise6.472p.398 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 2 SolutiontoExercise6.473p.398 notfactorable SolutiontoExercise6.474p.398 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 x +5 SolutiontoExercise6.475p.398 4 x 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise6.476p.399 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +3

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Chapter7 GraphingLinearEquationsand InequalitiesinOneandTwoVariables 7.1Objectives 1 Aftercompletingthechapter,youshould GraphingLinearEquationsandInequalitiesInOneVariableSection7.2 understandtheconceptofagraphandtherelationshipbetweenaxes,coordinatesystems,anddimension beabletoconstructone-dimensionalgraphs PlottingPointsinthePlaneSection7.3 befamiliarwiththeplane knowwhatismeantbythecoordinatesofapoint beabletoplotpointsintheplane GraphingLinearEquationsinTwoVariablesSection7.8 beabletorelatesolutionstoalinearequationtolines knowthegeneralformofalinearequation beabletoconstructthegraphofalineusingtheinterceptmethod beabletodistinguish,bytheirequations,slanted,horizontal,andverticallines TheSlope-InterceptFormofaLineSection7.5 bemorefamiliarwiththegeneralformofaline beabletorecognizetheslope-interceptformofaline beabletointerprettheslopeandinterceptofaline beabletousetheslopeformulatondtheslopeofaline GraphingEquationsinSlope-InterceptFormSection7.6 beabletousetheslopeandintercepttoconstructthegraphofaline FindingtheEquationofaLineSection7.7 1 Thiscontentisavailableonlineat. 411

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412 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES beabletondtheequationoflineusingeithertheslope-interceptformorthepoint-slopeformofa line GraphingLinearInequalitiesinTwoVariablesSection7.8 beabletolocatesolutionslinearinequalititesintwovariablesusinggraphicaltechniques 7.2GraphingLinearEquationsandInequalitiesinOneVariable 2 7.2.1Overview Graphs Axes,CoordinateSystems,andDimension GraphinginOneDimension 7.2.2Graphs Wehave,thusfarinourstudyofalgebra,developedandusedseveralmethodsforobtainingsolutionsto linearequationsinbothoneandtwovariables.Quiteoftenitishelpfultoobtainapictureofthesolutions toanequation.Thesepicturesarecalled graphs andtheycanrevealinformationthatmaynotbeevident fromtheequationalone. TheGraphofanEquation Thegeometricrepresentationpictureofthesolutionstoanequationiscalledthe graph oftheequation. 7.2.3Axes,CoordinateSystems,andDimension Axis Thebasicstructureofthegraphisthe axis .Itiswithrespecttotheaxisthatallsolutionstoanequation arelocated.Themostfundamentaltypeofaxisisthe numberline TheNumberLineisanAxis Wehavethefollowinggeneralrulesregardingaxes. NumberofVariablesandNumberofAxes Anequationinonevariablerequiresoneaxis. Anequationintwovariablesrequirestwoaxes. Anequationinthreevariablesrequiresthreeaxes. ...Anequationin n variablesrequires n axes. Weshallalwaysdrawanaxisasastraightline,andifmorethanoneaxisisrequired,weshalldrawthem sotheyareallmutuallyperpendicularthelinesformingtheaxeswillbeat 90 anglestooneanother. CoordinateSystem Asystemofaxesconstructedforgraphinganequationiscalleda coordinatesystem ThePhrase,GraphinganEquation Thephrase graphinganequation isusedfrequentlyandshouldbeinterpretedasmeaninggeometrically locatingthesolutionstoanequation. 2 Thiscontentisavailableonlineat.

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413 RelatingtheNumberofVariablesandtheNumberofAxes WewillnotstartactuallygraphingequationsuntilSectionSection7.3,butinthefollowingexampleswewill relate thenumberofvariablesinanequationtothenumberofaxesinthecoordinatesystem. 1.One-DimensionalGraphs: Ifwewishtographtheequation 5 x +2=17 ,wewouldneedtoconstruct acoordinatesystemconsistingofasingleaxisasinglenumberlinesincetheequationconsistsof onlyonevariable.Welabeltheaxiswiththevariablethatappearsintheequation. Wemightinterpretanequationinonevariableasgivinginformationinone-dimensionalspace.Since weliveinthree-dimensionalspace,one-dimensionalspacemightbehardtoimagine.Objectsinonedimensionalspacewouldhaveonlylength,nowidthordepth. 2.Two-DimensionalGraphs: Tographanequationintwovariablessuchas y =2 x )-285()]TJ/F8 9.9626 Tf 18.335 0 Td [(3 ,wewould needtoconstructacoordinatesystemconsistingoftwomutuallyperpendicularnumberlines axes. Wecalltheintersectionofthetwoaxesthe origin andlabelitwitha0.Thetwoaxesaresimply numberlines;onedrawnhorizontally,onedrawnvertically. Recallthatanequationintwovariablesrequiresasolutiontobeapairofnumbers.Thesolutionscan bewrittenasorderedpairs x;y .Sincetheequation y =2 x )-232()]TJ/F8 9.9626 Tf 17.813 0 Td [(3 involvesthevariables x and y ,we labeloneaxis x andtheotheraxis y .Inmathematicsitiscustomarytolabelthehorizontalaxiswith theindependentvariableandtheverticalaxiswiththedependentvariable. Wemightinterpretequationsintwovariablesasgivinginformationintwo-dimensionalspace.Objects intwo-dimensionalspacewouldhavelengthandwidth,butnodepth. 3.Three-DimensionalGraphs: Anequationinthreevariables,suchas 3 x 2 )-255()]TJ/F8 9.9626 Tf 18.038 0 Td [(4 y 2 +5 z =0 ,requires threemutuallyperpendicularaxes,oneforeachvariable.Wewouldconstructthefollowingcoordinate systemandgraph.

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414 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Wemightinterpretequationsinthreevariablesasgivinginformationaboutthree-dimensionalspace. 4.Four-DimensionalGraphs: Tographanequationinfourvariables,suchas 3 x )-231()]TJ/F8 9.9626 Tf 17.794 0 Td [(2 y +8 x )-231()]TJ/F8 9.9626 Tf 17.794 0 Td [(5 w = )-12()]TJ/F8 9.9626 Tf 15.731 0 Td [(7 ,wouldrequirefourmutuallyperpendicularnumberlines.Thesegraphsarelefttotheimagination. Wemightinterpretequationsinfourvariablesasgivinginformationinfour-dimensionalspace.Fourdimensionalobjectswouldhavelength,width,depth,andsomeotherdimension. BlackHoles Theseotherspacesarehardforustoimagine,buttheexistenceofblackholesmakesthepossibilityof otheruniversesofone-,two-,four-,or n -dimensionsnotentirelyunlikely.Althoughitmaybedicult forus-Dpeopletotravelaroundinanotherdimensionalspace,atleastwecouldbeprettysurethat ourmathematicswouldstillworksinceitisnotrestrictedtoonlythreedimensions! 7.2.4GraphinginOneDimension Graphingalinearequationinonevariableinvolvessolvingtheequation,thenlocatingthesolutionontheaxis numberline,andmarkingapointatthislocation.Wehaveobservedthatgraphsmayrevealinformation thatmaynotbeevidentfromtheoriginalequation.Thegraphsoflinearequationsinonevariabledonot yieldmuch,ifany,information,buttheyserveasafoundationtographsofhigherdimensiongraphsoftwo variablesandthreevariables. 7.2.5SampleSetA Example7.1 Graphtheequation 3 x )-222()]TJ/F8 9.9626 Tf 17.711 0 Td [(5=10 Solvetheequationfor x andconstructanaxis.Sincethereisonlyonevariable,weneedonly oneaxis.Labeltheaxis x 3 x )-222()]TJ/F8 9.9626 Tf 17.711 0 Td [(5=10 3 x =15 x =5 Example7.2 Graphtheequation 3 x +4+7 x )-222()]TJ/F8 9.9626 Tf 17.711 0 Td [(1+8=31 Solvingtheequationweget,

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415 10 x +11=31 10 x =20 x =2 7.2.6PracticeSetA Exercise7.1 Solutiononp.513. Graphtheequation 4 x +1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 7.2.7SampleSetB Example7.3 Graphthelinearinequality 4 x 12 Weproceedbysolvingtheinequality. 4 x 12 Divideeachsideby4. x 3 Asweknow,anyvaluegreaterthanorequalto3willsatisfytheoriginalinequality.Hencewe haveinnitelymanysolutionsand,thus,innitelymanypointstomarkoonourgraph. The closedcircle at3meansthat3isincludedasasolution.Allthepointsbeginningat3 andinthedirectionofthearrowaresolutions. Example7.4 Graphthelinearinequality )-222()]TJ/F8 9.9626 Tf 19.926 0 Td [(2 y )-222()]TJ/F8 9.9626 Tf 17.711 0 Td [(1 > 3 Werstsolvetheinequality. )-222()]TJ/F8 9.9626 Tf 19.925 0 Td [(2 y )-222()]TJ/F8 9.9626 Tf 17.711 0 Td [(1 > 3 )-222()]TJ/F8 9.9626 Tf 19.925 0 Td [(2 y> 4 y< )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Theinequalitysymbolreverseddirectionbecausewedividedby. Thus,allnumbersstrictlylessthan )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 willsatisfytheinequalityandarethussolutions. Since )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 itselfis not tobeincludedasasolution,wedrawan opencircle at )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 .Thesolutions aretotheleftof )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 sowedrawanarrowpointingtotheleftof )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 todenotetheregionofsolutions.

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416 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Example7.5 Graphtheinequality )-222()]TJ/F8 9.9626 Tf 19.926 0 Td [(2 y +1 < 1 Werecognizethisinequalityasa compoundinequality andsolveitbysubtracting1fromall threeparts. )-222()]TJ/F8 9.9626 Tf 19.925 0 Td [(2 y +1 < 1 )-222()]TJ/F8 9.9626 Tf 19.925 0 Td [(3 y< 0 Thus,thesolutionisallnumbersbetween )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 and0,moreprecisely,allnumbersgreaterthan orequalto )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 butstrictlylessthan0. Example7.6 Graphthelinearequation 5 x = )-222()]TJ/F8 9.9626 Tf 19.925 0 Td [(125 Thesolutionis x = )-222()]TJ/F8 9.9626 Tf 19.925 0 Td [(25 .Scalingtheaxisbyunitsof5ratherthan1,weobtain 7.2.8PracticeSetB Exercise7.2 Solutiononp.513. Graphtheinequality 3 x 18 Exercise7.3 Solutiononp.513. Graphtheinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 m +1 < 13 Exercise7.4 Solutiononp.513. Graphtheinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 < 5 Exercise7.5 Solutiononp.513. Graphthelinearequation )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 y =480 7.2.9Exercises Forproblems1-25,graphthelinearequationsandinequalities. Exercise7.6 Solutiononp.513. 4 x +7=19

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417 Exercise7.7 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=7 Exercise7.8 Solutiononp.513. 2 x +3=4 Exercise7.9 x +3=15 Exercise7.10 Solutiononp.513. 6 y +3= y +8 Exercise7.11 2 x =0 Exercise7.12 Solutiononp.513. 4+1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4=3 z Exercise7.13 x + 1 2 = 4 3 Exercise7.14 Solutiononp.513. 7 r = 1 4 Exercise7.15 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6= 2 5 Exercise7.16 Solutiononp.514. x +7 12

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418 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.17 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 < 3 Exercise7.18 Solutiononp.514. x +19 > 2 Exercise7.19 z +5 > 11 Exercise7.20 Solutiononp.514. 3 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 8 Exercise7.21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 t 10 Exercise7.22 Solutiononp.514. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 34 Exercise7.23 x 4 < 2 Exercise7.24 Solutiononp.514. y 7 3 Exercise7.25 2 y 9 4 Exercise7.26 Solutiononp.514. )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y 8 4

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419 Exercise7.27 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a 7 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise7.28 Solutiononp.514. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 < 0 Exercise7.29 6 x +4 7 Exercise7.30 Solutiononp.514. )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 )]TJ/F8 9.9626 Tf 18.265 0 Td [(8 7.2.10ExercisesforReview Exercise7.31 Section2.6 Simplify )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 x 8 y 2 3 Exercise7.32 Solutiononp.515. Section4.2 List,ifanyshouldappear,thecommonfactorsintheexpression 10 x 4 )]TJ/F8 9.9626 Tf 8.829 0 Td [(15 x 2 +5 x 6 Exercise7.33 Section5.7 Solvetheinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x +3 < )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +1 Exercise7.34 Solutiononp.515. Section5.8 Solvetheequation y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x +8 if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise7.35 Section5.8 Solvetheequation 2 y =5 x +7 if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 7.3PlottingPointsinthePlane 3 7.3.1Overview ThePlane CoordinatesofaPoint PlottingPoints 3 Thiscontentisavailableonlineat.

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420 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.3.2ThePlane OrderedPairs Wearenowinterestedinstudyinggraphsoflinearequationsintwovariables.Weknowthatsolutionsto equationsintwovariablesconsistofapairofvalues,onevalueforeachvariable.Wehavecalledthesepairs ofvalues orderedpairs. Sincewehaveapairofvaluestograph,wemusthaveapairofaxesnumber linesuponwhichthevaluescanbelocated. Origin Wedrawtheaxessotheyareperpendiculartoeachotherandsothattheyintersecteachotherattheir 0 's. Thispointiscalledthe origin. RectangularCoordinateSystem Thesetwolinesformwhatiscalleda rectangularcoordinatesystem. Theyalsodetermineaplane. xy -plane A plane isaatsurface,andaresultfromgeometrystatesthatthroughanytwointersectinglinesthe axesexactlyoneplaneatsurfacemaybepassed.Ifwearedealingwithalinearequationinthetwo variables x and y ,wesometimessaywearegraphingtheequationusingarectangularcoordinatesystem,or thatwearegraphingtheequationinthe xy -plane. Quadrant Noticethatthetwointersectingcoordinateaxesdividetheplaneintofourequalregions.Sincethereare fourregions,wecalleachonea quadrant andnumberthemcounterclockwiseusingRomannumerals. Recallthatwhenwerststudiedthenumberlineweobservedthefollowing: Foreachrealnumberthereexistsauniquepointonthenumberline,andforeachpointonthenumber linewecanassociateauniquerealnumber. Wehaveasimilarsituationfortheplane.

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421 Foreachorderedpair a;b ,thereexistsauniquepointintheplane,andtoeachpointintheplanewe canassociateauniqueorderedpair a;b ofrealnumbers. 7.3.3CoordinatesofaPoint CoordinatesofaPoint Thenumbersinanorderedpairthatareassociatedwithaparticularpointarecalledthe coordinatesof thepoint. The rstnumber intheorderedpairexpressesthepoint'shorizontaldistanceanddirection leftorrightfromtheorigin.The secondnumber expressesthepoint'sverticaldistanceanddirection upordownfromtheorigin. TheCoordinatesDetermineDistanceandDirection A positivenumber meansadirectiontothe rightorup .A negativenumber meansadirectiontothe leftordown 7.3.4PlottingPoints Sincepointsandorderedpairsaresocloselyrelated,thetwotermsaresometimesusedinterchangeably.The followingtwophraseshavethesamemeaning: 1.Plotthepoint a;b 2.Plottheorderedpair a;b PlottingaPoint Bothphrasesmean:Locate,intheplane,thepointassociatedwiththeorderedpair a;b anddrawamark atthatposition. 7.3.5SampleSetA Example7.7 Plottheorderedpair ; 6 Webeginattheorigin.Therstnumberintheorderedpair,2,tellsuswemove2unitstothe right +2 means2unitstotherightThesecondnumberintheorderedpair,6,tellsuswemove 6unitsup +6 means6unitsup.

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422 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.3.6PracticeSetA Exercise7.36 Solutiononp.515. Plottheorderedpairs. ; 3 ; ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 ; ; 1 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; 0 7.3.7Exercises Exercise7.37 Solutiononp.515. Plotthefollowingorderedpairs.DonotdrawthearrowsasinPracticeSetA. ; 2 ; ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 10 ; ; 5 ; ; 0 ; ; 0 ; )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(7 ; )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(3 2

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423 Exercise7.38 Asaccuratelyaspossible,statethecoordinatesofthepointsthathavebeenplottedonthe followinggraph. Exercise7.39 Solutiononp.515. Usingorderedpairnotation,whatarethecoordinatesoftheorigin? Exercise7.40 Weknowthatsolutionstolinearequationsintwovariablescanbeexpressedasorderedpairs. Hence,thesolutionscanberepresentedaspointsintheplane.Considerthelinearequation y = 2 x )]TJ/F8 9.9626 Tf 10.369 0 Td [(1 .Findatleasttensolutionstothisequationbychoosing x -valuesbetween )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 and5and computingthecorresponding y -values.Plotthesesolutionsonthecoordinatesystembelow.Fill inthetabletohelpyoukeeptrackoftheorderedpairs. x y Table7.1 Keepinginmindthatthereareinnitelymanyorderedpairsolutionsto y =2 x )]TJ/F8 9.9626 Tf 10.712 0 Td [(1 ,speculateonthegeometricstructureofthegraphof all thesolutions.Completethefollowingstatement: Thenameofthetypeofgeometricstructureofthegraphofallthesolutionstothelinear equation y =2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 seemstobe__________. Wheredoesthisgurecrossthe y -axis?Doesthisnumberappearintheequation y =2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ?

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424 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Placeyourpencilatanypointonthegureyoumayhavetoconnectthedotstoseethe gureclearly.Moveyourpencilexactly one unittotherighthorizontally.Togetbackonto thegure,youmustmoveyourpencileitherupordownaparticularnumberofunits.Howmany unitsmustyoumoveverticallytogetbackontothegure,anddoyouseethisnumberinthe equation y =2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ? Exercise7.41 Solutiononp.515. Considerthe xy -plane. Completethetablebywritingtheappropriateinequalities. I II III IV x> 0 x< 0 x x y> 0 y y y Table7.2 Inthefollowingproblems,thegraphsofpointsarecalled scatterdiagrams andarefrequently usedbystatisticianstodetermineifthereisarelationshipbetweenthetwovariablesunderconsideration.Therstcomponentoftheorderedpairiscalledthe inputvariable andthesecond componentiscalledthe outputvariable .Constructthescatterdiagrams.Determineifthere appearstobearelationshipbetweenthetwovariablesunderconsiderationbymakingthefollowing observations:Arelationshipmayexistif a.asonevariableincreases,theothervariableincreases b.asonevariableincreases,theothervariabledecreases Exercise7.42 Apsychologist,studyingtheeectsofaplaceboonassemblylineworkersataparticularindustrial site,notedthetimeittooktoassembleacertainitembeforethesubjectwasgiventheplacebo, x ,andthetimeittooktoassembleasimilaritemafterthesubjectwasgiventheplacebo, y .The psychologist'sdataare

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425 x y 10 8 12 9 11 9 10 7 14 11 15 12 13 10 Table7.3 Exercise7.43 Solutiononp.515. Thefollowingdatawereobtainedinanengineer'sstudyoftherelationshipbetweentheamount ofpressureusedtoformapieceofmachinery, x ,andthenumberofdefectivepiecesofmachinery produced, y x y 50 0 60 1 65 2 70 3 80 4 70 5 90 5 100 5 Table7.4

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426 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.44 Thefollowingdatarepresentthenumberofworkdaysmissedperyear, x ,bytheemployeesofan insurancecompanyandthenumberofminutestheyarrivelatefromlunch, y x y 1 3 6 4 2 2 2 3 3 1 1 4 4 4 6 3 5 2 6 1 Table7.5 Exercise7.45 Solutiononp.516. Amanufacturerofdentalequipmenthasthefollowingdataontheunitcostindollars, y ,ofa particularitemandthenumberofunits, x ,manufacturedforeachorder.

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427 x y 1 85 3 92 5 99 3 91 4 100 1 87 6 105 8 111 8 114 Table7.6 7.3.8ExercisesforReview Exercise7.46 Section2.7 Simplify 18 x 5 y 6 9 x 2 y 4 5 Exercise7.47 Solutiononp.516. Section4.3 Supplythemissingword.An isastatementthattwoalgebraicexpressions areequal. Exercise7.48 Section4.4 Simplifytheexpression 5 xy xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x +3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 xy xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 xy 2 Exercise7.49 Solutiononp.516. Section5.2 Identifytheequation x +2= x +1 asanidentity,acontradiction,oraconditional equation. Exercise7.50 Section7.2 Supplythemissingphrase.Asystemofaxesconstructedforgraphinganequation iscalleda .

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428 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.4GraphingLinearEquationsinTwoVariables 4 7.4.1Overview SolutionsandLines GeneralformofaLinearEquation TheInterceptMethodofGraphing GraphingUsinganyTwoorMorePoints Slanted,Horizontal,andVerticalLines 7.4.2SolutionsandLines Weknowthatsolutionstolinearequationsintwovariablescanbeexpressedasorderedpairs.Hence,the solutionscanberepresentedbypointintheplane.Wealsoknowthatthephrasegraphtheequationmeans tolocatethesolutiontothegivenequationintheplane.Considertheequation y )]TJ/F8 9.9626 Tf 10.115 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 .We'llgraph sixsolutionsorderedpairstothisequationonthecoordinatessystembelow.We'llndthesolutionsby choosing x -valuesfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 to +4 ,substitutingthemintotheequation y )]TJ/F8 9.9626 Tf 10.161 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ,andthensolvingto obtainthecorresponding y -values.Wecankeeptrackoftheorderedpairsbyusingatable. y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 If x = Then y = OrderedPairs )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 1 ; 1 3 3 ; 3 4 5 ; 5 Table7.7 Wehaveplottedonlysixsolutionstotheequation y )]TJ/F8 9.9626 Tf 9.874 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 .Thereare,asweknow,innitelymany solutions.Byobservingthesixpointswehaveplotted,wecanspeculateastothelocationofalltheother points.Thesixpointsweplottedseemtolieonastraightline.Thiswouldleadustobelievethatall theotherpointssolutionsalsolieonthatsameline.Indeed,thisistrue.Infact,thisispreciselywhy rst-degreeequationsarecalled linear equations. 4 Thiscontentisavailableonlineat.

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429 LinearEquationsProduceStraightLines Line l Linear 7.4.3GeneralFormofaLinearEquation GeneralFormofaLinearEquationinTwoVariables Thereisastandardforminwhichlinearequationsintwovariablesarewritten.Supposethat a b ,and c areanyrealnumbersandthat a and b cannotbothbezeroatthesametime.Then,thelinearequationin twovariables ax + by = c issaidtobein generalform Wemuststipulatethat a and b cannotbothequalzeroatthesametime,foriftheywerewewouldhave 0 x +0 y = c 0= c Thisstatementistrueonlyif c =0 .If c weretobeanyothernumber,wewouldgetafalsestatement. Now,wehavethefollowing: Thegraphingofallorderedpairsthatsolvealinearequationintwovariablesproducesastraightline. Thisimplies, Thegraphofalinearequationintwovariablesisastraightline. Fromthesestatementswecanconclude, Ifanorderedpairisasolutiontoalinearequationsintwovariables,thenitliesonthegraphofthe equation. Also, Anypointorderedpairsthatliesonthegraphofalinearequationintwovariablesisasolutiontothat equation. 7.4.4TheInterceptMethodofGraphing Whenwewanttographalinearequation,itiscertainlyimpracticaltographinnitelymanypoints.Since astraightlineisdeterminedbyonlytwopoints,weneedonlyndtwosolutionstotheequationalthough athirdpointishelpfulasacheck. Intercepts Whenalinearequationintwovariablesisgiveningeneralfrom, ax + by = c ,oftenthetwomostconvenient pointssolutionstonearecalledthe Intercepts: thesearethepointsatwhichthelineinterceptsthe coordinateaxes.Ofcourse,ahorizontalorverticallineinterceptsonlyoneaxis,sothismethoddoesnot

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430 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES apply.Horizontalandverticallinesareeasilyrecognizedastheycontainonly one variable.SeeSampleSet C. y -Intercept Thepointatwhichthelinecrossesthe y -axisiscalledthe y -intercept .The x -valueatthispointiszero sincethepointisneithertotheleftnorrightoftheorigin. x -Intercept Thepointatwhichthelinecrossesthe x -axisiscalledthe x -intercept andthe y -valueatthatpointiszero. The y -interceptcanbefoundbysubstitutingthevalue0for x intotheequationandsolvingfor y .The x -interceptcanbefoundbysubstitutingthevalue0for y intotheequationandsolvingfor x InterceptMethod Sincewearegraphinganequationbyndingtheintercepts,wecallthismethodthe interceptmethod 7.4.5SampleSetA Graphthefollowingequationsusingtheinterceptmethod. Example7.8 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Tondthe y -intercept,let x =0 and y = b b )]TJ/F8 9.9626 Tf 9.963 0 Td [(2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(0= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 b = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Thus,wehavethepoint ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 .So,if x =0 y = b = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Tondthe x -intercept,let y =0 and x = a 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Divideby-2. a = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a = 3 2 Thus,wehavethepoint )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(3 2 ; 0 .So,if x = a = 3 2 y =0 Constructacoordinatesystem,plotthesetwopoints,anddrawalinethroughthem.Keepin mindthateverypointonthislineisasolutiontotheequation y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 .

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431 Example7.9 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +3 y =3 Tondthe y -intercept,let x =0 and y = b )]TJ/F8 9.9626 Tf 7.748 0 Td [(2+3 b =3 0+3 b =3 3 b =3 b =1 Thus,wehavethepoint ; 1 .So,if x =0 y = b =1 Tondthe x -intercept,let y =0 and x = a )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 a +3=3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a +0=3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a =3 a = 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(3 2 Thus,wehavethepoint )]TJ/F14 9.9626 Tf 4.566 -8.069 Td [()]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 2 ; 0 .So,if x = a = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 2 y =0 Constructacoordinatesystem,plotthesetwopoints,anddrawalinethroughthem.Keepin mindthatallthesolutionstotheequation )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 x +3 y =3 arepreciselyonthisline. Example7.10 4 x + y =5 Tondthe y -intercept,let x =0 and y = b .

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432 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 4+ b =5 0+ b =5 b =5 Thus,wehavethepoint ; 5 .So,if x =0 y = b =5 Tondthe x -intercept,let y =0 and x = a 4 a +0=5 4 a =5 a = 5 4 Thus,wehavethepoint )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(5 4 ; 0 .So,if x = a = 5 4 y =0 Constructacoordinatesystem,plotthesetwopoints,anddrawalinethroughthem. 7.4.6PracticeSetA Exercise7.51 Solutiononp.516. Graph 3 x + y =3 usingtheinterceptmethod. 7.4.7GraphingUsinganyTwoorMorePoints Thegraphswehaveconstructedsofarhavebeendonebyndingtwoparticularpoints,theintercepts. Actually, any twopointswilldo.Wechosetousetheinterceptsbecausetheyareusuallytheeasiesttowork with.Inthenextexample,wewillgraphtwoequationsusingpointsotherthantheintercepts.We'lluse threepoints,theextrapointservingasacheck.

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433 7.4.8SampleSetB Example7.11 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Wecanndthreepointsbychoosingthree x -valuesandcomputingtondthecorresponding y -values.We'llputourresultsinatableforeaseofreading. Sincewearegoingtochoose x -valuesandthencomputetondthecorresponding y -values,it willbetoouradvantagetosolvethegivenequationfor y x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 Subtract x frombothsides. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 Dividebothsidesby )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 : y = 1 3 x + 10 3 x y x;y 1 If x =1 ,then y = 1 3 + 10 3 = 1 3 + 10 3 = 11 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 ; 11 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 If x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ,then y = 1 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+ 10 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ 10 3 = 7 3 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 7 3 3 If x =3 ,then y = 1 3 + 10 3 =1+ 10 3 = 13 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 ; 13 3 Table7.8 Thus,wehavethethreeorderedpairspoints, )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 ; 11 3 )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 7 3 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 ; 13 3 .Ifwewish,wecan changetheimproperfractionstomixednumbers, )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1 ; 3 2 3 )]TJ/F14 9.9626 Tf 4.567 -8.069 Td [()]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; 2 1 3 )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(3 ; 4 1 3 Example7.12 4 x +4 y =0 Wesolvefor y 4 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x y = )]TJ/F11 9.9626 Tf 7.748 0 Td [(x x y x;y 0 0 ; 0 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; 3 Table7.9

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434 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Noticethatthe x )]TJ/F15 9.9626 Tf 11.473 0 Td [(and y -interceptsarethesamepoint.Thustheinterceptmethoddoesnot provideenoughinformationtoconstructthisgraph. Whenanequationisgiveninthegeneralform ax + by = c ,usuallythemostecientapproach toconstructingthegraphistousetheinterceptmethod,whenitworks. 7.4.9PracticeSetB Graphthefollowingequations. Exercise7.52 Solutiononp.517. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 y =5 Exercise7.53 Solutiononp.517. x +2 y =6

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435 Exercise7.54 Solutiononp.517. 2 x + y =1 7.4.10Slanted,Horizontal,andVerticalLines Inallthegraphswehaveobservedsofar,thelineshavebeenslanted.Thiswillalwaysbethecasewhen both variablesappearintheequation.Ifonlyonevariableappearsintheequation,thenthelinewillbe eitherverticalorhorizontal.Toseewhy,let'sconsideraspeciccase: Usingthegeneralformofaline, ax + by = c ,wecanproduceanequationwithexactlyonevariableby choosing a =0 b =5 ,and c =15 .Theequation ax + by = c thenbecomes 0 x +5 y =15 Since 0 anynumber =0 ,theterm 0 x is 0 foranynumberthatischosenfor x Thus, 0 x +5 y =15 becomes 0+5 y =15 But, 0 istheadditiveidentityand 0+5 y =5 y 5 y =15 Then,solvingfor y weget y =3 Thisisanequationinwhichexactlyonevariableappears. Thismeansthatregardlessofwhichnumberwechoosefor x ,thecorresponding y -valueis3.Sincethe y -valueisalwaysthesameaswemovefromleft-to-rightthroughthe x -values,theheightofthelineabove the x -axisisalwaysthesameinthiscase,3units.Thistypeoflinemustbehorizontal. Anargumentsimilartotheoneabovewillshowthatiftheonlyvariablethatappearsis x ,wecanexpect togetaverticalline. 7.4.11SampleSetC Example7.13 Graph y =4 Theonlyvariableappearingis y .Regardlessofwhich x -valuewechoose,the y -valueisalways4. Allpointswitha y -valueof4satisfytheequation.Thuswegetahorizontalline4unitabovethe x -axis.

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436 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES x y x;y )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 4 0 4 ; 4 1 4 ; 4 2 4 ; 4 3 4 ; 4 4 4 ; 4 Table7.10 Example7.14 Graph x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Theonlyvariablethatappearsis x .Regardlessofwhich y -valuewechoose,the x -valuewillalways be )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 .Thus,wegetaverticallinetwounitstotheleftofthe y -axis. x y x;y )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 4 Table7.11

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437 7.4.12PracticeSetC Exercise7.55 Solutiononp.517. Graph y =2 Exercise7.56 Solutiononp.518. Graph x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Summarizingourresultswecanmakethefollowingobservations: 1.Whenalinearequationintwovariablesiswrittenintheform ax + by = c ,wesayitiswrittenin generalform .

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438 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 2.Tographanequationingeneralformitissometimesconvenienttousetheinterceptmethod. 3.Alinearequationinwhichbothvariablesappearwillgraphasaslantedline. 4.Alinearequationinwhichonlyonevariableappearswillgraphaseitheraverticalorhorizontalline. x = a graphsasaverticallinepassingthrough a onthe x -axis. y = b graphsasahorizontallinepassingthrough b onthe y -axis. 7.4.13Exercises Forthefollowingproblems,graphtheequations. Exercise7.57 Solutiononp.518. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x + y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise7.58 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y =6 Exercise7.59 Solutiononp.518. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x + y =4

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439 Exercise7.60 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y =5 Exercise7.61 Solutiononp.519. 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y =6 Exercise7.62 2 x +5 y =10

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440 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.63 Solutiononp.519. 3 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y =9 Exercise7.64 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +3 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise7.65 Solutiononp.519. y + x =1

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441 Exercise7.66 4 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12=0 Exercise7.67 Solutiononp.520. 2 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y +4=0 Exercise7.68 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +5 y =0

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442 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.69 Solutiononp.520. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x +4=0 Exercise7.70 0 x + y =3 Exercise7.71 Solutiononp.520. 0 x +2 y =2

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443 Exercise7.72 0 x + 1 4 y =1 Exercise7.73 Solutiononp.521. 4 x +0 y =16 Exercise7.74 1 2 x +0 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1

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444 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.75 Solutiononp.521. 2 3 x +0 y =1 Exercise7.76 y =3 Exercise7.77 Solutiononp.521. y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2

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445 Exercise7.78 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y =20 Exercise7.79 Solutiononp.522. x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Exercise7.80 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(9

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446 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.81 Solutiononp.522. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +4=0 Exercise7.82 Constructthegraphofallthepointsthathavecoordinates a;a ,thatis,foreachpoint,the x )]TJ/F15 9.9626 Tf -420.401 -11.956 Td [(and y -valuesarethesame. 7.4.13.1 CalculatorProblems Exercise7.83 Solutiononp.522. 2 : 53 x +4 : 77 y =8 : 45

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447 Exercise7.84 1 : 96 x +2 : 05 y =6 : 55 Exercise7.85 Solutiononp.523. 4 : 1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 : 6 y =15 : 5 Exercise7.86 626 : 01 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(506 : 73 y =2443 : 50

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448 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.4.14ExercisesforReview Exercise7.87 Solutiononp.523. Section2.3 Namethepropertyofrealnumbersthatmakes 4+ x = x +4 atruestatement. Exercise7.88 Section3.3 Supplythemissingword.Theabsolutevalueofanumber a ,denoted j a j ,isthe from a to 0 onthenumberline. Exercise7.89 Solutiononp.523. Section4.6 Findtheproduct x +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise7.90 Section5.4 Solvetheequation 3[3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+4 x ] )]TJ/F8 9.9626 Tf 9.963 0 Td [(24=0 Exercise7.91 Solutiononp.523. Section7.3 Supplythemissingword.Thecoordinateaxesdividetheplaneintofourequal regionscalled 7.5TheSlope-InterceptFormofaLine 5 7.5.1Overview TheGeneralFormofaLine TheSlope-InterceptFormofaLine SlopeandIntercept TheFormulafortheSlopeofaLine 7.5.2TheGeneralFormofaLine Wehaveseenthatthegeneralformofalinearequationintwovariablesis ax + by = c SectionSection7.4. Whenthisequationissolvedfor y ,theresultingformiscalledtheslope-interceptform.Let'sgeneratethis newform. 5 Thiscontentisavailableonlineat.

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449 ax + by = c Subtract ax frombothsides. by = )]TJ/F11 9.9626 Tf 7.748 0 Td [(ax + c Divide both sidesby b by b = )]TJ/F10 6.9738 Tf 6.227 0 Td [(ax b + c b by b = )]TJ/F10 6.9738 Tf 6.227 0 Td [(ax b + c b y = )]TJ/F10 6.9738 Tf 6.227 0 Td [(ax b + c b y = )]TJ/F10 6.9738 Tf 6.227 0 Td [(ax b + c b Thisequationisoftheform y = mx + b ifwereplace )]TJ/F10 6.9738 Tf 6.227 0 Td [(a b with m andconstant c b with b Note: The factthatwelet b = c b isunfortunateandoccursbeacuseoftheletterswehavechosentouseinthegeneral form.Theletter b occursonbothsidesoftheequalsignandmaynotrepresentthesamevalueatall.This problemisoneofthehistoricalconventionand,fortunately,doesnotoccurveryoften. Thefollowingexamplesillustratethisprocedure. Example7.15 Solve 3 x +2 y =6 for y 3 x +2 y =6 Subtract3 x frombothsides. 2 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 x +6 Dividebothsidesby2. y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 2 x +3 Thisequationisoftheform y = mx + b .Inthiscase, m = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(3 2 and b =3 Example7.16 Solve )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 x +5 y =20 for y )]TJ/F8 9.9626 Tf 7.748 0 Td [(15 x +5 y =20 5 y =15 x +20 y =3 x +4 Thisequationisoftheform y = mx + b .Inthiscase, m =3 and b =4 Example7.17 Solve 4 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =0 for y 4 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x y =4 x Thisequationisoftheform y = mx + b .Inthiscase, m =4 and b =0 .Noticethatwecan write y =4 x as y =4 x +0 7.5.3TheSlope-InterceptFormofaLine TheSlope-InterceptFormofaLine y = mx + b Alinearequationintwovariableswrittenintheform y = mx + b issaidtobein slope-interceptform. 7.5.4SampleSetA Thefollowingequations are inslope-interceptform: Example7.18 y =6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 : Inthiscase m =6 and b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 :

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450 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Example7.19 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +9 : Inthiscase m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 and b =9 : Example7.20 y = 1 5 x +4 : 8 Inthiscase m = 1 5 and b =4 : 8 : Example7.21 y =7 x: Inthiscase m =7 and b =0 sincewecanwrite y =7 x as y =7 x +0 : Thefollowingequations arenot inslope-interceptform: Example7.22 2 y =4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : Thecoecientof y is 2 : Tobeinslope-interceptform,thecoecientof y mustbe 1 : Example7.23 y +4 x =5 : Theequationisnotsolvedfor y: The x and y appearonthesamesideoftheequalsign. Example7.24 y +1=2 x: Theequationisnotsolvedfor y: 7.5.5PracticeSetA Thefollowingequationareinslope-interceptform.Ineachcase,specifytheslopeand y -intercept. Exercise7.92 Solutiononp.523. y =2 x +7; m = b = Exercise7.93 Solutiononp.523. y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x +2; m = b = Exercise7.94 Solutiononp.523. y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1; m = b = Exercise7.95 Solutiononp.523. y = 2 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10; m = b = Exercise7.96 Solutiononp.523. y = )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 8 x + 1 2 ; m = b = Exercise7.97 Solutiononp.523. y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x ; m = b = 7.5.6SlopeandIntercept Whentheequationofalineiswritteninslope-interceptform,twoimportantpropertiesofthelinecan beseen:the slope andthe intercept .Let'slookatthesetwopropertiesbygraphingseverallinesand observingthemcarefully.

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451 7.5.7SampleSetB Example7.25 Graphtheline y = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x y x;y 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 4 1 ; 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Table7.12 Lookingcarefullyatthisline,answerthefollowingtwoquestions. Problem1 Atwhatnumberdoesthislinecrossthe y -axis?Doyouseethisnumberintheequation? Solution Thelinecrossesthe y -axisat )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Problem2 Placeyourpencilatanypointontheline.Moveyourpencilexactly one unithorizontallytothe right.Now,howmanyunitsstraightupordownmustyoumoveyourpenciltogetbackonthe line?Doyouseethisnumberintheequation? Solution Aftermovinghorizontallyoneunittotheright,wemustmoveexactlyoneverticalunitup.This numberisthecoecientof x Example7.26 Graphtheline y = 2 3 x +1 x y x;y 0 1 ; 1 3 3 ; 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1

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452 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Table7.13 Lookingcarefullyatthisline,answerthefollowingtwoquestions. Problem1 Atwhatnumberdoesthislinecrossthe y -axis?Doyouseethisnumberintheequation? Solution Thelinecrossesthe y -axisat +1 Problem2 Placeyourpencilatanypointontheline.Moveyourpencilexactly one unithorizontallytothe right.Now,howmanyunitsstraightupordownmustyoumoveyourpenciltogetbackonthe line?Doyouseethisnumberintheequation? Solution Aftermovinghorizontallyoneunittotheright,wemustmoveexactly 2 3 unitupward.Thisnumber isthecoecientof x 7.5.8PracticeSetB Example7.27 Graphtheline y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +4 x y x;y 0 3 2 Table7.14

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453 Lookingcarefullyatthisline,answerthefollowingtwoquestions. Exercise7.98 Solutiononp.523. Atwhatnumberdoesthelinecrossthe y -axis?Doyouseethisnumberintheequation? Exercise7.99 Solutiononp.523. Placeyourpencilatanypointontheline.Moveyourpencilexactly one unithorizontallytothe right.Now,howmanyunitsstraightupordownmustyoumoveyourpenciltogetbackonthe line?Doyouseethisnumberintheequation? InthegraphsconstructedinSampleSetBandPracticeSetB,eachequationhadtheform y = mx + b .We cananswerthesamequestionsbyusingthisformoftheequationshowninthediagram. y -Intercept Exercise7.100 Atwhatnumberdoesthelinecrossthe y -axis?Doyouseethisnumberintheequation? Solution Ineachcase,thelinecrossesthe y -axisattheconstant b .Thenumber b isthenumberatwhich thelinecrossesthe y -axis,anditiscalledthe y -intercept.Theorderedpaircorrespondingtothe y -interceptis ;b : Exercise7.101 Placeyourpencilatanypointontheline.Moveyourpencilexactly one unithorizontallytothe right.Now,howmanyunitsstraightupordownmustyoumoveyourpenciltogetbackonthe line?Doyouseethisnumberintheequation? Solution Togetbackontheline,wemustmoveourpencilexactly m verticalunits.

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454 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Slope Thenumber m isthecoecientofthevariable x .Thenumber m iscalledthe slope ofthelineanditisthe numberofunitsthat y changeswhen x isincreasedby1unit.Thus,if x changesby1unit, y changesby m units. Sincetheequation y = mx + b containsboththeslopeofthelineandthe y -intercept,wecalltheform y = mx + b the slope-intercept form. TheSlope-InterceptFormoftheEquationofaLine Theslope-interceptformofastraightlineis y = mx + b Theslopeofthelineis m ,andthe y -interceptisthepoint ;b TheSlopeisaMeasureoftheSteepnessofaLine Theword slope isreallyquiteappropriate.Itgivesusameasureofthesteepnessoftheline.Considertwo lines,onewithslope 1 2 andtheotherwithslope3.Thelinewithslope3issteeperthanisthelinewithslope 1 2 .Imagineyourpencilbeingplacedatanypointonthelines.Wemakea1-unitincreaseinthe x -value bymovingthepencil one unittotheright.Togetbacktoonelineweneedonlymovevertically 1 2 unit, whereastogetbackontotheotherlineweneedtomovevertically3units. 7.5.9SampleSetC Findtheslopeandthe y -interceptofthefollowinglines. Example7.28 y =2 x +7 : Thelineisintheslope-interceptform y = mx + b: Theslopeis m ,thecoecientof x Therefore, m =2 : The y -interceptisthepoint ;b : Since b =7 ,the y -interceptis ; 7 :

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455 Slope :2 y -intercept : ; 7 Example7.29 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x +1 : Thelineisinslope-interceptform y = mx + b: Theslopeis m ,thecoecientof x .So, m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 : The y -interceptisthepoint ;b : Since b =1 ,the y -interceptis ; 1 : Slope : )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y -intercept : ; 1 Example7.30 3 x +2 y =5 : Theequationiswritteningeneralform.Wecanputtheequationinslope-interceptform bysolvingfor y 3 x +2 y =5 2 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 x +5 y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 2 x + 5 2 Nowtheequationisinslope-interceptform. Slope: )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(3 2 y -intercept: )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 5 2 7.5.10PracticeSetC Exercise7.102 Solutiononp.524. Findtheslopeand y -interceptoftheline 2 x +5 y =15 : 7.5.11TheFormulafortheSlopeofaLine Wehaveobservedthattheslopeisameasureofthesteepnessofaline.Wewishtodevelopaformulafor measuringthissteepness. Itseemsreasonabletodevelopaslopeformulathatproducesthefollowingresults: Steepnessofline 1 > steepnessofline2.

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456 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Consideralineonwhichweselectanytwopoints.We'lldenotethesepointswiththeorderedpairs x 1 ; y 1 and x 2 ; y 2 .Thesubscriptshelpustoidentifythepoints. x 1 ; y 1 istherstpoint.Subscript1indicatestherstpoint. x 2 ; y 2 isthesecondpoint.Subscript2indicatesthesecondpoint. Thedierencein x values x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 givesusthehorizontalchange,andthedierencein y values y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 givesustheverticalchange.Ifthelineisverysteep,thenwhengoingfromtherstpointtothesecond point,wewouldexpectalargeverticalchangecomparedtothehorizontalchange.Ifthelineisnotvery steep,thenwhengoingfromtherstpointtothesecondpoint,wewouldexpectasmallverticalchange comparedtothehorizontalchange. Wearecomparingchanges.Weseethatwearecomparing Theverticalchangetothehorizontalchange Thechangein y tothechangein x y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 to x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1

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457 Thisisacomparisonandisthereforea ratio .Ratioscanbeexpressedasfractions.Thus,ameasureof thesteepnessofalinecanbeexpressedasaratio. Theslopeofalineisdenedastheratio Slope = changein y changein x Mathematically,wecanwritethesechangesas Slope = y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 FindingtheSlopeofaLine Theslopeofanonverticallinepassingthroughthepoints x 1 ; y 1 and x 2 ; y 2 isfoundbytheformula m = y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 7.5.12SampleSetD Forthetwogivenpoints,ndtheslopeofthelinethatpassesthroughthem. Example7.31 ; 1 and ; 3 Lookinglefttorightonthelinewecanchoose x 1 ; y 1 tobe ; 1 ,and x 2 ; y 2 tobe ; 3 : Then, m = y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 = 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 = 2 1 =2 Thislinehasslope2.Itappearsfairlysteep.Whentheslopeiswritteninfractionform, 2= 2 1 wecansee,byrecallingtheslopeformula,thatas x changes1unittotherightbecauseofthe +1 y changes2unitsupwardbecauseofthe +2 m = changein y changein x = 2 1

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458 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Noticethataswelooklefttoright,thelinerises. Example7.32 ; 2 and ; 3 Lookinglefttorightonthelinewecanchoose x 1 ; y 1 tobe ; 2 and x 2 ; y 2 tobe ; 3 : Then, m = y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 = 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 = 1 2 Thislinehasslope 1 2 .Thus,as x changes2unitstotherightbecauseofthe +2 y changes1 unitupwardbecauseofthe +1 m = changein y changein x = 1 2 Noticethatinexamples1and2,bothlineshavepositiveslopes, +2 and + 1 2 ,andboth lines rise aswelooklefttoright. Example7.33 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; 4 and ; 1 Lookinglefttorightonthelinewecanchoose x 1 ; y 1 tobe )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 4 and x 2 ; y 2 tobe ; 1 .Then, m = y 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 = 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 1+2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 3 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1

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459 Thislinehasslope )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : Whentheslopeiswritteninfractionform, m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1= )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 +1 ,wecanseethatas x changes1 unittotherightbecauseofthe +1 y changes1unitdownwardbecauseofthe )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Noticealsothatthislinehasanegativeslopeanddeclinesaswelooklefttoright. Example7.34 ; 3 and ; 3 m = y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 = 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 = 0 4 =0 Thislinehas0slope.Thismeansithas no riseand,therefore,isahorizontalline.Thisdoes notmeanthatthelinehasnoslope,however. Example7.35 ; 4 and ; 0 Thisproblemshowswhytheslopeformulaisvalidonlyfornonverticallines. m = y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 = 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 0

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460 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Sincedivisionby0isundened,wesaythatverticallineshaveundenedslope.Sincethereisno realnumbertorepresenttheslopeofthisline,wesometimessaythatverticallineshave undened slope ,or noslope 7.5.13PracticeSetD Exercise7.103 Solutiononp.524. Findtheslopeofthelinepassingthrough ; 1 and ; 3 .Graphthislineonthegraphofproblem 2below. Exercise7.104 Solutiononp.524. Findtheslopeofthelinepassingthrough ; 4 and ; 5 .Graphthisline. Exercise7.105 Solutiononp.524. Comparethelinesofthefollowingproblems.Dothelinesappeartocross?Whatisitcalledwhen linesdonotmeetparallelorintersecting?Comparetheirslopes.Makeastatementaboutthe conditionoftheselinesandtheirslopes. Beforetryingsomeproblems,let'ssummarizewhatwehaveobserved. Exercise7.106 Theequation y = mx + b iscalledtheslope-interceptformoftheequationofaline.Thenumber m istheslopeofthelineandthepoint ;b isthe y -intercept. Exercise7.107 Theslope, m ; ofalineisdenedasthesteepnessoftheline,anditisthenumberofunitsthat y changeswhen x changes1unit.

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461 Exercise7.108 Theformulaforndingtheslopeofalinethroughanytwogivenpoints x 1 ;y 1 and x 2 ;y 2 is m = y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 1 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 1 Exercise7.109 Thefraction y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 1 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 1 representsthe Changein y Changein x : Exercise7.110 Aswelookatagraphfromlefttoright,lineswithpositivesloperiseandlineswithnegativeslope decline. Exercise7.111 Parallellineshavethesameslope. Exercise7.112 Horizontallineshave0slope. Exercise7.113 Verticallineshaveundenedslopeornoslope. 7.5.14Exercises Forthefollowingproblems,determinetheslopeand y -interceptofthelines. Exercise7.114 Solutiononp.524. y =3 x +4 Exercise7.115 y =2 x +9 Exercise7.116 Solutiononp.524. y =9 x +1 Exercise7.117 y =7 x +10 Exercise7.118 Solutiononp.524. y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x +5 Exercise7.119 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 x +8 Exercise7.120 Solutiononp.524. y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise7.121 y = )]TJ/F11 9.9626 Tf 7.748 0 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise7.122 Solutiononp.524. y = )]TJ/F11 9.9626 Tf 7.748 0 Td [(x +2 Exercise7.123 2 y =4 x +8 Exercise7.124 Solutiononp.524. 4 y =16 x +20 Exercise7.125 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y =15 x +55 Exercise7.126 Solutiononp.524. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y =12 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(27

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462 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.127 y = 3 5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 Exercise7.128 Solutiononp.524. y = 2 7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 Exercise7.129 y = )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 8 x + 2 3 Exercise7.130 Solutiononp.525. y = )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 5 x )]TJ/F7 6.9738 Tf 11.159 3.922 Td [(4 7 Exercise7.131 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y =5 x +8 Exercise7.132 Solutiononp.525. )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 x +1 Exercise7.133 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y = x +1 Exercise7.134 Solutiononp.525. )]TJ/F11 9.9626 Tf 7.749 0 Td [(y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +3 Exercise7.135 3 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y =7 Exercise7.136 Solutiononp.525. 5 x +3 y =6 Exercise7.137 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Exercise7.138 Solutiononp.525. )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +4 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Forthefollowingproblems,ndtheslopeofthelinethroughthepairsofpoints. Exercise7.139 ; 6 ; ; 9 Exercise7.140 Solutiononp.525. ; 3 ; ; 7 Exercise7.141 ; 5 ; ; 7 Exercise7.142 Solutiononp.525. ; 1 ; ; 8 Exercise7.143 ; 5 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise7.144 Solutiononp.525. )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; 1 ; ; 5 Exercise7.145 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 ; ; 1 Exercise7.146 Solutiononp.525. ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 1 Exercise7.147 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 ; 4 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; 0 Exercise7.148 Solutiononp.525. )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; 2 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 ; 6

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463 Exercise7.149 ; 12 ; ; 0 Exercise7.150 Solutiononp.525. ; 0 ; ; 6 Exercise7.151 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise7.152 Solutiononp.525. )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 Exercise7.153 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise7.154 Solutiononp.525. )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; 0 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Exercise7.155 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ; ; 0 Exercise7.156 Solutiononp.525. ; 3 ; ; 3 Exercise7.157 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 ; ; 7 Exercise7.158 Solutiononp.525. ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 ; ; 3 Exercise7.159 ; 2 ; ; 2 Exercise7.160 Solutiononp.525. ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 ; ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 Exercise7.161 Dolineswithapositivesloperiseordeclineaswelooklefttoright? Exercise7.162 Solutiononp.525. Dolineswithanegativesloperiseordeclineaswelooklefttoright? Exercise7.163 Makeastatementabouttheslopesofparallellines. 7.5.14.1 CalculatorProblems Forthefollowingproblems,determinetheslopeand y -interceptofthelines.Roundtotwodecimalplaces. Exercise7.164 Solutiononp.525. 3 : 8 x +12 : 1 y =4 : 26 Exercise7.165 8 : 09 x +5 : 57 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 42 Exercise7.166 Solutiononp.525. 10 : 813 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(17 : 0 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(45 : 99 Exercise7.167 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 : 003 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(92 : 388 y =0 : 008 Forthefollowingproblems,ndtheslopeofthelinethroughthepairsofpoints.Roundtotwodecimal places.

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464 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.168 Solutiononp.525. : 56 ; 9 : 37 ; : 16 ; 4 : 90 Exercise7.169 : 1 ; 8 : 9 ; : 7 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : 06 Exercise7.170 Solutiononp.525. : 89 ; 227 : 61 ; : 04 ; 227 : 61 Exercise7.171 : 00426 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 : 00404 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 00191 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(0 : 00404 Exercise7.172 Solutiononp.525. : 81 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(23 : 19 ; : 81 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(26 : 87 Exercise7.173 )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 0000567 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 : 0000567 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 00765 ; 0 : 00764 7.5.15ExercisesforReview Exercise7.174 Solutiononp.525. Section2.7 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 y 3 w 4 0 Exercise7.175 Section5.4 Solvetheequation 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2+4=0 Exercise7.176 Solutiononp.525. Section5.6 Whenfourtimesanumberisdividedbyve,andthatresultisdecreasedbyeight, theresultiszero.Whatistheoriginalnumber? Exercise7.177 Section5.8 Solve )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y +10= x +2 if x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise7.178 Solutiononp.526. Section7.4 Graphthelinearequation x + y =3 7.6GraphingEquationsinSlope-InterceptForm 6 7.6.1Overview UsingtheSlopeandIntercepttoGraphaLine 6 Thiscontentisavailableonlineat.

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465 7.6.2UsingtheSlopeandIntercepttoGraphaLine Whenalinearequationisgiveninthe generalform ax + by = c ,weobservedthatanecientgraphical approachwastheinterceptmethod.Welet x =0 andcomputedthecorrespondingvalueof y ,thenlet y =0 andcomputedthecorrespondingvalueof x Whenanequationiswritteninthe slope-interceptform y = mx + b ,therearealsoecientwaysof constructingthegraph.Oneway,butlessecient,istochoosetwoorthree x -valuesandcomputetond thecorresponding y -values.However,computationsaretedious,timeconsuming,andcanleadtoerrors. Anotherway,themethodlistedbelow,makesuseoftheslopeandthe y -interceptforgraphingtheline.It isquick,simple,andinvolvesnocomputations. GraphingMethod 1.Plotthe y -intercept ;b 2.Determineanotherpointbyusingtheslope m 3.Drawalinethroughthetwopoints. Recallthatwedenedtheslope m astheratio y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 1 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 1 .Thenumerator y 2 )]TJ/F11 9.9626 Tf 9.205 0 Td [(y 1 representsthenumberofunits that y changesandthedenominator x 2 )]TJ/F11 9.9626 Tf 9.156 0 Td [(x 1 representsthenumberofunitsthat x changes.Suppose m = p q Then p isthenumberofunitsthat y changesand q isthenumberofunitsthat x changes.Sincethese changesoccursimultaneously,startwithyourpencilatthe y -intercept,move p unitsintheappropriate verticaldirection,andthenmove q unitsintheappropriatehorizontaldirection.Markapointatthis location. 7.6.3SampleSetA Graphthefollowinglines. Example7.36 y = 3 4 x +2 Step1:The y -interceptisthepoint ; 2 .Thusthelinecrossesthe y -axis2unitsabovetheorigin. Markapointat ; 2 Step2:Theslope, m ,is 3 4 .Thismeansthatifwestartatanypointonthelineandmoveour pencil 3 unitsupandthen 4 unitstotheright,we'llbebackontheline.Startata knownpoint,the y -intercept ; 2 .Moveup 3 units,thenmove 4 unitstotheright. Markapointatthislocation.Notealsothat 3 4 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 .Thismeansthatifwestart atanypointonthelineandmoveourpencil 3 units down and 4 unitstothe left we'llbebackontheline.Notealsothat 3 4 = 3 4 1 .Thismeansthatifwestartatanypoint onthelineandmovetotheright 1 unit,we'llhavetomoveup 3 = 4 unittogetbackontheline.

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466 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Step3:Drawalinethroughbothpoints. Example7.37 y = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(1 2 x + 7 2 Step1:The y -interceptisthepoint )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 7 2 .Thusthelinecrossesthe y -axis 7 2 unitsabovetheorigin. Markapointat )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 7 2 ,or )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 3 1 2 Step2:Theslope, m ,is )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(1 2 .Wecanwrite )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(1 2 as )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 .Thus,westartataknownpoint,the y -intercept )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 3 1 2 ,move down oneunitbecauseofthe )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,thenmoveright 2 units. Markapointatthislocation.

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467 Step3:Drawalinethroughbothpoints. Example7.38 y = 2 5 x Step1:Wecanputthisequationintoexplicitslope-interceptbywritingitas y = 2 5 x +0 The y -interceptisthepoint ; 0 ,theorigin.Thislinegoesrightthroughtheorigin. Step2:Theslope, m ,is 2 5 .Startingattheorigin,wemoveup 2 units,thenmovetotheright 5 units.Markapointatthislocation.

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468 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Step3:Drawalinethroughthetwopoints. Example7.39 y =2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Step1:The y -interceptisthepoint ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 .Thusthelinecrossesthe y -axis 4 unitsbelowthe origin.Markapointat ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Step2:Theslope, m ,is2.Ifwewritetheslopeasafraction, 2= 2 1 ,wecanreadhowtomakethe changes.Startattheknownpoint ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ,moveup 2 units,thenmoveright 1 unit.Mark apointatthislocation. Step3:Drawalinethroughthetwopoints.

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469 7.6.4PracticeSetA Usethe y -interceptandtheslopetographeachline. Exercise7.179 Solutiononp.526. y = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 3 x +4 Exercise7.180 Solutiononp.526. y = 3 4 x 7.6.5Excercises Forthefollowingproblems,graphtheequations. Exercise7.181 Solutiononp.526. y = 2 3 x +1

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470 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.182 y = 1 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Exercise7.183 Solutiononp.527. y =5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise7.184 y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(6 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3

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471 Exercise7.185 Solutiononp.527. y = 3 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Exercise7.186 y = 1 5 x +2 Exercise7.187 Solutiononp.527. y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(8 3 x +4

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472 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.188 y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(10 3 x +6 Exercise7.189 Solutiononp.528. y =1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise7.190 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 x +1

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473 Exercise7.191 Solutiononp.528. y = x +2 Exercise7.192 y = 3 5 x Exercise7.193 Solutiononp.528. y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(4 3 x

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474 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.194 y = x Exercise7.195 Solutiononp.529. y = )]TJ/F11 9.9626 Tf 7.748 0 Td [(x Exercise7.196 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3

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475 Exercise7.197 Solutiononp.529. 6 x +10 y =30 Exercise7.198 x + y =0 7.6.6ExcersiseforReview Exercise7.199 Solutiononp.529. Section5.7 Solvetheinequality 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 .

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476 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.200 Section7.2 Graphtheinequality y +3 > 1 Exercise7.201 Solutiononp.529. Section7.4 Graphtheequation y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise7.202 Section7.5 Determinetheslopeand y -interceptoftheline )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x =16 Exercise7.203 Solutiononp.530. Section7.5 Findtheslopeofthelinepassingthroughthepoints )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 5 and ; 3 7.7FindingtheEquationofaLine 7 7.7.1Overview TheSlope-InterceptandPoint-SlopeForms 7.7.2TheSlope-InterceptandPoint-SlopeForms Inthepervioussectionswehavebeengivenanequationandhaveconstructedthelinetowhichitcorresponds. Now,however,supposewe'regivensomegeometricinformationaboutthelineandwewishtoconstructthe correspondingequation.Wewishtondtheequationofaline. Weknowthattheformulafortheslopeofalineis m = y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 1 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 1 : Wecanndtheequationofalineusing theslopeformulaineitheroftwoways: Example7.40 Ifwe'regiventheslope, m ,and any point x 1 ;y 1 ontheline,wecansubstitutethisinformation intotheformulaforslope. Let x 1 ;y 1 betheknownpointonthelineandlet x;y beanyotherpointontheline.Then 7 Thiscontentisavailableonlineat.

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477 m = y )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 1 x )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 1 Multiplybothsidesby x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 : m x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 = x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 y )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 1 x )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 1 m x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 = y )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 1 Forconvenience,we'llrewritetheequation : y )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 Sincethisequationwasderivedusingapointandtheslopeofaline,itiscalledthe point-slope formofaline. Example7.41 Ifwearegiventheslope, m ,y-intercept ; ;b ,wecansubstitutethisinformationintotheformula forslope. Let ;b bethey-interceptand x;y beanyotherpointontheline.Then, m = y )]TJ/F10 6.9738 Tf 6.226 0 Td [(b x )]TJ/F7 6.9738 Tf 6.227 0 Td [(0 m = y )]TJ/F10 6.9738 Tf 6.227 0 Td [(b x Multiplybothsidesby x m x = x y )]TJ/F10 6.9738 Tf 6.226 0 Td [(b x mx = y )]TJ/F11 9.9626 Tf 9.962 0 Td [(b Solvefor y: mx + b = y Forconvenience,we'llrewritethisequation. y = mx + b Sincethisequationwasderivedusingtheslopeandtheintercept,itwascalledthe slopeintercept formofaline. Wesummarizethesetwoderivationsasfollows. FormsoftheEquationofaLine Wecanndtheequationofalineifwe'regiveneitherofthefollowingsetsofinformation: 1.Theslope, m; andthe y -intercept, ;b ; bysubstitutingthesevaluesinto y = mx + b Thisistheslope-interceptform. 2.Theslope, m; andanypoint, x 1 ;y 1 ; bysubstitutingthesevaluesinto y )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 Thisisthepoint-slopeform. Noticethatbothformsrelyonknowingtheslope.Ifwearegiventwopointsonthelinewemaystillnd theequationofthelinepassingthroughthembyrstndingtheslopeoftheline,thenusingthepoint-slope form. Itiscustomarytouseeithertheslope-interceptformorthegeneralformforthenalformoftheline. Wewillusetheslope-interceptformasthenalform. 7.7.3SampleSetA Findtheequationofthelineusingthegiveninformation. Example7.42 m =6 ;y -intercept ; 4 Sincewe'regiventheslopeandthe y -intercept,we'llusetheslope-interceptform. m =6 ;b =4 : y = mx + b y =6 x +4 Example7.43 m = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 4 ;y -intercept )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 1 8 Sincewe'regiventheslopeandthe y -intercept,we'llusetheslope-interceptform. m = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 4 ;

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478 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES b = 1 8 : y = mx + b y = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 4 x + 1 8 Example7.44 m =2 ; thepoint ; 3 : Writetheequationinslope-interceptform. Sincewe'regiventheslopeandsomepoint,we'llusethepoint-slopeform. y )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 Let x 1 ;y 1 be,3. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Putthisequationinslope-interceptformbysolvingfor y: y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 y =2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Example7.45 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; thepoint )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 0 : Writetheequationinslope-interceptform. Sincewe'regiventheslopeandsomepoint,we'llusethepoint-slopeform. y )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 Let x 1 ;y 1 be-3,0. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(0= )]TJ/F8 9.9626 Tf 7.749 0 Td [(5[ x )]TJ/F8 9.9626 Tf 9.962 0 Td [( )]TJ/F8 9.9626 Tf 7.748 0 Td [(3] y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x +3 Solvefor y: y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(15 Example7.46 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; thepoint ; 7 : Writetheequationinslope-interceptform. We'regiventheslopeandapoint,butcarefulobservationrevealsthatthispointisactuallythe y -intercept.Thus,we'llusetheslope-interceptform.Ifwehadnotseenthatthispointwasthe y -interceptwewouldhaveproceededwiththepoint-slopeform.Thiswouldcreateslightlymore work,butstillgivethesameresult. Slope-interceptform Point-slopeform y = mx + b y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 x +7 y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +7 y )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(7= )]TJ/F11 9.9626 Tf 7.749 0 Td [(x y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +7 Example7.47 Thetwopoints ; 1 and ; 5 : Writetheequationinslope-interceptform. Sincewe'regiventwopoints,we'llndthesloperst. m = y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 1 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 1 = 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Now,wehavetheslopeandtwopoints.Wecanuseeitherpointandthepoint-slopeform.

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479 Using ; 1 Using ; 5 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x +16 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x +17 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x +12 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x +17 Table7.15 Wecanseethattheuseofeitherpointgivesthesameresult. 7.7.4PracticeSetA Findtheequationofeachlinegiventhefollowinginformation.Usetheslope-interceptformasthenalform oftheequation. Exercise7.204 Solutiononp.530. m =5 ;y -intercept ; 8 : Exercise7.205 Solutiononp.530. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 ;y -intercept ; 3 : Exercise7.206 Solutiononp.530. m =2 ;y -intercept ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 : Exercise7.207 Solutiononp.530. m =1 ;y -intercept ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : Exercise7.208 Solutiononp.530. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ;y -intercept ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 : Exercise7.209 Solutiononp.530. m =4 ; thepoint ; 2 : Exercise7.210 Solutiononp.530. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; thepoint )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 0 : Exercise7.211 Solutiononp.530. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; thepoint )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 : Exercise7.212 Solutiononp.530. Thetwopoints ; 1 and ; 5 : Exercise7.213 Solutiononp.530. Thetwopoints )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; 8 : 7.7.5SampleSetB Example7.48 Findtheequationofthelinepassingthroughthepoint ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 havingslope0. We'regiventheslopeandsomepoint,sowe'llusethepoint-slopeform.With m =0 and x 1 ;y 1 as ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 ; wehave

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480 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES y )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 y )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(7=0 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 y +7=0 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 Thisisahorizontalline. Example7.49 Findtheequationofthelinepassingthroughthepoint ; 3 giventhatthelineisvertical. Sincethelineisvertical,theslopedoesnotexist.Thus,wecannotuseeithertheslope-intercept formorthepoint-slopeform.Wemustrecallwhatweknowaboutverticallines.Theequationof thislineissimply x =1 : 7.7.6PracticeSetB Exercise7.214 Solutiononp.530. Findtheequationofthelinepassingthroughthepoint )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 9 havingslope0. Exercise7.215 Solutiononp.530. Findtheequationofthelinepassingthroughthepoint )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 6 giventhatthelineisvertical. 7.7.7SampleSetC Example7.50 Readingonlyfromthegraph,determinetheequationoftheline. Theslopeofthelineis 2 3 ; andthelinecrossesthe y -axisatthepoint ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 : Usingtheslopeinterceptformweget y = 2 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 7.7.8PracticeSetC Exercise7.216 Solutiononp.530. Readingonlyfromthegraph,determinetheequationoftheline.

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481 7.7.9Exercises Forthefollowingproblems,writetheequationofthelineusingthegiveninformationinslope-interceptform. Exercise7.217 Solutiononp.530. m =3 ;y -intercept ; 4 Exercise7.218 m =2 ;y -intercept ; 5 Exercise7.219 Solutiononp.530. m =8 ;y -intercept ; 1 Exercise7.220 m =5 ;y -intercept ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise7.221 Solutiononp.530. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ;y -intercept ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Exercise7.222 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ;y -intercept ; 0 Exercise7.223 Solutiononp.530. m = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(3 2 ;y -intercept ; 0 Exercise7.224 m =3 ; ; 4 Exercise7.225 Solutiononp.530. m =1 ; ; 8 Exercise7.226 m =2 ; ; 4 Exercise7.227 Solutiononp.530. m =8 ; ; 0 Exercise7.228 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; ; 0 Exercise7.229 Solutiononp.531. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; ; 0 Exercise7.230 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; ; 0

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482 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.231 Solutiononp.531. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; ; 1 Exercise7.232 ; 0 ; ; 2 Exercise7.233 Solutiononp.531. ; 0 ; ; 8 Exercise7.234 ; 1 ; ; 3 Exercise7.235 Solutiononp.531. ; 5 ; ; 4 Exercise7.236 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; ; 2 Exercise7.237 Solutiononp.531. ; 3 ; ; 3 Exercise7.238 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; 5 ; ; 5 Exercise7.239 Solutiononp.531. ; 1 ; ; 2 Exercise7.240 ; 7 ; ; 8 Exercise7.241 Solutiononp.531. ; 3 ; ; 5 Exercise7.242 ; 0 ; ; 1 Exercise7.243 Solutiononp.531. )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; 4 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise7.244 ; 6 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise7.245 Solutiononp.531. ; 12 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 Exercise7.246 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; ; 0 Forthefollowingproblems,readonlyfromthegraphanddeterminetheequationofthelines. Exercise7.247 Solutiononp.531.

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483 Exercise7.248 Exercise7.249 Solutiononp.531. Exercise7.250

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484 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.251 Solutiononp.531. Exercise7.252 Exercise7.253 Solutiononp.531. 7.7.10ExercisesforReview Exercise7.254 Section7.2 Graphtheequation x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3=0 :

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485 Exercise7.255 Solutiononp.531. Section7.4 Supplythemissingword.Thepointatwhichalinecrossesthe y -axisiscalledthe Exercise7.256 Section7.5 Supplythemissingword.The ofalineisameasureofthesteepnessof theline. Exercise7.257 Solutiononp.531. Section7.5 Findtheslopeofthelinethatpassesthroughthepoints ; 0 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 : Exercise7.258 Section7.6 Graphtheequation 3 y =2 x +3 : 7.8GraphingLinearInequalitiesinTwoVariables 8 7.8.1Overview LocationofSolutions MethodofGraphing 7.8.2LocationofSolutions Inourstudyoflinearequationsintwovariables,weobservedthat all thesolutionstotheequation,and onlythesolutionstotheequation,werelocatedonthegraphoftheequation.Wenowwishtodetermine thelocationofthesolutionstolinearinequalitiesintwovariables.Linearinequalitiesintwovariablesare inequalitiesoftheforms: ax + by cax + by c ax + byc Half-Planes Astraightlinedrawnthroughtheplanedividestheplaneintotwo half-planes BoundaryLine Thestraightlineiscalledthe boundaryline 8 Thiscontentisavailableonlineat.

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486 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoanInequalityinTwoVariables Recallthatwhenworkingwithlinearequationsintwovariables,weobservedthatorderedpairsthat producedtruestatementswhensubstitutedintoanequationwerecalledsolutionstothatequation.Wecan makeasimilarstatementforinequalitiesintwovariables.Wesaythataninequalityintwovariableshasa solutionwhenapairofvalueshasbeenfoundsuchthatwhenthesevaluesaresubstitutedintotheinequality atruestatementresults. TheLocationofSolutionsinthePlane Aswithequations,solutionstolinearinequalitieshaveparticularlocationsintheplane.Allsolutionstoa linearinequalityintwovariablesarelocatedinoneandonlyinoneentirehalf-plane.Forexample,consider theinequality 2 x +3 y 6 Allthesolutionstotheinequality 2 x +3 y 6 lieintheshadedhalf-plane. Example7.51 Point A ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 isasolutionsince 2 x +3 y 6 2+3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 6? 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 6? )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 6 : True Example7.52 Point B ; 5 isnotasolutionsince

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487 2 x +3 y 6 2+3 6? 4+15 6? 19 6 : False 7.8.3MethodofGraphing Themethodofgraphinglinearinequalitiesintwovariablesisasfollows: 1.Graphtheboundarylineconsidertheinequalityasanequation,thatis,replacetheinequalitysign withanequalsign. a.Iftheinequalityis or ,drawtheboundaryline solid .Thismeansthatpointsonthelineare solutionsandarepartofthegraph. b.Iftheinequalityis < or > ,drawtheboundaryline dotted .Thismeansthatpointsontheline are not solutionsandare not partofthegraph. 2.Determinewhichhalf-planetoshadebychoosingatestpoint. a.If,whensubstituted,thetestpointyieldsatruestatement,shadethehalf-planecontainingit. b.If,whensubstituted,thetestpointyieldsafalsestatement,shadethehalf-planeontheopposite sideoftheboundaryline. 7.8.4SampleSetA Example7.53 Graph 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y )]TJ/F8 9.9626 Tf 19.958 0 Td [(4 1.Graphtheboundaryline.Theinequalityis sowe'lldrawtheline solid .Considerthe inequalityasanequation. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 .1 x y x;y 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 2 0 ; 2 )]TJ/F13 6.9738 Tf 5.761 -4.147 Td [()]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 ; 0 Table7.16

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488 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 2.Chooseatestpoint.Theeasiestoneis ; 0 .Substitute ; 0 intotheoriginalinequality. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F8 9.9626 Tf 18.265 0 Td [(4 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 )]TJ/F8 9.9626 Tf 18.265 0 Td [(4? 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 )]TJ/F8 9.9626 Tf 18.265 0 Td [(4? 0 )]TJ/F8 9.9626 Tf 18.264 0 Td [(4 : True .2 Shadethehalf-planecontaining ; 0 Example7.54 Graph x + y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 < 0 1.Graphtheboundaryline: x + y )]TJ/F8 9.9626 Tf 9.755 0 Td [(3=0 .Theinequalityis < sowe'lldrawtheline dotted 2.Chooseatestpoint,say ; 0 x + y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 < 0 0+0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 < 0? )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 < 0 : True .3 Shadethehalf-planecontaining ; 0 .

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489 Example7.55 Graph y 2 x 1.Graphtheboundaryline y =2 x .Theinequalityis ,sowe'lldrawtheline solid 2.Chooseatestpoint,say ; 0 y 2 x 0 2? 0 0 : True Shadethehalf-planecontaining ; 0 .Wecan't! ; 0 isrightontheline!Pickanothertest point,say ; 6 y 2 x 6 2? 6 2 : False Shadethehalf-planeontheoppositesideoftheboundaryline.

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490 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Example7.56 Graph y> 2 1.Graphtheboundaryline y =2 .Theinequalityis > sowe'lldrawtheline dotted 2.Wedon'treallyneedatestpoint.Whereis y> 2? Above theline y =2! Anypointabove thelineclearlyhasa y -coordinategreaterthan2. 7.8.5PracticeSetA Solvethefollowinginequalitiesbygraphing.

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491 Exercise7.259 Solutiononp.531. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +2 y 4 Exercise7.260 Solutiononp.531. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y< 4 Exercise7.261 Solutiononp.532. 3 x + y> 0 Exercise7.262 Solutiononp.532. x 1

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492 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.8.6Exercises Solvetheinequalitiesbygraphing. Exercise7.263 Solutiononp.532. y
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493 Exercise7.266 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +5 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 < 0 Exercise7.267 Solutiononp.533. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +4 y> )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 Exercise7.268 2 x +5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 0

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494 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.269 Solutiononp.533. y 4 Exercise7.270 x 2 Exercise7.271 Solutiononp.534. x 0

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495 Exercise7.272 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y< 0 Exercise7.273 Solutiononp.534. x +3 y 0 Exercise7.274 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +4 y> 0

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496 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.8.7ExercisesforReview Exercise7.275 Solutiononp.534. Section7.2 Graphtheinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +5 )]TJ/F8 9.9626 Tf 18.265 0 Td [(1 Exercise7.276 Section7.2 Supplythemissingword.Thegeometricrepresentationpictureofthesolutions toanequationiscalledthe oftheequation. Exercise7.277 Solutiononp.534. Section7.5 Supplythedenominator: m = y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 1 ? Exercise7.278 Section7.6 Graphtheequation y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x +2 Exercise7.279 Solutiononp.534. Section7.7 Writetheequationofthelinethathasslope4andpassesthroughthepoint )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 2 .

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497 7.9SummaryofKeyConcepts 9 7.9.1SummaryofKeyConcepts GraphofaFunctionSection7.2 Thegeometricrepresentationpictureofthesolutionstoanequationiscalledthe graph oftheequation. AxisSection7.2 An axis isthemostbasicstructureofagraph.Inmathematics,thenumberlineisusedasanaxis. NumberofVariablesandtheNumberofAxesSection7.2 Anequationinonevariablerequiresoneaxis.One-dimension. Anequationintwovariablerequirestwoaxes.Two-dimensions. Anequationinthreevariablerequiresthreeaxes.Three-dimensions. Anequationin n variablerequires n axes. n -dimensions. CoordinateSystemSection7.2 Asystemofaxesthatisconstructedforgraphinganequationiscalleda coordinatesystem GraphinganEquationSection7.2 Thephrase graphinganequation isinterpretedasmeaninggeometricallylocatingthesolutionstothat equation. UsesofaGraphSection7.2 Agraphmayrevealinformationthatmaynotbeevidentfromtheequation. RectangularCoordinateSystem xy -PlaneSection7.3 A rectangularcoordinatesystem isconstructedbyplacingtwonumberlinesat 90 angles.Theselines formaplanethatisreferredtoasthe xy -plane. OrderedPairsandPointsSection7.3 Foreachorderedpair a;b ; thereexistsauniquepointintheplane,andforeachpointintheplanewecan associateauniqueorderedpair a;b ofrealnumbers. GraphsofLinearEquationsSection7.4 Whengraphed,alinearequationproducesastraightline. GeneralFormofaLinearEquationinTwoVariablesSection7.4 The generalform ofalinearequationintwovariablesis ax + by = c; where a and b arenotboth0. Graphs,OrderedPairs,Solutions,andLinesSection7.4 Thegraphingofallorderedpairsthatsolvealinearequationintwovariablesproducesastraightline. Thegraphofalinearequationintwovariablesisastraightline. Ifanorderedpairisasolutiontoalinearequationintwovariables,thenitliesonthegraphoftheequation. Anypointorderedpairthatliesonthegraphofalinearequationintwovariablesisasolutiontothat equation. InterceptSection7.4 An intercept isapointwherealineinterceptsacoordinateaxis. InterceptMethodSection7.4 The interceptmethod isamethodofgraphingalinearequationintwovariablesbyndingtheintercepts, thatis,byndingthepointswherethelinecrossesthe x -axisandthe y -axis. Slanted,Vertical,andHorizontalLinesSection7.4 Anequationinwhichbothvariablesappearwillgraphasa slanted line. Alinearequationinwhichonlyonevariableappearswillgraphaseithera vertical or horizontal line. 9 Thiscontentisavailableonlineat.

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498 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES x = a graphsasaverticallinepassingthrough a onthe x -axis. y = b graphsasahorizontallinepassingthrough b onthe y -axis. SlopeofaLineSection7.5 Theslopeofalineisameasureoftheline'ssteepness.If x 1 ;y 1 and x 2 ;y 2 areanytwopointsonaline, theslopeofthelinepassingthroughthesepointscanbefoundusingtheslopeformula. m = y 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 1 x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 = verticalchange horizontalchange SlopeandRiseandDeclineSection7.5 Movinglefttoright,lineswithpositivesloperise,andlineswithnegativeslopedecline. GraphinganEquationGiveninSlope-InterceptFormSection7.6 Anequationwritteninslopeinterceptformcanbegraphedby 1.Plottingthe y -intercept ;b 2.Determininganotherpointusingtheslope, m 3.Drawingalinethroughthesetwopoints. FormsofEquationsofLinesSection6.7 Generalform Slope-interceptform point-slopefrom ax + by = cy = mx + by )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 1 = m x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 Tousethisform,the slopeand y -intercept areneeded. Tousethisform,the slopeandonepoint, ortwopoints,areneeded. Half-PlanesandBoundaryLinesSection7.8 Astraightlinedrawnthroughtheplanedividestheplaneintotwo half-planes .Thestraightlineiscalled a boundaryline SolutiontoanInequalityinTwoVariablesSection7.8 Asolutiontoaninequalityintwovariablesisapairofvaluesthatproduceatruestatementwhensubstituted intotheinequality. LocationofSolutionstoInequalitiesinTwoVariablesSection7.8 Allsolutionstoalinearinequalityintwovariablesarelocatedinone,andonlyone,half-plane. 7.10ExerciseSupplement 10 7.10.1ExerciseSupplement 7.10.1.1GraphingLinearEquationsandInequalitiesinOneVariableSection7.2 Forthefollowingproblems,graphtheequationsandinequalities. Exercise7.280 Solutiononp.534. 6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(18=6 Exercise7.281 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3= )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 10 Thiscontentisavailableonlineat.

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499 Exercise7.282 Solutiononp.535. 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1=2 Exercise7.283 10 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 < 4 Exercise7.284 Solutiononp.535. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y +1 5 Exercise7.285 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 a 12 2 Exercise7.286 Solutiononp.535. 3 x +4 12 Exercise7.287 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F8 9.9626 Tf 18.264 0 Td [(11 Exercise7.288 Solutiononp.535. 0 < )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y +9 9 Exercise7.289 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 c 2 +1=7 7.10.1.2PlottingPointsinthePlaneSection7.3 Exercise7.290 Solutiononp.535. Drawacoordinatesystemandplotthefollowingorderedpairs. ; 1 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; ; 3 ; ; 0 ; )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(5 ; )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(2 3 Exercise7.291 Asaccuratelyaspossible,statethecoordinatesofthepointsthathavebeenplottedonthegraph.

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500 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES 7.10.1.3GraphingLinearEquationsinTwoVariablesSection7.4 Exercise7.292 Solutiononp.535. Whatisthegeometricstructureofthegraphofallthesolutionstothelinearequation y =4 x )]TJ/F8 9.9626 Tf 9.51 0 Td [(9 ? 7.10.1.4GraphingLinearEquationsinTwoVariablesSection7.4-GraphingEquationsin Slope-InterceptFormSection7.6 Forthefollowingproblems,graphtheequations. Exercise7.293 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(x =2 Exercise7.294 Solutiononp.535. y + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=0 Exercise7.295 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +3 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise7.296 Solutiononp.536. 2 y + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(8=0 Exercise7.297 4 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y =12 Exercise7.298 Solutiononp.536. 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +12=0 Exercise7.299 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise7.300 Solutiononp.536. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=0 Exercise7.301 x =4 Exercise7.302 Solutiononp.537. x +1=0 Exercise7.303 x =0 Exercise7.304 Solutiononp.537. y =0 7.10.1.5TheSlope-InterceptFormofaLineSection7.5 Exercise7.305 Writetheslope-interceptformofastraightline. Exercise7.306 Solutiononp.537. Theslopeofastraightlineisa ofthesteepnessoftheline. Exercise7.307 Writetheformulafortheslopeofalinethatpassesthroughthepoints x 1 ;y 1 and x 2 ;y 2 Forthefollowingproblems,determinetheslopeand y -interceptofthelines.

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501 Exercise7.308 Solutiononp.537. y =4 x +10 Exercise7.309 y =3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 Exercise7.310 Solutiononp.537. y =9 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise7.311 y = )]TJ/F11 9.9626 Tf 7.748 0 Td [(x +2 Exercise7.312 Solutiononp.537. y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise7.313 y = x Exercise7.314 Solutiononp.538. y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 x Exercise7.315 3 y =4 x +9 Exercise7.316 Solutiononp.538. 4 y =5 x +1 Exercise7.317 2 y =9 x Exercise7.318 Solutiononp.538. 5 y +4 x =6 Exercise7.319 7 y +3 x =10 Exercise7.320 Solutiononp.538. 6 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x =24 Exercise7.321 5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(15=0 Exercise7.322 Solutiononp.538. 3 y +3 x =1 Exercise7.323 7 y +2 x =0 Exercise7.324 Solutiononp.538. y =4 Forthefollowingproblems,ndtheslope,ifitexists,ofthelinethroughthegivenpairsofpoints. Exercise7.325 ; 2 ; ; 3 Exercise7.326 Solutiononp.538. ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise7.327 ; 5 ; ; 4 Exercise7.328 Solutiononp.538. ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; ; 3 Exercise7.329 ; 0 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(8 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(5

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502 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.330 Solutiononp.538. )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; 1 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 7 Exercise7.331 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise7.332 Solutiononp.538. ; 7 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Exercise7.333 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 1 ; ; 1 Exercise7.334 Solutiononp.538. )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 3 ; 3 4 ; )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(2 9 ; )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(5 6 Exercise7.335 Movinglefttoright,lineswith sloperisewhilelineswith slopedecline. Exercise7.336 Solutiononp.538. Comparetheslopesofparallellines. 7.10.1.6FindingtheEquationofaLineSection7.7 Forthefollowingproblems,writetheequationofthelineusingthegiveninformation.Writetheequation inslope-interceptform. Exercise7.337 Slope= 4 ;y -intercept= 5 Exercise7.338 Solutiononp.538. Slope= 3 ;y -intercept= )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 Exercise7.339 Slope= 1 ;y -intercept= 8 Exercise7.340 Solutiononp.538. Slope= 1 ;y -intercept= )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise7.341 Slope= )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 ;y -intercept= 1 Exercise7.342 Solutiononp.538. Slope= )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 ;y -intercept= )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Exercise7.343 Slope= 2 ;y -intercept= 0 Exercise7.344 Solutiononp.538. Slope= )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ;y -intercept= 0 Exercise7.345 m =3 ; ; 1 Exercise7.346 Solutiononp.538. m =2 ; ; 5 Exercise7.347 m =6 ; ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2

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503 Exercise7.348 Solutiononp.538. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise7.349 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 Exercise7.350 Solutiononp.538. m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; ; 2 Exercise7.351 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; ; 0 Exercise7.352 Solutiononp.539. ; 3 ; ; 5 Exercise7.353 ; 4 ; ; 1 Exercise7.354 Solutiononp.539. ; 1 ; ; 3 Exercise7.355 ; 6 ; ; 2 Exercise7.356 Solutiononp.539. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 1 ; ; 3 Exercise7.357 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Exercise7.358 Solutiononp.539. ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Exercise7.359 ; 1 ; ; 1 Exercise7.360 Solutiononp.539. )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; 7 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 7 Exercise7.361 ; 1 ; ; 3 Exercise7.362 Solutiononp.539. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 5 Exercise7.363 ; 4 ; ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Exercise7.364 Solutiononp.539. ; 2 ; ; 0 Forthefollowingproblems,readingonlyfromthegraph,determinetheequationoftheline.

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504 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.365 Exercise7.366 Solutiononp.539. Exercise7.367

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505 Exercise7.368 Solutiononp.539. Exercise7.369 Exercise7.370 Solutiononp.539. 7.10.1.7GraphingLinearInequalitiesinTwoVariablesSection7.8 Forthefollowingproblems,graphtheinequalities. Exercise7.371 y x +2

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506 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.372 Solutiononp.539. y< )]TJ/F7 6.9738 Tf 11.159 3.922 Td [(1 2 x +3 Exercise7.373 y> 1 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 Exercise7.374 Solutiononp.539. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +3 y )]TJ/F8 9.9626 Tf 18.265 0 Td [(6

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507 Exercise7.375 2 x +5 y 20 Exercise7.376 Solutiononp.539. 4 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y +12 > 0 Exercise7.377 y )]TJ/F8 9.9626 Tf 18.264 0 Td [(2

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508 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.378 Solutiononp.540. x< 3 Exercise7.379 y 0 7.11ProciencyExam 11 7.11.1ProciencyExam Forthefollowingproblems,constructacoordinatesystemandgraphtheinequality. 11 Thiscontentisavailableonlineat.

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509 Exercise7.380 Solutiononp.540. Section7.2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x +4 > )]TJ/F8 9.9626 Tf 9.963 0 Td [(14 Exercise7.381 Solutiononp.540. Section7.2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8
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510 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.388 Solutiononp.541. Section7.5 Considerthegraphof y = 2 7 x +16 .Ifweweretoplaceourpencilatanypointon thelineandthenmoveithorizontally 7 unitstotheright,howmanyunitsandinwhatdirection wouldwehavetomoveourpenciltogetbackontheline? Forthefollowingtwoproblems,ndtheslope,ifitexists,ofthelinecontainingthefollowingpoints. Exercise7.389 Solutiononp.541. Section7.5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 and ; 8 Exercise7.390 Solutiononp.541. Section7.5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 10 Exercise7.391 Solutiononp.541. Section7.5 Determinetheslopeand y )]TJ/F15 9.9626 Tf 9.962 0 Td [(interceptoftheline 3 y +2 x +1=0 Exercise7.392 Solutiononp.541. Section7.5 Aswelookatagraphlefttoright,dolineswithapositivesloperiseordecline? Forthefollowingproblems,ndtheequationofthelineusingtheinformationprovided.Writetheequation inslope-interceptform. Exercise7.393 Solutiononp.541. Section7.7 Slope=4, y -intercept = )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise7.394 Solutiononp.541. Section7.7 Slope= )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(3 2 y -intercept = 4 3 Exercise7.395 Solutiononp.541. Section7.7 slope= 2 3 ; passesthrough )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 2 Exercise7.396 Solutiononp.541. Section7.7 slope=7,passesthrough,0. Exercise7.397 Solutiononp.541. Section7.7 passesthroughthepoints ; 2 and ; 1 Forthefollowingproblems,graphtheequationofinequality. Exercise7.398 Solutiononp.541. Section7.4-Section7.6 y = 1 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2

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511 Exercise7.399 Solutiononp.541. Section7.4-Section7.6 5 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +15=0 Exercise7.400 Solutiononp.542. Section7.4-Section7.6 4 x + y =8 Exercise7.401 Solutiononp.542. Section7.4Section7.6 3 2 y +2=0 Exercise7.402 Solutiononp.542. Section7.4Section7.6 x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2

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512 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES Exercise7.403 Solutiononp.543. Section7.9 2 x +3 y> 6 Exercise7.404 Solutiononp.543. Section7.7 Readingonlyfromthegraph,determinetheequationoftheline.

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513 SolutionstoExercisesinChapter7 SolutiontoExercise7.1p.415 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise7.2p.416 x 6 SolutiontoExercise7.3p.416 m> )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise7.4p.416 2 x< 10 SolutiontoExercise7.5p.416 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(80 SolutiontoExercise7.6p.416 x =3 SolutiontoExercise7.8p.417 x = 1 2 SolutiontoExercise7.10p.417 y =1 SolutiontoExercise7.12p.417 z = 1 3

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514 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.14p.417 r = 1 28 SolutiontoExercise7.16p.417 x 5 SolutiontoExercise7.18p.418 x> )]TJ/F8 9.9626 Tf 9.963 0 Td [(17 SolutiontoExercise7.20p.418 m 5 SolutiontoExercise7.22p.418 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(5 SolutiontoExercise7.24p.418 y 21 SolutiontoExercise7.26p.418 y )]TJ/F7 6.9738 Tf 19.461 3.923 Td [(32 5 SolutiontoExercise7.28p.419 2 x< 3 SolutiontoExercise7.30p.419 3 x< 5

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515 SolutiontoExercise7.32p.419 5 x 2 SolutiontoExercise7.34p.419 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 18 SolutiontoExercise7.36p.422 Noticethatthedottedlinesonthegraphareonlyforillustrationandshouldnotbeincludedwhenplotting points. SolutiontoExercise7.37p.422 SolutiontoExercise7.39p.423 Coordinatesoftheoriginare ; 0 SolutiontoExercise7.41p.424 I II III IV x> 0 x< 0 x< 0 x> 0 y> 0 y> 0 y< 0 y< 0 Table7.17

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516 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.43p.425 Yes,theredoesappeartobearelation. SolutiontoExercise7.45p.426 Yes,theredoesappeartobearelation. SolutiontoExercise7.47p.427 equation SolutiontoExercise7.49p.427 contradiction SolutiontoExercise7.51p.432 When x =0 y =3 ;when y =0 x =1

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517 SolutiontoExercise7.52p.434 SolutiontoExercise7.53p.434 SolutiontoExercise7.54p.435

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518 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.55p.437 SolutiontoExercise7.56p.437 SolutiontoExercise7.57p.438

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519 SolutiontoExercise7.59p.438 SolutiontoExercise7.61p.439 SolutiontoExercise7.63p.440

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520 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.65p.440 SolutiontoExercise7.67p.441 SolutiontoExercise7.69p.442

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521 SolutiontoExercise7.71p.442 SolutiontoExercise7.73p.443 SolutiontoExercise7.75p.444 x = 3 2 SolutiontoExercise7.77p.444 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2

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522 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.79p.445 SolutiontoExercise7.81p.446

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523 SolutiontoExercise7.83p.446 SolutiontoExercise7.85p.447 SolutiontoExercise7.87p.448 commutativepropertyofaddition SolutiontoExercise7.89p.448 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(19 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(14 SolutiontoExercise7.91p.448 quadrants SolutiontoExercise7.92p.450 m =2 ;b =7 SolutiontoExercise7.93p.450 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ;b =2 SolutiontoExercise7.94p.450 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ;b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise7.95p.450 m = 2 3 ;b = )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 SolutiontoExercise7.96p.450 m = )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 8 ;b = 1 2 SolutiontoExercise7.97p.450 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ;b =0 SolutiontoExercise7.98p.453 Thelinecrossesthe y -axisat +4 .Aftermovinghorizontally1unittotheright,wemustmoveexactly3 unitsdownward.

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524 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.99p.453 SolutiontoExercise7.102p.455 Solvingfor y weget y = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 5 x +3 : Now, m = )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 5 and b =3 : SolutiontoExercise7.103p.460 m = 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 = 2 4 = 1 2 : SolutiontoExercise7.104p.460 Thelinehasslope 1 2 SolutiontoExercise7.105p.460 Thelinesappeartobeparallel.Parallellineshavethesameslope,andlinesthathavethesameslopeare parallel. SolutiontoExercise7.114p.461 slope =3 ; y -intercept = ; 4 SolutiontoExercise7.116p.461 slope =9 ; y -intercept = ; 1 SolutiontoExercise7.118p.461 slope = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; y -intercept = ; 5 SolutiontoExercise7.120p.461 slope = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; y -intercept = ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise7.122p.461 slope = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; y -intercept = ; 2 SolutiontoExercise7.124p.461 slope =4 ; y -intercept = ; 5 SolutiontoExercise7.126p.461 slope = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ; y -intercept = ; 9

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525 SolutiontoExercise7.128p.462 slope = 2 7 ; y -intercept = ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise7.130p.462 slope = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(4 5 ; y -intercept = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(4 7 SolutiontoExercise7.132p.462 slope = 6 5 ; y -intercept = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; )]TJ/F7 6.9738 Tf 10.93 3.923 Td [(1 10 SolutiontoExercise7.134p.462 slope = 1; y -intercept = ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise7.136p.462 slope = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(5 3 ; y -intercept = ; 2 SolutiontoExercise7.138p.462 slope = 1 4 ; y -intercept = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(1 4 SolutiontoExercise7.140p.462 m = 4 3 SolutiontoExercise7.142p.462 m = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(7 4 SolutiontoExercise7.144p.462 m =2 SolutiontoExercise7.146p.462 m = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(7 6 SolutiontoExercise7.148p.462 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise7.150p.463 m =1 SolutiontoExercise7.152p.463 m =2 SolutiontoExercise7.154p.463 m =2 SolutiontoExercise7.156p.463 m =0 horizontalline y =3 SolutiontoExercise7.158p.463 Noslope verticallineat x =8 SolutiontoExercise7.160p.463 m =0 horizontallineat y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 SolutiontoExercise7.162p.463 decline SolutiontoExercise7.164p.463 slope = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 31 y )]TJ/F15 9.9626 Tf 9.963 0 Td [(intercept = ; 0 : 35 SolutiontoExercise7.166p.463 slope =0 : 64 y )]TJ/F15 9.9626 Tf 9.963 0 Td [(intercept = ; 2 : 71 SolutiontoExercise7.168p.463 m =1 : 31 SolutiontoExercise7.170p.464 m =0 horizontallineat y =227 : 61 SolutiontoExercise7.172p.464 Noslope verticalline x =88 : 81 SolutiontoExercise7.174p.464 1 if xyw 6 =0

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526 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.176p.464 10 SolutiontoExercise7.178p.464 SolutiontoExercise7.179p.469 SolutiontoExercise7.180p.469

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527 SolutiontoExercise7.181p.469 SolutiontoExercise7.183p.470 SolutiontoExercise7.185p.471

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528 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.187p.471 SolutiontoExercise7.189p.472 SolutiontoExercise7.191p.473

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529 SolutiontoExercise7.193p.473 SolutiontoExercise7.195p.474 SolutiontoExercise7.197p.475 SolutiontoExercise7.199p.475 x 1

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530 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.201p.476 SolutiontoExercise7.203p.476 m = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 3 SolutiontoExercise7.204p.479 y =5 x +8 SolutiontoExercise7.205p.479 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 x +3 SolutiontoExercise7.206p.479 y =2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 SolutiontoExercise7.207p.479 y = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise7.208p.479 y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 SolutiontoExercise7.209p.479 y =4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 SolutiontoExercise7.210p.479 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 SolutiontoExercise7.211p.479 y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 SolutiontoExercise7.212p.479 y =2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 SolutiontoExercise7.213p.479 y =3 x +20 SolutiontoExercise7.214p.480 y =9 SolutiontoExercise7.215p.480 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise7.216p.480 y = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 3 x +4 SolutiontoExercise7.217p.481 y =3 x +4 SolutiontoExercise7.219p.481 y =8 x +1 SolutiontoExercise7.221p.481 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise7.223p.481 y = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(3 2 x SolutiontoExercise7.225p.481 y = x +5

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531 SolutiontoExercise7.227p.481 y =8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(32 SolutiontoExercise7.229p.481 y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +6 SolutiontoExercise7.231p.481 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +1 SolutiontoExercise7.233p.482 y = 8 5 x SolutiontoExercise7.235p.482 y = x +3 SolutiontoExercise7.237p.482 y =3 horizontalline SolutiontoExercise7.239p.482 x =4 verticalline SolutiontoExercise7.241p.482 y = x SolutiontoExercise7.243p.482 y = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(9 5 x + 2 5 SolutiontoExercise7.245p.482 y = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise7.247p.482 y = 2 5 x +1 SolutiontoExercise7.249p.483 y = 1 4 x +1 SolutiontoExercise7.251p.484 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise7.253p.484 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise7.255p.485 y -intercept SolutiontoExercise7.257p.485 m =1 SolutiontoExercise7.259p.490

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532 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.260p.491 SolutiontoExercise7.261p.491 SolutiontoExercise7.262p.491

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533 SolutiontoExercise7.263p.492 SolutiontoExercise7.265p.492 SolutiontoExercise7.267p.493

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534 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.269p.494 SolutiontoExercise7.271p.494 SolutiontoExercise7.273p.495 SolutiontoExercise7.275p.496 SolutiontoExercise7.277p.496 m = y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 1 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 1 SolutiontoExercise7.279p.496 y =4 x +6 SolutiontoExercise7.280p.498 x =4

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535 SolutiontoExercise7.282p.499 x = 3 5 SolutiontoExercise7.284p.499 y )]TJ/F8 9.9626 Tf 18.265 0 Td [(2 SolutiontoExercise7.286p.499 x 8 3 SolutiontoExercise7.288p.499 0 y< 3 SolutiontoExercise7.290p.499 SolutiontoExercise7.292p.500 astraightline

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536 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.294p.500 SolutiontoExercise7.296p.500 SolutiontoExercise7.298p.500

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537 SolutiontoExercise7.300p.500 SolutiontoExercise7.302p.500 SolutiontoExercise7.304p.500 SolutiontoExercise7.306p.500 measure SolutiontoExercise7.308p.500 slope: 4 y -intercept: ; 10 SolutiontoExercise7.310p.501 slope: 9 y -intercept: ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1

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538 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.312p.501 slope: )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 y -intercept: ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 SolutiontoExercise7.314p.501 slope: )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 y -intercept: ; 0 SolutiontoExercise7.316p.501 slope: 5 4 y -intercept: )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 1 4 SolutiontoExercise7.318p.501 slope: )]TJ/F7 6.9738 Tf 12.851 3.922 Td [(4 5 y -intercept: )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(0 ; 6 5 SolutiontoExercise7.320p.501 slope: 2 y -intercept: ; 4 SolutiontoExercise7.322p.501 slope: )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 y -intercept: )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(0 ; 1 3 SolutiontoExercise7.324p.501 slope: 0 y -intercept: ; 4 SolutiontoExercise7.326p.501 slope: )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise7.328p.501 slope: 7 2 SolutiontoExercise7.330p.502 slope: 3 2 SolutiontoExercise7.332p.502 NoSlope SolutiontoExercise7.334p.502 slope: 57 4 SolutiontoExercise7.336p.502 Theslopesofparallellinesareequal. SolutiontoExercise7.338p.502 y =3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 SolutiontoExercise7.340p.502 y = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise7.342p.502 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise7.344p.502 y = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x SolutiontoExercise7.346p.502 y =2 x +3 SolutiontoExercise7.348p.502 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 x +7

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539 SolutiontoExercise7.350p.503 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +2 SolutiontoExercise7.352p.503 y =2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise7.354p.503 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +13 SolutiontoExercise7.356p.503 y = 2 5 x + 11 5 SolutiontoExercise7.358p.503 y = 2 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 SolutiontoExercise7.360p.503 y =7 zeroslope SolutiontoExercise7.362p.503 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 noslope SolutiontoExercise7.364p.503 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +2 SolutiontoExercise7.366p.504 y = 2 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise7.368p.505 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise7.370p.505 y =1 SolutiontoExercise7.372p.506 SolutiontoExercise7.374p.506

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540 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.376p.507 SolutiontoExercise7.378p.508 SolutiontoExercise7.380p.509 x< 3 SolutiontoExercise7.381p.509 )]TJ/F8 9.9626 Tf 7.749 0 Td [(14
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541 SolutiontoExercise7.384p.509 straightline SolutiontoExercise7.385p.509 generalform SolutiontoExercise7.386p.509 slope-intercept SolutiontoExercise7.387p.509 Itliesontheline. SolutiontoExercise7.388p.510 2unitsup SolutiontoExercise7.389p.510 3 2 SolutiontoExercise7.390p.510 noslope;verticallineat x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise7.391p.510 slope = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(2 3 ;y -interceptis )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(0 ; )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(1 3 SolutiontoExercise7.392p.510 rise SolutiontoExercise7.393p.510 y =4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise7.394p.510 y = )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(3 2 x + 4 3 SolutiontoExercise7.395p.510 y = 2 3 x + 8 3 SolutiontoExercise7.396p.510 y =7 x SolutiontoExercise7.397p.510 y = 1 3 x + 1 3 SolutiontoExercise7.398p.510 y = 1 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2

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542 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES SolutiontoExercise7.399p.511 SolutiontoExercise7.400p.511 4 x + y =8 SolutiontoExercise7.401p.511 SolutiontoExercise7.402p.511 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2

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543 SolutiontoExercise7.403p.512 SolutiontoExercise7.404p.512 y = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(1 3 x +3

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544 CHAPTER7.GRAPHINGLINEAREQUATIONSANDINEQUALITIESIN ONEANDTWOVARIABLES

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Chapter8 RationalExpressions 8.1Objectives 1 Aftercompletingthischapter,youshould RationalExpressionsSection8.2 beabletorecognizearationalexpression befamilarwiththeequalityandnegativepropertiesoffractions ReducingRationalExpressionsSection8.3 understandandbeabletousetheprocessofreducingrationalexpressions MultiplyingandDividingRationalExpressionsSection8.4 beabletomultiplyanddividerationalexpressions BuildingRationalExpressionsandtheLCDSection8.5 understandandbeabletousetheprocessofbuildingrationalexpressionsandknowwhyitisoften necessarytobuildthem beabletondtheLCDofoneormoreexpressions AddingandSubtractingRationalExpressionsSection8.6 befamiliarwiththebasicruleforaddingandsubtractingrationalexpressions beabletoaddandsubtractfractionswiththesameandwithdierentdenominators RationalEquationsSection8.7 beabletoidentifyrationalequations understandandbeabletousethemethodofsolvingrationalexpressions beabletorecognizeextraneoussolutions ApplicationsSection8.8 beabletousetheve-stepmethodtosolvevariousappliedproblems ComplexRationalExpressionsSection8.9 1 Thiscontentisavailableonlineat. 545

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546 CHAPTER8.RATIONALEXPRESSIONS beabletodistinguishbetweensimpleandcomplexfractions beabletosimplifycomplexfractionsusingthecombine-divideandtheLCD-multiply-dividemethod DividingPolynomialsSection8.10 beabletodivideapolynomialbyamonomial understandtheprocessandbeabletodivideapolynomialbyapolynomial 8.2RationalExpressions 2 8.2.1Overview RationalExpressions Zero-FactorProperty TheEqualityPropertyofFractions TheNegativePropertyofFractions 8.2.2RationalExpressions Inarithmeticitisnotedthatafractionisaquotientoftwowholenumbers.Theexpression a b ,where a and b areanytwowholenumbersand b 6 =0 ,iscalledafraction.Thetopnumber, a ,iscalledthenumerator, andthebottomnumber, b ,iscalledthedenominator. SimpleAlgebraicFraction Wedeneasimplealgebraicfractioninasimilarmanner.Ratherthanrestrictingourselvesonlytonumbers, weusepolynomialsforthenumeratoranddenominator.Anothertermforasimplealgebraicfractionisa rationalexpression .Arationalexpressionisanexpressionoftheform P Q ,where P and Q areboth polynomialsand Q neverrepresentsthezeropolynomial. RationalExpression A rationalexpression isanalgebraicexpressionthatcanbewrittenasthequotientoftwopolynomials. Examples1arerationalexpressions: Example8.1 x +9 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 isarationalexpression: P is x +9 and Q is x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Example8.2 x 3 +5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x +1 x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 isarationalexpression: P is x 3 +5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x +1 and Q is x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 Example8.3 3 8 isarationalexpression: P is3and Q is8. Example8.4 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 isarationalexpressionsince 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 canbewrittenas 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 1 : P is 4 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 and Q is1. Example8.5 p 5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 is not arationalexpressionsince p 5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 isnotapolynomial. Intherationalexpression P Q P iscalledthenumeratorand Q iscalledthedenominator. DomainofaRationalExpression Sincedivisionbyzeroisnotdened,wemustbecarefultonotethevaluesforwhichtherationalexpression isvalid.Thecollectionofvaluesforwhichtherationalexpressionisdenediscalledthe domain ofthe rationalexpression.RecallourstudyofthedomainofanequationinSectionSection4.8. 2 Thiscontentisavailableonlineat.

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547 FindingtheDomainofaRationalExpression Tondthedomainofarationalexpressionwemustask,"Whatvalues,ifany,ofthevariablewillmake thedenominatorzero?"Tondthesevalues,wesetthedenominatorequaltozeroandsolve.Ifanyzeroproducingvaluesareobtained,theyarenotincludedinthedomain.Allotherrealnumbersareincludedin thedomainunlesssomehavebeenexcludedforparticularsituationalreasons. 8.2.3Zero-FactorProperty Sometimestondthedomainofarationalexpression,itisnecessarytofactorthedenominatorandusethe zero-factorproperty ofrealnumbers. Zero-factorProperty Iftworealnumbers a and b aremultipliedtogetherandtheresultingproductis0,thenatleastoneofthe factorsmustbezero,thatis,either a =0 b =0 ,orboth a =0 and b =0 Thefollowingexamplesillustratetheuseofthezero-factorproperty. Example8.6 Whatvaluewillproducezerointheexpression 4 x ?Bythezero-factorproperty,if 4 x =0 ,then x =0 Example8.7 Whatvaluewillproducezerointheexpression 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 ?Bythezero-factorproperty,if 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=0 ,then x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=0 x =6 Thus, 8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=0 when x =6 Example8.8 Whatvalueswillproducezerointheexpression x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +5 ?Bythezero-factorproperty,if x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +5=0 ,then x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3=0 or x +5=0 x =3 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Thus, x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +5=0 when x =3 or x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Example8.9 Whatvalueswillproducezerointheexpression x 2 +6 x +8 ?Wemustfactor x 2 +6 x +8 toput itintothezero-factorpropertyform. x 2 +6 x +8= x +2 x +4 Now, x +2 x +4=0 when x +2=0 or x +4=0 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Thus, x 2 +6 x +8=0 when x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 or x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 .

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548 CHAPTER8.RATIONALEXPRESSIONS Example8.10 Whatvalueswillproducezerointheexpression 6 x 2 )]TJ/F8 9.9626 Tf 9.701 0 Td [(19 x )]TJ/F8 9.9626 Tf 9.7 0 Td [(7 ?Wemustfactor 6 x 2 )]TJ/F8 9.9626 Tf 9.7 0 Td [(19 x )]TJ/F8 9.9626 Tf 9.701 0 Td [(7 to putitintothezero-factorpropertyform. 6 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(19 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7= x +1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 Now, x +1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7=0 when 3 x +1=0 or 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=0 3 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 x =7 x = )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 x = 7 2 Thus, 6 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(19 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7=0 when x = )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 or 7 2 8.2.4SampleSetA Findthedomainofthefollowingexpressions. Example8.11 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Thedomainisthecollectionofallrealnumbersexcept1.Oneisnotincluded,forif x =1 divisionbyzeroresults. Example8.12 3 a 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Ifweset 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 equaltozero,wendthat a =4 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(8=0 2 a =8 a =4 Thus4mustbeexcludedfromthedomainsinceitwillproducedivisionbyzero.Thedomainis thecollectionofallrealnumbersexcept4. Example8.13 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Setting x +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6=0 ,wendthat x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 and x =6 .Boththesevaluesproduce divisionbyzeroandmustbeexcludedfromthedomain.Thedomainisthecollectionofallreal numbersexceptand6. Example8.14 9 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F52 6.9738 Tf 6.226 0 Td [(15 Setting x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(15=0 ,weget

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549 x +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5=0 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 5 Thus, x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 and x =5 producedivisionbyzeroandmustbeexcludedfromthedomain.The domainisthecollectionofallrealnumbersexceptand5. Example8.15 2 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +10 Setting x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +10=0 ,weget x =0 ; 1 ; 3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 .Thesenumbersmustbeexcludedfromthedomain.Thedomainisthecollectionofallrealnumbersexcept0,1,3, Example8.16 8 b +7 b +1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Setting b +1 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(2=0 ,weget b = )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(1 2 2 3 .Thedomainisthecollectionofallreal numbersexcept )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 2 and 2 3 Example8.17 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 +1 Novalueof x isexcludedsinceforanychoiceof x ,thedenominatorisneverzero.The domainisthecollectionofallrealnumbers. Example8.18 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 6 Novalueof x isexcludedsinceforanychoiceof x ,thedenominatorisneverzero.The domainisthecollectionofallrealnumbers. 8.2.5PracticeSetA Findthedomainofeachofthefollowingrationalexpressions. Exercise8.1 Solutiononp.638. 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 Exercise8.2 Solutiononp.638. 5 x x x +4 Exercise8.3 Solutiononp.638. 2 x +1 x +2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x Exercise8.4 Solutiononp.638. 5 a +2 a 2 +6 a +8 Exercise8.5 Solutiononp.638. 12 y 3 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 Exercise8.6 Solutiononp.638. 2 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 m 2 +3 Exercise8.7 Solutiononp.638. k 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 5

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550 CHAPTER8.RATIONALEXPRESSIONS 8.2.6TheEqualityPropertyofFractions Fromourexperiencewitharithmeticwemayrecalltheequalitypropertyoffractions.Let a b c d bereal numberssuchthat b 6 =0 and d 6 =0 EqualityPropertyofFractions 1.If a b = c d ,then ad = bc 2.If ad = bc ,then a b = c d Twofractionsareequalwhentheircross-productsareequal. Weseethispropertyinthefollowingexamples: Example8.19 2 3 = 8 12 ,since 2 12 =3 8 Example8.20 5 y 2 = 15 y 2 6 y ,since 5 y 6 y =2 15 y 2 and 30 y 2 =30 y 2 Example8.21 Since 9 a 4= 18 a 2 9 a 18 a = 2 4 8.2.7TheNegativePropertyofFractions Ausefulpropertyoffractionsisthe negativepropertyoffractions NegativePropertyofFractions Thenegativesignofafractionmaybeplaced 1.infrontofthefraction, )]TJ/F10 6.9738 Tf 8.944 3.923 Td [(a b 2.inthenumeratorofthefraction, )]TJ/F10 6.9738 Tf 6.227 0 Td [(a b 3.inthedenominatorofthefraction, a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b Allthreefractionswillhavethesamevalue,thatis, )]TJ/F10 6.9738 Tf 8.944 3.923 Td [(a b = )]TJ/F10 6.9738 Tf 6.226 0 Td [(a b = a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b Thenegativepropertyoffractionsisillustratedbythefractions )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(3 4 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 4 = 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Toseethis,consider )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(3 4 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 4 .Isthiscorrect? Bytheequalitypropertyoffractions, )]TJ/F8 9.9626 Tf 9.409 0 Td [( 4= )]TJ/F15 9.9626 Tf 7.749 0 Td [(12and )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 4= )]TJ/F15 9.9626 Tf 7.748 0 Td [(12.Thus, )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(3 4 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 4 .Convinceyourselfthattheothertwofractionsareequalaswell. Thissamepropertyholdsforrationalexpressionsandnegativesigns.Thispropertyisoftenquitehelpful insimplifyingarationalexpressionasweshallneedtodoinsubsequentsections. Ifeitherthenumeratorordenominatorofafractionorafractionitselfisimmediatelyprecededbya negativesign,itisusuallymostconvenienttoplacethenegativesigninthenumeratorforlateroperations.

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551 8.2.8SampleSetB Example8.22 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 isbestwrittenas )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 4 Example8.23 )]TJ/F10 6.9738 Tf 8.945 4.444 Td [(y 9 isbestwrittenas )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 9 Example8.24 )]TJ/F10 6.9738 Tf 10.93 3.922 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 couldbewrittenas )]TJ/F7 6.9738 Tf 6.227 0 Td [( x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 ,whichwouldthenyield )]TJ/F10 6.9738 Tf 6.226 0 Td [(x +4 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Example8.25 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x : Factorout )]TJ/F15 9.9626 Tf 9.962 0 Td [(1fromthedenominator. )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(+ x Anegativedividedbyanegativeisapositive. 5 10+ x Example8.26 )]TJ/F7 6.9738 Tf 14.317 3.922 Td [(3 7 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x : Rewritethis. )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 7 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x Factorout )]TJ/F15 9.9626 Tf 9.962 0 Td [(1fromthedeno min ator. )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 6.226 0 Td [( )]TJ/F7 6.9738 Tf 6.226 0 Td [(7+ x Anegativedividedbyanegativeispositive : 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7+ x Rewrite : 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 Thisexpressionseemslesscumbersomethandoestheoriginalfewerminussigns. 8.2.9PracticeSetB Fillinthemissingterm. Exercise8.8 Solutiononp.638. )]TJ/F7 6.9738 Tf 14.203 3.923 Td [(5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.9 Solutiononp.638. )]TJ/F10 6.9738 Tf 12.058 3.923 Td [(a +2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a +3 = a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.10 Solutiononp.638. )]TJ/F7 6.9738 Tf 14.203 3.922 Td [(8 5 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y = y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 8.2.10Exercises Forthefollowingproblems,ndthedomainofeachoftherationalexpressions. Exercise8.11 Solutiononp.638. 6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.12 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 Exercise8.13 Solutiononp.638. )]TJ/F52 6.9738 Tf 6.227 0 Td [(11 x x +1

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552 CHAPTER8.RATIONALEXPRESSIONS Exercise8.14 x + 10 x +4 Exercise8.15 Solutiononp.638. x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.16 x +7 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 Exercise8.17 Solutiononp.638. )]TJ/F10 6.9738 Tf 6.226 0 Td [(x +4 x 2 )]TJ/F52 6.9738 Tf 6.227 0 Td [(36 Exercise8.18 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a +5 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.19 Solutiononp.638. 2 b b b +6 Exercise8.20 3 b +1 b b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b +5 Exercise8.21 Solutiononp.638. 3 x +4 x x )]TJ/F52 6.9738 Tf 6.226 0 Td [(10 x +1 Exercise8.22 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x Exercise8.23 Solutiononp.638. 6 a a 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a Exercise8.24 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 a 2 +6 a +8 Exercise8.25 Solutiononp.638. )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 b +3 Exercise8.26 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x +2 Exercise8.27 Solutiononp.638. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y )]TJ/F52 6.9738 Tf 6.227 0 Td [(20 Exercise8.28 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 2 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.29 Solutiononp.638. 2 x +7 6 x 3 + x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x Exercise8.30 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +4 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 x 2 +12 x Forthefollowingproblems,showthatthefractionsareequivalent. Exercise8.31 Solutiononp.638. )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 5 and )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(3 5 Exercise8.32 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 7 and )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(2 7 Exercise8.33 Solutiononp.638. )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(1 4 and )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 Exercise8.34 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 3 and )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(2 3 Exercise8.35 Solutiononp.638. )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 10 and 9 )]TJ/F52 6.9738 Tf 6.227 0 Td [(10

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553 Forthefollowingproblems,llinthemissingterm. Exercise8.36 )]TJ/F7 6.9738 Tf 14.317 3.923 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.37 Solutiononp.638. )]TJ/F7 6.9738 Tf 14.261 3.923 Td [(2 x +7 = x +7 Exercise8.38 )]TJ/F7 6.9738 Tf 9 3.922 Td [(3 x +4 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.39 Solutiononp.638. )]TJ/F7 6.9738 Tf 9 3.922 Td [(2 x +7 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.40 )]TJ/F10 6.9738 Tf 10.93 3.922 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.41 Solutiononp.638. )]TJ/F10 6.9738 Tf 10.93 3.923 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 = 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.42 )]TJ/F10 6.9738 Tf 12.113 3.923 Td [(x +5 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = x +3 Exercise8.43 Solutiononp.639. )]TJ/F10 6.9738 Tf 12.113 3.923 Td [(a +1 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 = a +6 Exercise8.44 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +2 = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.45 Solutiononp.639. y + 10 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = y +6 8.2.11ExercisesForReview Exercise8.46 Section3.7 Write 15 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 y 4 5 x 2 y )]TJ/F6 4.9813 Tf 5.397 0 Td [(7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 sothatonlypositiveexponentsappear. Exercise8.47 Solutiononp.639. Section5.7 Solvethecompoundinequality 1 6 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 < 13 Exercise8.48 Section6.8 Factor 8 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(18 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 Exercise8.49 Solutiononp.639. Section6.8 Factor x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 x +36 Exercise8.50 Section7.2 Supplythemissingword.Thephrase"graphinganequation"isinterpretedas meaning"geometricallylocatethe toanequation." 8.3ReducingRationalExpressions 3 8.3.1Overview TheLogicBehindTheProcess TheProcess 3 Thiscontentisavailableonlineat.

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554 CHAPTER8.RATIONALEXPRESSIONS 8.3.2TheLogicBehindTheProcess Whenworkingwithrationalexpressions,itisoftenbesttowritetheminthesimplestpossibleform.For example,therationalexpression x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 x +8 canbereducedtothesimplerexpression x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 forall x except x =2 ; 4 FromourdiscussionofequalityoffractionsinSectionSection8.2,weknowthat a b = c d when ad = bc Thisfactallowsustodeducethat,if k 6 =0 ; ak bk = a b ; since akb = abk recallthecommutativepropertyof multiplication.Butthisfactmeansthatifafactorinthiscase, k iscommontoboththenumeratorand denominatorofafraction,wemayremoveitwithoutchangingthevalueofthefraction. ak bk = a k b k = a b 8.3.2.1Cancelling Theprocessofremovingcommonfactorsiscommonlycalled cancelling Example8.27 16 40 canbereducedto 2 5 .Process: 16 40 = 2 2 2 2 2 2 2 5 Removethethreefactorsof1; 2 2 2 2 2 2 : 2 5 = 2 5 Noticethatin 2 5 ,thereisnofactorcommontothenumeratoranddenominator. Example8.28 111 148 canbereducedto 3 4 .Process: 111 148 = 3 37 4 37 Removethefactorof1; 37 37 3 4 3 4 Noticethatin 3 4 ,thereisnofactorcommontothenumeratoranddenominator. Example8.29 3 9 canbereducedto 1 3 .Process: 3 9 = 3 1 3 3 Removethefactorof1; 3 3 1 3 = 1 3

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555 Noticethatin 1 3 thereisnofactorcommontothenumeratoranddenominator. Example8.30 5 7 cannotbereducedsincetherearenofactorscommontothenumeratoranddenominator. Problems1,2,and3shownabovecouldallbereduced.Theprocessineachreduction includedthefollowingsteps: 1.Boththenumeratoranddenominatorwerefactored. 2.Factorsthatwerecommontoboththenumeratoranddenominatorwerenotedandremoved bydividingthemout. Weknowthatwecandividebothsidesofanequationbythesamenonzeronumber,butwhyshouldwebe abletodivideboththenumeratoranddenominatorofafractionbythesamenonzeronumber?Thereason isthatanynonzeronumberdividedbyitselfis1,andthatifanumberismultipliedby1,itisleftunchanged. Considerthefraction 6 24 .Multiplythisfractionby1.Thisiswritten 6 24 1 .But1canberewrittenas 1 6 1 6 6 24 1 6 1 6 = 6 1 6 24 1 6 = 1 4 Theanswer, 1 4 ,isthereducedform.Noticethatin 1 4 thereisnofactorcommontoboththenumeratorand denominator.Thisreasoningprovidesjusticationforthefollowingrule. Cancelling Multiplyingordividingthenumeratoranddenominatorbythesamenonzeronumberdoesnotchangethe valueofafraction. 8.3.3TheProcess Wecannowstateaprocessforreducingarationalexpression. ReducingaRationalExpression 1.Factorthenumeratoranddenominatorcompletely. 2.Dividethenumeratoranddenominatorbyallfactorstheyhaveincommon,thatis,removeallfactors of1. ReducedtoLowestTerms 1.Arationalexpressionissaidtobe reducedtolowestterms whenthenumeratoranddenominator have no factorsincommon. 8.3.4SampleSetA Reducethefollowingrationalexpressions. Example8.31 15 x 20 x : Factor. 15 x 20 x = 5 3 x 5 2 2 x Thefactorsthatarecommontoboththenumeratorand denominatorare5and x .Divideeachby 5 x 3 x 2 2 x = 3 4 ;x 6 =0 Itishelpfultodrawalinethroughthedivided-outfactors :

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556 CHAPTER8.RATIONALEXPRESSIONS Example8.32 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +8 : Factor. x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Thefactorthatiscommontoboththenumerator anddenominatoris x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : Divideeachby x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 ;x 6 =2 ; 4 Theexpression x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 isthereducedformsincethereareno factors commontoboththe numeratoranddenominator.Althoughthereisan x inboth,itisa commonterm ,nota commonfactor ,andthereforecannotbedividedout. CAUTIONThisisacommonerror: x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = 2 4 is incorrect! Example8.33 a +2 b 6 a +12 b : Factor. a +2 b 6 a +2 b = a +2 b 6 a +2 b = 1 6 ;a 6 = )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b Since a +2 b isacommonfactortoboththenumeratoranddenominator,wedividebothby a +2 b Since a +2 b a +2 b =1 ,weget1inthenumerator. Sometimeswemayreducearationalexpressionbyusingthedivisionruleofexponents. Example8.34 8 x 2 y 5 4 xy 2 : Factorandusetherule a n a m = a n )]TJ/F10 6.9738 Tf 6.226 0 Td [(m : 8 x 2 y 5 4 xy 2 = 2 2 2 2 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 =2 xy 3 ;x 6 =0 ;y 6 =0 Example8.35 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 x 3 a x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(36 2 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 x : Factor. )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 x 3 a x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(36 2 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 x = )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 2 x 3 a x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 2 x x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 2 x 3 a x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 2 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +1 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 a x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +1 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 a x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;x 6 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; 6

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557 Example8.36 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 2 +2 x +8 : Sinceitismostconvenienttohavetheleadingtermsofa polynomialpositive,factorout )]TJ/F15 9.9626 Tf 9.963 0 Td [(1fromthedenominator. x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 )]TJ/F7 6.9738 Tf 6.227 0 Td [( x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Rewritethis. )]TJ/F10 6.9738 Tf 8.944 3.922 Td [(x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 Factor. )]TJETq1 0 0 1 144.422 599.516 cm[]0 d 0 J 0.339 w 0 0 m 24.056 0 l SQBT/F7 6.9738 Tf 144.422 593.1 Td [( x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +2 )]TJ/F10 6.9738 Tf 8.945 3.923 Td [(x +3 x +2 = )]TJ/F7 6.9738 Tf 6.227 0 Td [( x +3 x +2 = )]TJ/F10 6.9738 Tf 6.226 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 ;x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 4 Example8.37 a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b b )]TJ/F10 6.9738 Tf 6.227 0 Td [(a : Thenumeratoranddenominatorhavethesametermsbutthey occurwithoppositesigns.Factor )]TJ/F15 9.9626 Tf 9.963 0 Td [(1fromthedenominator. a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [(b + a = a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b )]TJ/F7 6.9738 Tf 6.226 0 Td [( a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b = )]TJETq1 0 0 1 241.715 498.993 cm[]0 d 0 J 0.339 w 0 0 m 17.165 0 l SQBT/F7 6.9738 Tf 241.715 492.577 Td [( a )]TJ/F10 6.9738 Tf 6.226 0 Td [(b a )]TJ/F10 6.9738 Tf 6.226 0 Td [(b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ;a 6 = b 8.3.5PracticeSetA Reduceeachofthefollowingfractionstolowestterms. Exercise8.51 Solutiononp.639. 30 y 35 y Exercise8.52 Solutiononp.639. x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x 2 +5 x +6 Exercise8.53 Solutiononp.639. x +2 b 4 x +8 b Exercise8.54 Solutiononp.639. 18 a 3 b 5 c 7 3 ab 3 c 5 Exercise8.55 Solutiononp.639. )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a 4 +75 a 2 2 a 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 a 2 +30 a Exercise8.56 Solutiononp.639. x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x +4 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 2 +12 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(32 Exercise8.57 Solutiononp.639. 2 x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x 8.3.6Excercises Forthefollowingproblems,reduceeachrationalexpressiontolowestterms. Exercise8.58 Solutiononp.639. 6 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Exercise8.59 8 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 Exercise8.60 Solutiononp.639. 9 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(21

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558 CHAPTER8.RATIONALEXPRESSIONS Exercise8.61 10 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise8.62 Solutiononp.639. 7 7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(14 Exercise8.63 6 6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 Exercise8.64 Solutiononp.639. 2 y 2 8 y Exercise8.65 4 x 3 2 x Exercise8.66 Solutiononp.639. 16 a 2 b 3 2 ab 2 Exercise8.67 20 a 4 b 4 4 ab 2 Exercise8.68 Solutiononp.639. x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +3 x +5 Exercise8.69 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y +6 Exercise8.70 Solutiononp.639. a +6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 a +2 Exercise8.71 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m +4 Exercise8.72 Solutiononp.639. y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.73 x +7 x +8 x +8 x +7 Exercise8.74 Solutiononp.639. )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 x 2 x +4 4 x Exercise8.75 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a +5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a +9 Exercise8.76 Solutiononp.639. 6 x 2 y 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 xy x +4 Exercise8.77 22 a 4 b 6 c 7 a +2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 4 c a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.78 Solutiononp.639. x +10 3 x +10 Exercise8.79 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 7 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.80 Solutiononp.639. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 2 x +6 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 x +6 Exercise8.81 a +1 5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 7 a +1 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4

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559 Exercise8.82 Solutiononp.639. y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 6 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 Exercise8.83 x +10 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +10 2 Exercise8.84 Solutiononp.639. a +6 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 6 a +6 5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 2 Exercise8.85 m +7 4 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 5 m +7 7 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 2 Exercise8.86 Solutiononp.640. a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.87 b +6 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 4 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.88 Solutiononp.640. 8 x +2 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 6 2 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 2 Exercise8.89 14 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 2 Exercise8.90 Solutiononp.640. x 2 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 Exercise8.91 x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 Exercise8.92 Solutiononp.640. x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 x +21 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise8.93 x 2 +10 x +24 x 2 +6 x Exercise8.94 Solutiononp.640. x 2 +9 x +14 x 2 +7 x Exercise8.95 6 b 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(b 6 b 2 +11 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.96 Solutiononp.640. 3 b 2 +10 b +3 3 b 2 +7 b +2 Exercise8.97 4 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 b 2 +5 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.98 Solutiononp.640. 16 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 4 a 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.99 20 x 2 +28 xy +9 y 2 4 x 2 +4 xy + y 2 Forthefollowingproblems,reduceeachrationalexpressionifpossible.Ifnotpossible,statetheanswerin lowestterms. Exercise8.100 Solutiononp.640. x +3 x +4 Exercise8.101 a +7 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.102 Solutiononp.640. 3 a +6 3

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560 CHAPTER8.RATIONALEXPRESSIONS Exercise8.103 4 x +12 4 Exercise8.104 Solutiononp.640. 5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.105 6 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.106 Solutiononp.640. 8 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.107 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise8.108 Solutiononp.640. )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +10 10 Exercise8.109 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x Exercise8.110 Solutiononp.640. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 3 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a Exercise8.111 x 3 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x x Exercise8.112 Solutiononp.640. y 4 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y y Exercise8.113 a 5 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 2 a Exercise8.114 Solutiononp.640. a 6 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 4 a 3 Exercise8.115 4 b 2 +3 b b Exercise8.116 Solutiononp.640. 2 a 3 +5 a a Exercise8.117 a a 3 + a Exercise8.118 Solutiononp.640. x 4 x 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x Exercise8.119 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F10 6.9738 Tf 6.226 0 Td [(a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a

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561 8.3.7ExcercisesForReview Exercise8.120 Solutiononp.640. Section3.7 Write 4 4 a 8 b 10 4 2 a 6 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 sothatonlypositiveexponentsappear. Exercise8.121 Section6.6 Factor y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(16 Exercise8.122 Solutiononp.640. Section6.8 Factor 10 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(17 x +3 Exercise8.123 Section7.4 Supplythemissingword.Anequationexpressedintheform ax + by = c issaidto beexpressedin form. Exercise8.124 Solutiononp.640. Section8.2 Findthedomainoftherationalexpression 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 8.4MultiplyingandDividingRationalExpressions 4 8.4.1Overview MultiplicationOfRationalExpressions DivisionOfRationalExpressions 8.4.2MultiplicationOfRationalExpressions Rationalexpressionsaremultipliedtogetherinmuchthesamewaythatarithmeticfractionsaremultiplied together.Tomultiplyrationalnumbers,wedothefollowing: Denition8.1:MethodforMultiplyingRationalNumbers 1.Reduceeachfractiontolowestterms. 2.Multiplythenumeratorstogether. 3.Multiplythedenominatorstogether. Rationalexpressionsaremultipliedtogetherusingexactlythesamethreesteps.Sincerationalexpressions tendtobelongerthanarithmeticfractions,wecansimplifythemultiplicationprocessbyaddingonemore step. Denition8.2:MethodforMultiplyingRationalExpressions 1.Factorallnumeratorsanddenominators. 2.Reducetolowesttermsrstbydividingoutallcommonfactors.Itisperfectlylegitimateto cancelthenumeratorofonefractionwiththedenominatorofanother. 3.Multiplynumeratorstogether. 4.Multiplydenominators.Itisoftenconvenient,butnotnecessary,toleavedenominatorsin factoredform. 4 Thiscontentisavailableonlineat.

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562 CHAPTER8.RATIONALEXPRESSIONS 8.4.3SampleSetA Performthefollowingmultiplications. Example8.38 3 4 1 2 = 3 1 4 2 = 3 8 Example8.39 8 9 1 6 = 4 9 1 3 = 4 1 9 3 = 4 27 Example8.40 3 x 5 y 7 12 y = 1 x 5 y 7 4 y = x 7 5 y 4 y = 7 x 20 y 2 Example8.41 x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +7 x +4 Divideoutthecommonfactor x +4 : x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +7 x +4 Multiplynumeratorsanddenominatorstogether. x +7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Example8.42 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x 2 +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 : Factor. x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +1 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Divideoutthecommonfactors x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 and x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 : x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +1 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Multiply : x +3 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +6 or x 2 +4 x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +6 or x 2 +4 x +3 x 2 +5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Eachofthesethreeformsisanacceptableformofthesameanswer : Example8.43 2 x +6 8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 : Factor. 2 x +3 8 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 Divideoutthecommonfactors2, x +3 and x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : 1 x +3 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Multiply : x +2 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 or x +2 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 Boththeseformsareacceptableformsofthesameanswer :

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563 Example8.44 3 x 2 x +7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 : Rewrite 3 x 2 as 3 x 2 1 : 3 x 2 1 x +7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Multiply : 3 x 2 x +7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Example8.45 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +9 : x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 1 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Example8.46 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 +8 x +15 4 x +20 x 2 +2 x : Factorfromtherstnumerator : )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( x 2 +3 x +2 x 2 +8 x +15 4 x +20 x 2 +2 x Factor : )]TJ/F7 6.9738 Tf 6.227 0 Td [( x +1 x +2 x +3 x +5 4 x +5 x x +2 Multiply. )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x x +3 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x x +3 or )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x 2 +3 x 8.4.4PracticeSetA Performeachmultiplication. Exercise8.125 Solutiononp.640. 5 3 6 7 Exercise8.126 Solutiononp.640. a 3 b 2 c 2 c 5 a 5 Exercise8.127 Solutiononp.640. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y 2 +1 y +1 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.128 Solutiononp.640. x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x 2 +7 x +6 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 x +20 Exercise8.129 Solutiononp.641. x 2 +6 x +8 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +8 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 8.4.5DivisionOfRationalExpressions Todivideonerationalexpressionbyanother,werstinvertthedivisorthenmultiplythetwoexpressions. Symbolically,ifwelet P;Q;R; and S representpolynomials,wecanwrite P Q R S = P Q S R = P S Q R

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564 CHAPTER8.RATIONALEXPRESSIONS 8.4.6SampleSetB Performthefollowingdivisions. Example8.47 6 x 2 5 a 2 x 10 a 3 : Invertthedivisorandmultiply. 3 x a 2 a 2 x = 3 x 2 a 2 1 =6 a 2 x Example8.48 x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 +9 x +20 x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Invertandmultiply. x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 +9 x +20 Factor. x +5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +5 x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 Example8.49 x +7 12 x +21 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 : Write 4 x +7 as 4 x +7 1 : 4 x +7 1 12 x +21 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Invertandmultiply. 4 x +7 1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 12 x +21 Factor. x +7 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 3 x +7 = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 3 8.4.7PracticeSetB Performeachdivision. Exercise8.130 Solutiononp.641. 8 m 2 n 3 a 5 b 2 2 m 15 a 7 b 2 Exercise8.131 Solutiononp.641. x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x 2 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 +4 x +3 Exercise8.132 Solutiononp.641. 6 a 2 +17 a +12 3 a +2 a +3 8.4.8Excercises Forthefollowingproblems,performthemultiplicationsanddivisions. Exercise8.133 Solutiononp.641. 4 a 3 5 b 3 b 2 a Exercise8.134 9 x 4 4 y 3 10 y x 2 Exercise8.135 Solutiononp.641. a b b a Exercise8.136 2 x 5 y 5 y 2 x

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565 Exercise8.137 Solutiononp.641. 12 a 3 7 28 15 a Exercise8.138 39 m 4 16 4 13 m 2 Exercise8.139 Solutiononp.641. 18 x 6 7 1 4 x 2 Exercise8.140 34 a 6 21 42 17 a 5 Exercise8.141 Solutiononp.641. 16 x 6 y 3 15 x 2 25 x 4 y Exercise8.142 27 a 7 b 4 39 b 13 a 4 b 2 16 a 5 Exercise8.143 Solutiononp.641. 10 x 2 y 3 7 y 5 49 y 15 x 6 Exercise8.144 22 m 3 n 4 11 m 6 n 33 mn 4 mn 3 Exercise8.145 Solutiononp.641. )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 p 2 q 7 a 3 b 2 21 a 5 b 3 2 p Exercise8.146 )]TJ/F7 6.9738 Tf 6.226 0 Td [(25 m 4 n 3 14 r 3 s 3 21 rs 4 10 mn Exercise8.147 Solutiononp.641. 9 a 3 a 2 Exercise8.148 10 b 2 4 b 3 Exercise8.149 Solutiononp.641. 21 a 4 5 b 2 14 a 15 b 3 Exercise8.150 42 x 5 16 y 4 21 x 4 8 y 3 Exercise8.151 Solutiononp.641. 39 x 2 y 2 55 p 2 13 x 3 y 15 p 6 Exercise8.152 14 mn 3 25 n 6 32 m 20 m 2 n 3 Exercise8.153 Solutiononp.641. 12 a 2 b 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 xy 4 6 a 2 15 x 2 Exercise8.154 24 p 3 q 9 mn 3 10 pq )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 n 2 Exercise8.155 Solutiononp.641. x +8 x +1 x +2 x +8 Exercise8.156 x +10 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.157 Solutiononp.641. 2 x +5 x +8 x +8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.158 y +2 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2

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566 CHAPTER8.RATIONALEXPRESSIONS Exercise8.159 Solutiononp.641. x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 4 Exercise8.160 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 x 5 x +1 Exercise8.161 Solutiononp.641. a +2 b a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 a +8 b 3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.162 6 m +2 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 4 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.163 Solutiononp.641. x 3 4 ab x Exercise8.164 y 4 3 x 2 y 2 Exercise8.165 Solutiononp.641. 2 a 5 6 a 2 4 b Exercise8.166 16 x 2 y 3 10 xy 3 Exercise8.167 Solutiononp.641. 21 m 4 n 2 3 mn 2 7 n Exercise8.168 x +8 x +2 x +8 Exercise8.169 Solutiononp.641. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.170 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 3 a +2 2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Exercise8.171 Solutiononp.641. b +1 4 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 3 b +1 Exercise8.172 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b 2 +2 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b 2 +2 2 Exercise8.173 Solutiononp.642. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 2 Exercise8.174 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.175 Solutiononp.642. y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.176 y +6 3 y +6 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.177 Solutiononp.642. a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 b 4 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 b 2 a + b Exercise8.178 x 2 +3 x +2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 2 x +2 Exercise8.179 Solutiononp.642. 6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(42 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise8.180 3 a +3 b a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 9 a +9 b a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(10

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567 Exercise8.181 Solutiononp.642. a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a 2 +6 a +9 Exercise8.182 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 b +6 b 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 b +20 Exercise8.183 Solutiononp.642. m 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 m +3 m 2 +5 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 m 2 +4 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 m 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 m +6 Exercise8.184 r 2 +7 r +10 r 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 r 2 +6 r +5 r 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.185 Solutiononp.642. 2 a 2 +7 a +3 3 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 a +6 a 2 +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.186 6 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 x 2 +7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.187 Solutiononp.642. x 3 y )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 2 y 2 x 2 y )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 2 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y x )]TJ/F10 6.9738 Tf 6.227 0 Td [(xy Exercise8.188 4 a 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a 2 b 2 15 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 4 ab )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 2 Exercise8.189 Solutiononp.642. x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +3 Exercise8.190 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 x +8 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x +8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.191 Solutiononp.642. 2 a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b a + b a +3 b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b 2 a )]TJ/F10 6.9738 Tf 6.226 0 Td [(b Exercise8.192 3 a a +1 2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 2 5 a +5 15 a +30 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(20 Exercise8.193 Solutiononp.642. )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a 2 4 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 b 3 15 a Exercise8.194 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x 3 5 y 2 20 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x Exercise8.195 Solutiononp.642. )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 x 2 y 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 xy Exercise8.196 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 a 3 3 b 2 a 6 b 2 Exercise8.197 Solutiononp.642. )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 a +2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a +2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.198 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 2 +2 x +3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise8.199 Solutiononp.642. )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 x 2 +4 x +1 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.200 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a +15 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 Exercise8.201 Solutiononp.642. )]TJ/F10 6.9738 Tf 6.226 0 Td [(b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b +14 3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F10 6.9738 Tf 6.226 0 Td [(b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 )]TJ/F10 6.9738 Tf 6.227 0 Td [(b +8 Exercise8.202 3 a +6 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(24 6 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 3 a +15

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568 CHAPTER8.RATIONALEXPRESSIONS Exercise8.203 Solutiononp.642. 4 x +12 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 7 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 2 x +2 Exercise8.204 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 2 + b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F10 6.9738 Tf 6.227 0 Td [(b +2 b +5 Exercise8.205 Solutiononp.642. 3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 6 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.206 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 b +4 8 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(28 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 b +1 2 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.207 Solutiononp.642. x 2 +4 x +3 x 2 +5 x +4 x +3 Exercise8.208 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise8.209 Solutiononp.642. 3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 x +18 x 2 +5 x +6 x +2 8.4.9ExercisesForReview Exercise8.210 Section3.3 If a< 0 ,then j a j = Exercise8.211 Solutiononp.642. Section4.4 Classifythepolynomial 4 xy +2 y asamonomial,binomial,ortrinomial.Stateits degreeandwritethenumericalcoecientofeachterm. Exercise8.212 Section4.5 Findtheproduct: y 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 y +1 Exercise8.213 Solutiononp.642. Section5.5 Translatethesentencefourlessthantwicesomenumberistwomorethanthe numberintoanequation. Exercise8.214 Section8.3 Reducethefraction x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 8.5BuildingRationalExpressionsandtheLCD 5 8.5.1Overview TheProcess TheReasonForBuildingRationalExpressions TheLeastCommonDenominatorLCD 5 Thiscontentisavailableonlineat.

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569 8.5.2TheProcess Recall,fromSectionSection8.2,theequalitypropertyoffractions. EqualityPropertyOfFractions If a b = c d ; then ad = bc: Usingthefactthat 1= b b ;b 6 =0 ,andthat1isthemultiplicativeidentity,itfollowsthatif P Q isarational expression,then P Q b b = Pb Qb ;b 6 =0 Thisequationassertsthatarationalexpressioncanbetransformedintoanequivalentrationalexpression bymultiplyingboththenumeratoranddenominatorbythesamenonzeronumber. ProcessofBuildingRationalExpressions Thisprocessisknownastheprocessof buildingrationalexpressions anditisexactlytheoppositeof reducingrationalexpressions.Theprocessisshownintheseexamples: Example8.50 3 4 canbebuiltto 12 16 since 3 4 1= 3 4 4 4 = 3 4 4 4 = 12 16 Example8.51 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 5 canbebuiltto )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 10 since )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 5 1= )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 5 2 2 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 5 2 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 10 Example8.52 3 7 canbebuiltto 3 xy 7 xy since 3 7 1= 3 7 xy xy = 3 xy 7 xy Example8.53 4 a 3 b canbebuiltto 4 a 2 a +1 3 ab a +1 since 4 a 3 b 1= 4 a 3 b a a +1 a a +1 = 4 a 2 a +1 3 ab a +1 Supposewe'regivenarationalexpression P Q andwishtobuilditintoarationalexpressionwithdenominator Qb 2 ,thatis, P Q ? Qb 2 Sincewechangedthedenominator,wemustcertainlychangethenumeratorinthesameway.To determinehowtochangethenumeratorweneedtoknowhowthedenominatorwaschanged.Sinceone rationalexpressionisbuiltintoanotherequivalentexpressionbymultiplicationby1,therstdenominator musthavebeenmultipliedbysomequantity.Observationof P Q ? Qb 2 tellsusthat Q wasmultipliedby b 2 .Hence,wemustmultiplythenumerator P by b 2 .Thus, P Q = Pb 2 Qb 2 Quiteoftenasimplecomparisonoftheoriginaldenominatorwiththenewdenominatorwilltellusthe factorbeingused.However,therewillbetimeswhenthefactorisunclearbysimpleobservation.Weneed amethodforndingthefactor. Observethefollowingexamples;thentrytospeculateonthemethod.

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570 CHAPTER8.RATIONALEXPRESSIONS Example8.54 3 4 = ? 20 : Theoriginaldenominator4wasmultipliedby5toyield20.Whatarithmeticprocesswill yield5using4and20? Example8.55 9 10 = ? 10 y : Theoriginaldenominator10wasmultipliedby y toyield 10 y Example8.56 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 xy 2 a 3 b = ? 16 a 5 b 3 : Theoriginaldenominator 2 a 3 b wasmultipliedby 8 a 2 b 2 toyield 16 a 5 b 3 : Example8.57 5 ax a +1 2 = ? 4 a +1 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 : Theoriginaldenominator a +1 2 wasmultipliedby 4 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 toyield 4 a +1 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Todeterminethequantitythattheoriginaldenominatorwasmultipliedbytoyieldthenewdenominator, weask,"WhatdidImultiplytheoriginaldenominatorbytogetthenewdenominator?"Wendthisfactor bydividingtheoriginaldenominatorintothenewdenominator. Itispreciselythisquantitythatwemultiplythenumeratorbytobuildtherationalexpression. 8.5.3SampleSetA Determine N ineachofthefollowingproblems. Example8.58 8 3 = N 15 : Theoriginaldenominatoris3andthenew denominatois15.Dividetheoriginal denominatorintothenewdenominatorand multiplythenumerator8bythisresult. 15 3=5 Then, 8 5=40 : So, 8 3 = 40 15 and N =40 : Checkbyreducing 40 15 : Example8.59 2 x 5 b 2 y = N 20 b 5 y 4 : Theoriginaldenominatoris 5 b 2 y andthenew denominatoris 20 b 5 y 4 .Dividetheoriginal denominatorintothenewdenominatorand multiplythenumerator 2 x bythisresult. 20 b 5 y 4 5 b 2 y =4 b 3 y 3 So, 2 x 4 b 3 y 3 =8 b 3 xy 3 : Thus, 2 x 5 b 2 y = 8 b 3 xy 3 20 b 5 y 4 and N =8 b 3 xy 3 :

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571 Example8.60 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a a +2 = N a +2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 : Thenewdenominatordividedbytheoriginaldenominatoris a +2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 a +2 = a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Multiply )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 a by a )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 : )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 a 2 +42 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a a +2 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a 2 +42 a a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 and N = )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 a 2 +42 a: Example8.61 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = N a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 : Thenewdenominatordividedbytheoriginaldenominatoris a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = a +4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = a +4 Multiply )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 by a +4 : )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 a +4= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a 2 +3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(9 a +12 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 a +12 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 and N = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 a +12 Example8.62 7 x = N x 2 y 3 : Write 7 x as 7 x 1 : 7 x 1 = N x 2 y 3 Nowwecanseeclearlythattheoriginaldenominator 1wasmultipliedby x 2 y 3 : Weneedtomultiplythe numerator 7 x by x 2 y 3 : 7 x = 7 x x 2 y 3 x 2 y 3 7 x = 7 x 3 y 3 x 2 y 3 and N =7 x 3 y 3 : Example8.63 5 x x +3 = 5 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(20 x N : Thesameprocessworksinthiscase : Dividetheoriginal numerator 5 x intothenewnumerator 5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(20 x: 5 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(20 x 5 x = x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x = x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 Multiplythedenominatorby x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 : x +3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 5 x x +3 = 5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(20 x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 and N =5 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 :

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572 CHAPTER8.RATIONALEXPRESSIONS Example8.64 4 x 3 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x = N x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 : Thetwodenominatorshavenearlythesameterms ; eachhas theoppositesign : Factor )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 fromtheoriginaldenominator : 3 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3+ x = )]TJ/F8 9.9626 Tf 9.409 0 Td [( x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 4 x 3 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x = 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [( x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 and N = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 x: Itisimportanttonotethatwe factored )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 fromtheoriginaldenominator.We didnot multiplyitby )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 .Hadwemultipliedonlythedenominatorby )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 wewouldhavehadtomultiply thenumeratorby )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 also. 8.5.4PracticeSetA Determine N Exercise8.215 Solutiononp.642. 3 8 = N 48 Exercise8.216 Solutiononp.642. 9 a 5 b = N 35 b 2 x 3 Exercise8.217 Solutiononp.642. )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = N y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.218 Solutiononp.643. a +7 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = N a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 Exercise8.219 Solutiononp.643. 4 a = N 6 a 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.220 Solutiononp.643. )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x = N 8 x 3 y 3 z 5 Exercise8.221 Solutiononp.643. 6 ab b +3 = N b 2 +6 b +9 Exercise8.222 Solutiononp.643. 3 m m +5 = 3 m 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 m N Exercise8.223 Solutiononp.643. )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 r 2 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 r 3 +8 r 2 N Exercise8.224 Solutiononp.643. )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 ab 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = N 4 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a

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573 8.5.5TheReasonForBuildingRationalExpressions BuildingRationalExpressions Normally,whenwewritearationalexpression,wewriteitinreducedform.Thereasonforbuildingrational expressionsistomakeadditionandsubtractionofrationalexpressionsconvenientsimpler. Toaddorsubtracttwoormorerationalexpressionsthey must havethe samedenominator Buildingrationalexpressionsallowsustotransformfractionsintofractionswiththesamedenominatorswhichwecanthenaddorsubtract.Themostconvenientnewdenominatoristhe leastcommon denominator LCDofthegivenfractions. 8.5.6TheLeastCommonDenominatorLCD Inarithmetic,the leastcommondenominator isthesmallestleastquantitythateachofthegiven denominatorswilldivideintowithoutaremainder.Foralgebraicexpressions,theLCDisthepolynomialof leastdegree divisiblebyeachdenominator.Someexamplesareshownbelow. Example8.65 3 4 ; 1 6 ; 5 12 : TheLCDis12since12isthesmallestnumberthat4,6,and12willdivideintowithouta remainder. Example8.66 1 3 ; 5 6 ; 5 8 ; 7 12 : TheLCDis24since24isthesmallestnumberthat3,6,8,and12willdivideintowithoutaremainder. Example8.67 2 x ; 3 x 2 : TheLCDis x 2 since x 2 isthesmallestquantitythat x and x 2 willdivideintowithouta remainder. Example8.68 5 a 6 a 2 b ; 3 a 8 ab 3 : TheLCDis 24 a 2 b 3 since 24 a 2 b 3 isthesmallestquantitythat 6 a 2 b and 8 ab 3 willdivideinto withoutaremainder. Example8.69 2 y y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; 4 y 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 3 ; y y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : TheLCDis y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 since y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 isthesmallestquantitythat y )]TJ/F8 9.9626 Tf 10.476 0 Td [(6 ; y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 3 and y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 willdivideintowithoutaremainder. We'llnowproposeanddemonstrateamethodforobtainingtheLCD. MethodforObtainingtheLCD 1.Factoreachdenominator.Useexponentsforrepeatedfactors.Itisusuallynotnecessarytofactor numericalquantities. 2.Writedowneach dierent factorthatappears.Ifafactorappearsmorethanonce,useonlythefactor withthehighestexponent. 3.TheLCDistheproductofthefactorswritteninstep2.

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574 CHAPTER8.RATIONALEXPRESSIONS 8.5.7SampleSetB FindtheLCD. Example8.70 1 x ; 3 x 3 ; 2 4 y 1.Thedenominatorsarealreadyfactored. 2.Notethat x appearsas x and x 3 .Useonlythe x withthehigherexponent, x 3 .Theterm 4 y appears,sowemustalsouse 4 y 3.TheLCDis 4 x 3 y Example8.71 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 ; 2 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 1.Onlythethirddenominatorneedstobefactored. x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +2= x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Nowthethreedenominatorsare x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 ; x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 ,and x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : 2.Notethat x )]TJ/F8 9.9626 Tf 10.315 0 Td [(1 appearsas x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 ;x )]TJ/F8 9.9626 Tf 10.316 0 Td [(1 ,and x )]TJ/F8 9.9626 Tf 10.315 0 Td [(1 : Useonlythe x )]TJ/F8 9.9626 Tf 10.316 0 Td [(1 withthehighest exponent, x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 .Alsoappearingare x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 and x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : 3.TheLCDis x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : Example8.72 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 6 a 4 ; 3 4 a 3 b ; 1 3 a 3 b +5 1.Thedenominatorsarealreadyfactored. 2.WecanseethattheLCDofthenumbers6,4,and3is12.Wealsoneed a 4 b ,and b +5 3.TheLCDis 12 a 4 b b +5 : Example8.73 9 x ; 4 8 y 1.Thedenominatorsarealreadyfactored. 2. x; 8 y: 3.TheLCDis 8 xy .

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575 8.5.8PracticeSetB FindtheLCD. Exercise8.225 Solutiononp.643. 3 x 2 ; 4 x 5 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 xy Exercise8.226 Solutiononp.643. x +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 ; x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +1 Exercise8.227 Solutiononp.643. 2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 m m +1 2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 ; 12 m 2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 3 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.228 Solutiononp.643. 1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +9 Exercise8.229 Solutiononp.643. 3 4 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 y ; 8 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 y +4 ; 10 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 y 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 y 2 8.5.9SampleSetC Changethegivenrationalexpressionsintorationalexpressionshavingthesamedenominator. Example8.74 3 x 2 ; 4 x : TheLCD,byinspection,is x 2 .Rewriteeachexpression with x 2 asthenewdenominator. x 2 ; x 2 Determinethenumerators.In 3 x 2 ,thedenominatorwasnot changedsoweneednotchangethenumerator. 3 x 2 ; x 2 Inthesecondfraction,theoriginaldenominatorwas x Wecanseethat x mustbemultipliedby x tobuilditto x 2 Sowemustalsomultiplythenumerator4by x .Thus,4 x =4 x 3 x 2 ; 4 x x 2 Example8.75 4 b b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b b +3 : Byinspection,theLCDis b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 b +3 : Rewriteeachfractionwithnewdenominator b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 b +3 : b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b +3 ; b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b +3 Thedenominatoroftherstrationalexpressionhasbeenmultiplied by b +3 ; sothenumerator 4 b mustbemultipliedby b +3 : 4 b b +3=4 b 2 +12 b 4 b 2 +12 b b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b +3 ; b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b +3 Thedenominatorofthesecondrationalexpressionhasbeenmultiplied by b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ,sothenumerator )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 b mustbemultipliedby b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 b b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 b 2 +2 b 4 b 2 +12 b b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b +3 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b 2 +2 b b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b +3 Example8.76

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576 CHAPTER8.RATIONALEXPRESSIONS 6 x x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(8 x +15 ; )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 x 2 x 2 )]TJ/F54 7.9701 Tf 6.586 0 Td [(7 x +12 : WerstndtheLCD : Factor : 6 x x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(5 ; )]TJ/F54 7.9701 Tf 6.587 0 Td [(2 x 2 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 TheLCDis x )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(5 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 : Rewriteeachofthese fractionswithnewdenominator x )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(5 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 : x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 ; x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 BycomparingthedenominatoroftherstfractionwiththeLCD weseethatwemustmultiplythenumerator 6 x by x )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 : 6 x x )]TJ/F89 11.9552 Tf 11.956 0 Td [(4=6 x 2 )]TJ/F89 11.9552 Tf 11.955 0 Td [(24 x 6 x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(24 x x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 ; x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 BycomparingthedenominatorofthesecondfractionwiththeLCD, weseethatwemustmultiplythenumerator )]TJ/F89 11.9552 Tf 11.956 0 Td [(2 x 2 by x )]TJ/F89 11.9552 Tf 11.955 0 Td [(5 : )]TJ/F89 11.9552 Tf 9.298 0 Td [(2 x 2 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(5= )]TJ/F89 11.9552 Tf 9.298 0 Td [(2 x 3 +10 x 2 6 x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(24 x x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 ; )]TJ/F54 7.9701 Tf 6.587 0 Td [(2 x 3 +10 x 2 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 Theseexampleshavebeendonestep-by-stepandincludeexplanations.Thismakestheprocess seemfairlylong.Inpractice,however,theprocessismuchquicker. Example8.77 6 ab a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a +4 ; a + b a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 a +16 6 ab a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 ; a + b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 LCD = a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 : 6 ab a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 ; a + b a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 Example8.78 x +1 x 3 +3 x 2 ; 2 x x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x ; x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +4 x +1 x 2 x +3 ; 2 x x x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 ; x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 LCD = x 2 x +3 x +2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 2 : x +1 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 x 2 x +3 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 ; 2 x 2 x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 x +3 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 ; x 2 x +3 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 x +3 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 8.5.10PracticeSetC Changethegivenrationalexpressionsintorationalexpressionswiththesamedenominators. Exercise8.230 Solutiononp.643. 4 x 3 ; 7 x 5 Exercise8.231 Solutiononp.643. 2 x x +6 ; x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.232 Solutiononp.643. )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(b ; 4 b b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.233 Solutiononp.643. 8 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.234 Solutiononp.643. 10 x x 2 +8 x +16 ; 5 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16

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577 Exercise8.235 Solutiononp.643. )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 ab 2 a 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a 2 ; 6 b a 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 3 ; )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 a +4 8.5.11Exercises Forthefollowingproblems,replace N withtheproperquantity. Exercise8.236 Solutiononp.643. 3 x = N x 3 Exercise8.237 4 a = N a 2 Exercise8.238 Solutiononp.643. )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x = N xy Exercise8.239 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 m = N ms Exercise8.240 Solutiononp.643. 6 a 5 = N 10 b Exercise8.241 a 3 z = N 12 z Exercise8.242 Solutiononp.643. x 2 4 y 2 = N 20 y 4 Exercise8.243 b 3 6 a = N 18 a 5 Exercise8.244 Solutiononp.643. )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a 5 x 2 y = N 15 x 3 y 3 Exercise8.245 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 z 7 a 3 b = N 21 a 4 b 5 Exercise8.246 Solutiononp.643. 8 x 2 y 5 a 3 = N 25 a 3 x 2 Exercise8.247 2 a 2 = N a 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.248 Solutiononp.644. 5 x 3 = N x 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.249 2 a b 2 = N b 3 )]TJ/F10 6.9738 Tf 6.226 0 Td [(b Exercise8.250 Solutiononp.644. 4 x a = N a 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a 2 Exercise8.251 6 b 3 5 a = N 10 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(30 a Exercise8.252 Solutiononp.644. 4 x 3 b = N 3 b 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 b Exercise8.253 2 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = N m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 m +2 Exercise8.254 Solutiononp.644. 3 s s +12 = N s +12 s )]TJ/F7 6.9738 Tf 6.226 0 Td [(7

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578 CHAPTER8.RATIONALEXPRESSIONS Exercise8.255 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = N a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.256 Solutiononp.644. a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = N a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.257 b +7 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = N b )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 b +6 Exercise8.258 Solutiononp.644. 5 m 2 m +1 = N m +1 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.259 4 a +6 = N a 2 +5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.260 Solutiononp.644. 9 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = N b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 b +8 Exercise8.261 3 b b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = N b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 b +24 Exercise8.262 Solutiononp.644. )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 = N x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(21 Exercise8.263 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 m m +6 = N m 2 +10 m +24 Exercise8.264 Solutiononp.644. 4 y y +1 = N y 2 +9 y +8 Exercise8.265 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 = N x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.266 Solutiononp.644. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y +3 = N y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 Exercise8.267 a +5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = N a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(25 Exercise8.268 Solutiononp.644. z )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 z +4 = N z 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 Exercise8.269 4 2 a +1 = N 2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.270 Solutiononp.644. 1 3 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = N 3 b 2 +11 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.271 a +2 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = N 2 a 2 +9 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.272 Solutiononp.644. )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 4 x +3 = N 4 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(13 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Exercise8.273 b +2 3 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = N 6 b 2 +7 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.274 Solutiononp.644. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = N 12 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(11 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise8.275 3 x +2 = 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 N Exercise8.276 Solutiononp.644. 4 y +6 = 4 y +8 N

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579 Exercise8.277 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 N Exercise8.278 Solutiononp.644. )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 a a +3 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(40 a N Exercise8.279 y +1 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 = y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 N Exercise8.280 Solutiononp.644. x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +9 = x 2 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(20 N Exercise8.281 3 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x = N x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.282 Solutiononp.644. 7 a 5 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a = N a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.283 )]TJ/F10 6.9738 Tf 6.226 0 Td [(m +1 3 )]TJ/F10 6.9738 Tf 6.227 0 Td [(m = N m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.284 Solutiononp.644. k +6 10 )]TJ/F10 6.9738 Tf 6.227 0 Td [(k = N k )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 Forthefollowingproblems,convertthegivenrationalexpressionstorationalexpressionshavingthesame denominators. Exercise8.285 2 a ; 3 a 4 Exercise8.286 Solutiononp.644. 5 b 2 ; 4 b 3 Exercise8.287 8 z ; 3 4 z 3 Exercise8.288 Solutiononp.644. 9 x 2 ; 1 4 x Exercise8.289 2 a +3 ; 4 a +1 Exercise8.290 Solutiononp.644. 2 x +5 ; 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.291 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 ; 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.292 Solutiononp.644. 10 y +2 ; 1 y +8 Exercise8.293 4 a 2 ; a a +4 Exercise8.294 Solutiononp.644. )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 b 2 ; b 2 b +5 Exercise8.295 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; 5 b 4 b Exercise8.296 Solutiononp.644. 10 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; 2 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a Exercise8.297 4 x 2 +2 x ; 1 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.298 Solutiononp.645. x +1 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; x +4 x 2 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2

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580 CHAPTER8.RATIONALEXPRESSIONS Exercise8.299 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x +20 ; 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 Exercise8.300 Solutiononp.645. )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 2 +5 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; b +6 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.301 b +2 b 2 +6 b +8 ; b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b 2 +8 b +12 Exercise8.302 Solutiononp.645. x +7 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 ; x +3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise8.303 2 a 2 + a ; a +3 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.304 Solutiononp.645. x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x 2 +7 x +6 ; 2 x x 2 +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 Exercise8.305 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 x 2 +5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 ; x: )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 5 x 2 +16 x +3 Exercise8.306 Solutiononp.645. 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 5 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x Exercise8.307 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 ; )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 6 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a Exercise8.308 Solutiononp.645. 6 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x ; 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.309 k 5 )]TJ/F10 6.9738 Tf 6.227 0 Td [(k ; 3 k k )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise8.310 Solutiononp.645. 2 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 ; 7 8 )]TJ/F10 6.9738 Tf 6.227 0 Td [(m 8.5.12ExcercisesForReview Exercise8.311 Section6.4 Factor m 2 x 3 + mx 2 + mx: Exercise8.312 Solutiononp.645. Section6.7 Factor y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 y +21 : Exercise8.313 Section7.7 Writetheequationofthelinethatpassesthroughthepoints ; 1 and ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 Expresstheequationinslope-interceptform. Exercise8.314 Solutiononp.645. Section8.3 Reduce y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 : Exercise8.315 Section8.4 Findthequotient: x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +9 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 x 2 +2 x :

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581 8.6AddingandSubtractingRationalExpressions 6 8.6.1Overview BasicRule FractionswiththeSameDenominator FractionswithDierentDenominators 8.6.2BasicRule Wearenowinapositiontostudytheprocessofaddingandsubtractingrationalexpressions.Thereisamost basicruletowhichwemuststrictlyadhereifwewishtoconvenientlyaddorsubtractrationalexpressions. Toaddorsubtractrationalexpressionsconveniently,theyshouldhavethesamedenominators. Thus,toaddorsubtracttwoormorerationalexpressionsconveniently,wemustensurethattheyall havethesamedenominator.ThedenominatorthatismostconvenientistheLCD. 8.6.3FractionsWithTheSameDenominator TheRuleforAddingandSubtractingRationalExpressions Toaddorsubtracttwoormorerationalexpressionswiththesamedenominators,addorsubtractthe numeratorsandplacetheresultovertheLCD.Reduceifnecessary.Symbolically, a c + b c = a + b c a c )]TJ/F11 9.9626 Tf 11.176 6.74 Td [(b c = a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b c Notethatwecombine only thenumerators. 8.6.4SampleSetA Addorsubtractthefollowingrationalexpressions. Example8.79 1 6 + 3 6 Thedenominatorsarethesame.Addthenumerators. 1 6 + 3 6 = 1+3 6 = 4 6 Reduce. 1 6 + 3 6 = 2 3 Example8.80 5 x + 8 x Thedenominatorsarethesame.Addthenumerators. 5 x + 8 x = 5+8 x = 13 x Example8.81 2 ab y 2 w )]TJ/F7 6.9738 Tf 14.485 3.923 Td [(5 b y 2 w Thedenominatorsarethesame.Subtractthenumerators. 2 ab y 2 w )]TJ/F7 6.9738 Tf 14.486 3.922 Td [(5 b y 2 w = 2 ab )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b y 2 w 6 Thiscontentisavailableonlineat.

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582 CHAPTER8.RATIONALEXPRESSIONS Example8.82 3 x 2 + x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 + x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Thedenominatorsarethesame.Addthenumerators. 3 x 2 + x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 + x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = 3 x 2 + x +2+ x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 = 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Example8.83 5 y +3 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 )]TJ/F7 6.9738 Tf 11.213 4.444 Td [(2 y +4 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Thedenominatorsarethesame.Subtractthenumerators. But becareful tosubtractthe entire numerator.Useparentheses! 5 y +3 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 )]TJ/F7 6.9738 Tf 11.213 4.445 Td [(2 y +4 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = 5 y +3 )]TJ/F7 6.9738 Tf 6.227 0 Td [( y +4 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = 5 y +3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Note : 5 y +3 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F7 6.9738 Tf 26.861 4.444 Td [(2 y +4 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 | {z } Observethispart : Theterm )]TJ/F7 6.9738 Tf 11.213 4.445 Td [(2 y +4 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 couldbewrittenas + )]TJ/F7 6.9738 Tf 6.227 0 Td [( y +4 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Acommonmistakeistowrite )]TJ/F8 9.9626 Tf 8.945 6.74 Td [(2 y +4 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 as )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y +4 2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 Thisis not correct,asthenegativesignisnotbeingappliedtotheentirenumerator. Example8.84 3 x 2 +4 x +5 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 + 2 x 2 + x +6 x 2 +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 )]TJ/F10 6.9738 Tf 13.089 3.923 Td [(x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x 2 +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Factorthedenominatorstodetermineifthey'rethesame. 3 x 2 +4 x +5 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 + 2 x 2 + x +6 x +6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F10 6.9738 Tf 15.386 3.922 Td [(x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Thedenominatorsarethesame.Combinethenumeratorsbeingcarefultonotethenegativesign. 3 x 2 +4 x +5+2 x 2 + x +6 )]TJ/F8 9.9626 Tf 6.227 -0.748 Td [( x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +6 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 3 x 2 +4 x +5+2 x 2 + x +6 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 2 +4 x +6 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 4 x 2 +9 x +17 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2

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583 8.6.5PracticeSetA AddorSubtractthefollowingrationalexpressions. Exercise8.316 Solutiononp.645. 4 9 + 2 9 Exercise8.317 Solutiononp.645. 3 b + 2 b Exercise8.318 Solutiononp.645. 5 x 2 y 2 )]TJ/F7 6.9738 Tf 12.989 3.923 Td [(3 x 2 y 2 Exercise8.319 Solutiononp.645. x + y x )]TJ/F10 6.9738 Tf 6.226 0 Td [(y + 2 x +3 y x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y Exercise8.320 Solutiononp.645. 4 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +4 3 x +10 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(x 2 +2 x +5 3 x +10 Exercise8.321 Solutiononp.645. x x +1 x x +3 + 3 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +7 2 x 2 +3 x Exercise8.322 Solutiononp.645. 4 x +3 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F7 6.9738 Tf 22.701 3.923 Td [(8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.323 Solutiononp.645. 5 a 2 + a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 2 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 + 2 a 2 +3 a +4 2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 a + a 2 +2 2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 a Exercise8.324 Solutiononp.645. 8 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +8 + 2 x 2 +3 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +8 )]TJ/F7 6.9738 Tf 13.51 3.923 Td [(5 x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 8.6.6FractionswithDierentDenominators 8.6.7SampleSetB AddorSubtractthefollowingrationalexpressions. Example8.85 4 a 3 y + 2 a 9 y 2 : Thedenominatorsare not thesame.FindtheLCD.Byinspection,theLCDis 9 y 2 : 9 y 2 + 2 a 9 y 2 Thedenominatoroftherstrationalexpressionhasbeenmultipliedby 3 y; sothenumeratormustbemultipliedby 3 y: 4 a 3 y =12 ay 12 ay 9 y 2 + 2 a 9 y 2 Thedenominatorsarenowthesame.Addthenumerators. 12 ay +2 a 9 y 2 Example8.86

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584 CHAPTER8.RATIONALEXPRESSIONS 3 b b +2 + 5 b b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 : Thedenominatorsare not thesame.TheLCDis b +2 b )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 : b +2 b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 + b +2 b )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 Thedenominatoroftherstrationalexpressionhasbeenmultipliedby b )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 ; sothenumeratormustbemultipliedby b )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 : 3 b b )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 3 b b )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 b +2 b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 + b +2 b )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 Thedenominatorofthesecondrationalexpressionhasbeenmultipliedby b +2 ; sothenumeratormustbemultipliedby b +2 : 5 b b +2 3 b b )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 b +2 b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 + 5 b b +2 b +2 b )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 Thedenominatorsarenowthesame.Addthenumerators. 3 b b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3+5 b b +2 b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 b +2 = 3 b 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(9 b +5 b 2 +10 b b )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 b +2 = 8 b 2 + b b )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 b )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 Example8.87 x +3 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 + x )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 4 x +4 : Thedenominatorsare not thesame. FindtheLCD. x +3 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 + x )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 4 x +1 TheLCDis x +1 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 + 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 Thedenominatoroftherstrationalexpressionhasbeenmultipliedby 4 x +1 so thenumeratormustbemultipliedby 4 x +1 : 4 x +3 x +1 4 x +3 x +1 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 + 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 Thedenominatorofthesecondrationalexpressionhasbeenmultipliedby x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 sothenumeratormustbemultipliedby x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 : x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(2 4 x +3 x +1 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 + x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 Thedenominatorsarenowthesame. Addthenumerators. 4 x +3 x +1+ x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(2 4 x +1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 4 x 2 +4 x +3 + x 2 )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x +2 4 x +1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 4 x 2 +16 x +12+ x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x +2 4 x +1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 = 5 x 2 +13 x +14 4 x +1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 Example8.88

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585 x +5 x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(7 x +12 + 3 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(2 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 DeterminetheLCD. x +5 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 + 3 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x +1 TheLCDis x )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(3 x +1 : x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x +1 + x )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x +1 Therstnumeratormustbemultipliedby x +1 andthesecondby x )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 : x +5 x +1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x +1 + x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x +1 Thedenominatorsarenowthesame.Addthenumerators. x +5 x +1+ x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x +1 x 2 +6 x +5+3 x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(13 x +4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(3 x +1 4 x 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(7 x +9 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(3 x +1 Example8.89 a +4 a 2 +5 a +6 )]TJ/F56 7.9701 Tf 25.002 4.707 Td [(a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(5 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(24 DeterminetheLCD. a +4 a +3 a +2 )]TJ/F56 7.9701 Tf 27.396 4.707 Td [(a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a +3 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 TheLCDis a +3 a +2 a )]TJ/F89 11.9552 Tf 11.955 0 Td [(8 : a +3 a +2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 )]TJETq1 0 0 1 180.821 452.111 cm[]0 d 0 J 0.478 w 0 0 m 65.716 0 l SQBT/F54 7.9701 Tf 180.821 444.999 Td [( a +3 a +2 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(8 Therstnumeratormustbemultipliedby a )]TJ/F89 11.9552 Tf 11.955 0 Td [(8 andthesecondby a +2 : a +4 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(8 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 )]TJ/F54 7.9701 Tf 24.103 5.699 Td [( a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a +2 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(8 Thedenominatorsarenowthesame.Subtractthenumerators. a +4 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 )]TJ/F54 7.9701 Tf 6.587 0 Td [( a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a +2 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 a 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(32 )]TJ/F89 11.9552 Tf 6.586 -0.996 Td [( a 2 )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(8 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(8 a 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(32 )]TJ/F56 7.9701 Tf 6.586 0 Td [(a 2 +2 a +8 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 )]TJ/F54 7.9701 Tf 6.587 0 Td [(2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(24 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 Factor )]TJ/F89 11.9552 Tf 11.955 0 Td [(2 fromthenumerator. )]TJ/F54 7.9701 Tf 6.586 0 Td [(2 a +12 a +3 a +2 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(8 Example8.90 3 x 7 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x + 5 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 : Thedenominatorsare nearly thesame.Theydieronlyinsign. Ourtechniqueistofactor )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 fromoneofthem. 3 x 7 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x = 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [( x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Factor )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 fromtherstterm. 3 x 7 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x + 5 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 + 5 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +5 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = 2 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 8.6.8PracticeSetB AddorSubtractthefollowingrationalexpressions. Exercise8.325 Solutiononp.645. 3 x 4 a 2 + 5 x 12 a 3 Exercise8.326 Solutiononp.645. 5 b b +1 + 3 b b )]TJ/F7 6.9738 Tf 6.226 0 Td [(2

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586 CHAPTER8.RATIONALEXPRESSIONS Exercise8.327 Solutiononp.645. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 a +2 + a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a +3 Exercise8.328 Solutiononp.645. 4 x +1 x +3 )]TJ/F10 6.9738 Tf 11.214 3.923 Td [(x +5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.329 Solutiononp.645. 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 y + 3 y +1 y +4 Exercise8.330 Solutiononp.646. a )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a +2 + a +2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +8 Exercise8.331 Solutiononp.646. 6 b 2 +6 b +9 )]TJ/F7 6.9738 Tf 24.707 3.922 Td [(2 b 2 +4 b +4 Exercise8.332 Solutiononp.646. x x +4 )]TJ/F10 6.9738 Tf 13.144 3.923 Td [(x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.333 Solutiononp.646. 5 x 4 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x + 7 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 8.6.9SampleSetC Combinethefollowingrationalexpressions. Example8.91 3+ 7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : Rewritetheexpression. 3 1 + 7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 TheLCDis x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + 7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + 7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3+7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 3 x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Example8.92 3 y +4 )]TJ/F10 6.9738 Tf 11.158 4.444 Td [(y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 : Rewritetheexpression. 3 y +4 1 )]TJ/F10 6.9738 Tf 11.158 4.444 Td [(y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 TheLCDis y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 : y +4 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F10 6.9738 Tf 11.158 4.444 Td [(y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = y +4 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( y 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = 3 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(14 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(24 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y 2 + y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 = 2 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(13 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(27 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 8.6.10PracticeSetC Exercise8.334 Solutiononp.646. Simplify 8+ 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 : Exercise8.335 Solutiononp.646. Simplify 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(a 2 +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a +3 :

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587 8.6.11Exercises Forthefollowingproblems,addorsubtracttherationalexpressions. Exercise8.336 Solutiononp.646. 3 8 + 1 8 Exercise8.337 1 9 + 4 9 Exercise8.338 Solutiononp.646. 7 10 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 5 Exercise8.339 3 4 )]TJ/F7 6.9738 Tf 13.144 3.922 Td [(5 12 Exercise8.340 Solutiononp.646. 3 4 x + 5 4 x Exercise8.341 2 7 y + 3 7 y Exercise8.342 Solutiononp.646. 6 y 5 x + 8 y 5 x Exercise8.343 9 a 7 b + 3 a 7 b Exercise8.344 Solutiononp.646. 15 n 2 m )]TJ/F7 6.9738 Tf 12.231 3.923 Td [(6 n 2 m Exercise8.345 8 p 11 q )]TJ/F7 6.9738 Tf 13.041 4.445 Td [(3 p 11 q Exercise8.346 Solutiononp.646. y +4 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 + y +8 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 Exercise8.347 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y +4 + y +7 y +4 Exercise8.348 Solutiononp.646. a +6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + 3 a +5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.349 5 a +1 a +7 + 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +7 Exercise8.350 Solutiononp.646. x +1 5 x + x +3 5 x Exercise8.351 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +2 + a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a +2 Exercise8.352 Solutiononp.646. b +1 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 + b +2 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.353 a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F10 6.9738 Tf 11.213 3.923 Td [(a +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.354 Solutiononp.646. b +7 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.355 2 b +3 b +1 )]TJ/F10 6.9738 Tf 11.158 3.922 Td [(b )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 b +1 Exercise8.356 Solutiononp.646. 3 y +4 y +8 )]TJ/F7 6.9738 Tf 11.158 4.444 Td [(2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y +8

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588 CHAPTER8.RATIONALEXPRESSIONS Exercise8.357 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 + 3 a +5 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 Exercise8.358 Solutiononp.646. 8 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(15 x +7 x +2 Exercise8.359 7 2 x 2 + 1 6 x 3 Exercise8.360 Solutiononp.646. 2 3 x + 4 6 x 2 Exercise8.361 5 6 y 3 )]TJ/F7 6.9738 Tf 17.234 3.922 Td [(2 18 y 5 Exercise8.362 Solutiononp.646. 2 5 a 2 )]TJ/F7 6.9738 Tf 17.249 3.923 Td [(1 10 a 3 Exercise8.363 3 x +1 + 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.364 Solutiononp.646. 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 + 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.365 2 a a +1 )]TJ/F7 6.9738 Tf 14.216 3.923 Td [(3 a a +4 Exercise8.366 Solutiononp.646. 6 y y +4 + 2 y y +3 Exercise8.367 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 + x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.368 Solutiononp.646. x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +2 Exercise8.369 a +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F10 6.9738 Tf 11.213 3.923 Td [(a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.370 Solutiononp.646. y +1 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F10 6.9738 Tf 11.213 4.445 Td [(y +4 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.371 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 + x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.372 Solutiononp.647. y +2 y +1 y +6 + y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y +6 Exercise8.373 2 a +1 a +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(a +2 a +3 Exercise8.374 Solutiononp.647. 3 a +5 a +4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.375 2 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 + 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.376 Solutiononp.647. 4 a a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 + 3 a +1 Exercise8.377 3 y y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 y +12 )]TJ/F10 6.9738 Tf 14.313 4.444 Td [(y 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.378 Solutiononp.647. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 +6 x +8 + x +3 x 2 +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8

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589 Exercise8.379 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 a 2 +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 + a +2 a 2 +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.380 Solutiononp.647. b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b 2 +9 b +20 + b +4 b 2 + b )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Exercise8.381 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y 2 +4 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 )]TJ/F10 6.9738 Tf 22.332 4.444 Td [(y +3 y 2 +6 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 Exercise8.382 Solutiononp.647. x +3 x 2 +9 x +14 )]TJ/F10 6.9738 Tf 13.102 3.922 Td [(x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.383 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 + x +3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x +6 + 2 x x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +2 Exercise8.384 Solutiononp.647. 4 x x 2 +6 x +8 + 3 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Exercise8.385 y +2 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 )]TJ/F10 6.9738 Tf 20.346 4.445 Td [(y +3 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y +4 Exercise8.386 Solutiononp.647. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 a +18 + a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 )]TJ/F10 6.9738 Tf 18.377 3.923 Td [(a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.387 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y 2 +6 y + y +4 y 2 +5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.388 Solutiononp.647. a +1 a 3 +3 a 2 )]TJ/F10 6.9738 Tf 13.332 3.922 Td [(a +6 a 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a Exercise8.389 4 3 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 b )]TJ/F7 6.9738 Tf 21.705 3.922 Td [(2 6 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 b Exercise8.390 Solutiononp.647. 3 2 x 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 4 + )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 8 x 3 +24 x 2 Exercise8.391 x +2 12 x 3 + x +1 4 x 2 +8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 )]TJ/F10 6.9738 Tf 28.402 3.923 Td [(x +3 16 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(32 x +16 Exercise8.392 Solutiononp.647. 2 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 )]TJ/F10 6.9738 Tf 19.388 3.923 Td [(x +1 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 8 x 3 Exercise8.393 4+ 3 x +2 Exercise8.394 Solutiononp.647. 8+ 2 x +6 Exercise8.395 1+ 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise8.396 Solutiononp.647. 3+ 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.397 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+ 4 x x +5 Exercise8.398 Solutiononp.647. )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ 3 a a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.399 6 )]TJ/F7 6.9738 Tf 14.217 4.445 Td [(4 y y +2 Exercise8.400 Solutiononp.647. 2 x + x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +1 Exercise8.401 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y + 4 y 2 +2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y +3

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590 CHAPTER8.RATIONALEXPRESSIONS Exercise8.402 Solutiononp.647. x +2+ x 2 +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.403 b +6+ 2 b +5 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.404 Solutiononp.647. 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise8.405 4 y +5 y +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Exercise8.406 Solutiononp.647. 2 y 2 +11 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y +4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y Exercise8.407 5 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y +1 y 2 + y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise8.408 Solutiononp.647. 4 a 3 +2 a 2 + a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a 2 +11 a +28 +3 a Exercise8.409 2 x 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x + 6 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.410 Solutiononp.647. 5 m 6 )]TJ/F10 6.9738 Tf 6.227 0 Td [(m + 3 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.411 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a +7 8 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a + 2 a +1 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise8.412 Solutiononp.647. )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y +4 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y )]TJ/F7 6.9738 Tf 18.402 3.922 Td [(9 5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.413 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(m )]TJ/F7 6.9738 Tf 17.806 3.923 Td [(2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 8.6.12ExercisesForReview Exercise8.414 Solutiononp.647. Section2.7 Simplify )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 y 2 z 5 6 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 yz 2 : Exercise8.415 Section3.7 Write 6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b 4 c )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 c 3 sothatonlypositiveexponentsappear. Exercise8.416 Solutiononp.647. Section7.6 Constructthegraphof y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x +4 :

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591 Exercise8.417 Section8.4 Findtheproduct: x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 +6 x +5 x 2 +5 x +6 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 : Exercise8.418 Solutiononp.648. Section8.5 Replace N withtheproperquantity: x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = N x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x +10 : 8.7RationalEquations 7 8.7.1Overview RationalEquations TheLogicBehindTheProcess TheProcess ExtraneousSolutions 8.7.2RationalEquations 8.7.2.1RationalEquations Whenonerationalexpressionissetequaltoanotherrationalexpression,a rationalequation results. Someexamplesofrationalequationsarethefollowingexceptfornumber5: Example8.93 3 x 4 = 15 2 Example8.94 x +1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Example8.95 5 a 2 =10 Example8.96 3 x + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +1 = 6 5 x Example8.97 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +1 isarational expression ,notarationalequation. 7 Thiscontentisavailableonlineat.

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592 CHAPTER8.RATIONALEXPRESSIONS 8.7.3TheLogicBehindTheProcess Itseemsmostreasonablethatanequationwithoutanyfractionswouldbeeasiertosolvethananequation withfractions.Ourgoal,then,istoconvertanyrationalequationtoanequationthatcontainsnofractions. Thisiseasilydone. Todevelopthismethod,let'sconsidertherationalequation 1 6 + x 4 = 17 12 TheLCDis12.Weknowthatwecanmultiplybothsidesofanequationbythesamenonzeroquantity, sowe'llmultiplybothsidesbytheLCD,12. 12 1 6 + x 4 =12 17 12 Nowdistribute12toeachtermontheleftsideusingthedistributiveproperty. 12 1 6 +12 x 4 =12 17 12 Nowdividetoeliminatealldenominators. 2 1+3 x =17 2+3 x =17 Nowtherearenomorefractions,andwecansolvethisequationusingourprevioustechniquestoobtain 5asthesolution. 8.7.4TheProcess WehaveclearedtheequationoffractionsbymultiplyingbothsidesbytheLCD.Thisdevelopmentgenerates thefollowingrule. ClearinganEquationofFractions Toclearanequationoffractions,multiplybothsidesoftheequationbytheLCD. WhenmultiplyingbothsidesoftheequationbytheLCD,weusethedistributivepropertytodistribute theLCDtoeachterm.Thismeanswecansimplifytheaboverule. ClearinganEquationofFractions Toclearanequationoffractions,multiply every termonbothsidesoftheequationbytheLCD. Thecompletemethodforsolvingarationalequationis 1.Determineallthevaluesthatmustbeexcludedfromconsiderationbyndingthevaluesthatwill producezerointhedenominatorandthus,divisionbyzero.Theseexcludedvaluesarenotinthedomain oftheequationandarecallednondomainvalues. 2.CleartheequationoffractionsbymultiplyingeverytermbytheLCD. 3.Solvethisnonfractionalequationforthevariable.Checktoseeifanyofthesepotentialsolutionsare excludedvalues. 4.Checkthesolutionbysubstitution.

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593 8.7.5ExtraneousSolutions ExtraneousSolutions Potentialsolutionsthathavebeenexcludedbecausetheymakeanexpressionundenedorproduceafalse statementforanequationarecalled extraneoussolutions. Extraneoussolutionsarediscarded.Ifthere arenootherpotentialsolutions,theequationhasnosolution. 8.7.6SampleSetA Solvethefollowingrationalequations. Example8.98 3 x 4 = 15 2 : Sincethedenominatorsareconstants,therearenoexcludedvalues. Novaluesmustbeexcluded.TheLCDis4.Multiplyeachtermby4. 4 3 x 4 =4 15 2 3 x = 2 15 3 x =2 15 3 x =30 x =10 10isnotanexcludedvalue.Checkitasasolution. Check : 3 x 4 = 15 2 3 4 = 15 2 Isthiscorrect? 30 4 = 15 2 Isthiscorrect? 15 2 = 15 2 Yes,thisiscorrect. 10 isthesolution. Example8.99

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594 CHAPTER8.RATIONALEXPRESSIONS 4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 = 2 x +6 : 1and )]TJ/F71 11.9552 Tf 11.955 0 Td [(6arenondomainvalues.Excludethemfromconsideration. TheLCDis x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 x +6 : Multiplyeverytermby x )]TJ/F89 11.9552 Tf 11.956 0 Td [(1 x +6 : x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 x +6 4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 = x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 x +6 2 x +6 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 x +6 4 x )]TJ/F54 7.9701 Tf 6.586 0 Td [(1 = x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 x +6 2 x +6 4 x +6=2 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(1 Solvethisnonfractionalequation. 4 x +24=2 x )]TJ/F89 11.9552 Tf 11.955 0 Td [(2 2 x = )]TJ/F89 11.9552 Tf 9.298 0 Td [(26 x = )]TJ/F89 11.9552 Tf 9.298 0 Td [(13 )]TJ/F89 11.9552 Tf 9.298 0 Td [(13 isnotanexcludedvalue.Checkitasasolution. Check : 4 x )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 = 2 x +6 4 )]TJ/F54 7.9701 Tf 6.586 0 Td [(13 )]TJ/F54 7.9701 Tf 6.587 0 Td [(1 = 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(13+6 Isthiscorrect? 4 )]TJ/F54 7.9701 Tf 6.587 0 Td [(14 = 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(7 Isthiscorrect? 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(7 = 2 )]TJ/F54 7.9701 Tf 6.587 0 Td [(7 Yes,thisiscorrect. )]TJ/F89 11.9552 Tf 9.298 0 Td [(13 isthesolution. Example8.100 4 a a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 =2+ 16 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 : 4isanondomainvalue.Excludeitfromconsideration. TheLCDis a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 : Multiplyeverytermby a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 : a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 4 a a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 =2 a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4+ a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 16 a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 4 a a )]TJ/F54 7.9701 Tf 6.587 0 Td [(4 =2 a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4+ a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4 16 a )]TJ/F54 7.9701 Tf 6.586 0 Td [(4 4 a =2 a )]TJ/F89 11.9552 Tf 11.955 0 Td [(4+16 Solvethisnonfractionalequation. 4 a =2 a )]TJ/F89 11.9552 Tf 11.955 0 Td [(8+16 4 a =2 a +8 2 a =8 a =4 Thisvalue, a =4 ; hasbeenexcludedfromconsideration.Itisnottobeconsideredasasolution.Itisextraneous. Astherearenootherpotentialsolutionstoconsider,weconcludethatthisequationhas nosolution 8.7.7PracticeSetA Solvethefollowingrationalequations. Exercise8.419 Solutiononp.648. 2 x 5 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 6

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595 Exercise8.420 Solutiononp.648. 3 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 3 a +8 a +3 Exercise8.421 Solutiononp.648. 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 +2= y y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 8.7.8SampleSetB Solvethefollowingrationalequations. Example8.101 3 x + 4 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 4 x 2 + x +5 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x : Factoralldenominatorstondany excludedvaluesandtheLCD. 3 x + 4 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 4 x 2 + x +5 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Nondomainvaluesare0and1. Excludethemfromconsideration. TheLCDis x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Multiplyeach termby x x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 andsimplify. x x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 3 x + x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 4 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = x x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 4 x 2 + x +5 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+4 x x =4 x 2 + x +5 Solvethisnonfractionalequation toobtainthepotentialsolutions. 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+4 x 2 =4 x 2 + x +5 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3= x +5 2 x =8 x =4 4isnotanexcludedvalue.Checkitasasolution. Check : 3 x + 4 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 4 x 2 + x +5 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x 3 4 + 4 4 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 4 4 2 +4+5 16 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Isthiscorrect? 3 4 + 16 3 = 64+4+5 12 Isthiscorrect? 9 12 + 64 12 = 73 12 Isthiscorrect? 73 12 = 73 12 Yes,thisiscorrect. 4isthesolution. Thezero-factorpropertycanbeusedtosolvecertaintypesofrationalequations.Westudied thezero-factorpropertyinSection7.1,andyoumayrememberthatitstatesthatif a and b are realnumbersandthat a b =0 ; theneitherorboth a =0 or b =0 : Thezero-factorpropertyis usefulinsolvingthefollowingrationalequation.

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596 CHAPTER8.RATIONALEXPRESSIONS Example8.102 3 a 2 )]TJ/F7 6.9738 Tf 11.333 3.923 Td [(2 a =1 : Zeroisanexcludedvalue. TheLCDis a 2 Multiplyeach termby a 2 andsimplify. a 2 3 a 2 )]TJETq1 0 0 1 170.427 614.07 cm[]0 d 0 J 0.398 w 0 0 m 13.61 0 l SQBT/F8 9.9626 Tf 170.427 605.203 Td [( a 2 2 a =1 a 2 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a = a 2 Solvethisnonfractionalquadratic equation.Setitequaltozero. 0= a 2 +2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 0= a +3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ;a =1 Checktheseassolutions. Check : If a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3: 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 2 )]TJ/F7 6.9738 Tf 14.271 3.923 Td [(2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 =1 Isthiscorrect? 3 9 + 2 3 =1 Isthiscorrect? 1 3 + 2 3 =1 Isthiscorrect? 1=1 Yes,thisiscorrect. a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 checksandisasolution. If a =1: 3 2 )]TJ/F7 6.9738 Tf 11.158 3.922 Td [(2 1 =1 Isthiscorrect? 3 1 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 1 =1 Isthiscorrect? 1=1 Yes,thisiscorrect. a =1 checksandisasolution. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 and1arethesolutions. 8.7.9PracticeSetB Exercise8.422 Solutiononp.648. Solvetheequation a +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = a +1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : Exercise8.423 Solutiononp.648. Solvetheequation 1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 16.475 3.923 Td [(1 x +1 = 2 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 :

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597 8.7.10Section7.6Exercises Forthefollowingproblems,solvetherationalequations. Exercise8.424 Solutiononp.648. 32 x = 16 3 Exercise8.425 54 y = 27 4 Exercise8.426 Solutiononp.648. 8 y = 2 3 Exercise8.427 x 28 = 3 7 Exercise8.428 Solutiononp.648. x +1 4 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 Exercise8.429 a +3 6 = a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 Exercise8.430 Solutiononp.648. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 6 = y +1 4 Exercise8.431 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 8 = x +5 6 Exercise8.432 Solutiononp.648. a +6 9 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 6 =0 Exercise8.433 y +11 4 = y +8 10 Exercise8.434 Solutiononp.648. b +1 2 +6= b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 3 Exercise8.435 m +3 2 +1= m )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 5 Exercise8.436 Solutiononp.648. a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 2 +4= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise8.437 b +11 3 +8=6 Exercise8.438 Solutiononp.648. y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y +2 = y +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.439 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +2 Exercise8.440 Solutiononp.648. 3 m +1 2 m = 4 3 Exercise8.441 2 k +7 3 k = 5 4 Exercise8.442 Solutiononp.648. 4 x +2 =1 Exercise8.443 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 =1

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598 CHAPTER8.RATIONALEXPRESSIONS Exercise8.444 Solutiononp.648. a 3 + 10+ a 4 =6 Exercise8.445 k +17 5 )]TJ/F10 6.9738 Tf 11.158 3.923 Td [(k 2 =2 k Exercise8.446 Solutiononp.648. 2 b +1 3 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = 1 4 Exercise8.447 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a +4 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 9 Exercise8.448 Solutiononp.648. x x +3 )]TJ/F10 6.9738 Tf 16.257 3.922 Td [(x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = 10 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.449 3 y y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + 2 y y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 = 5 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 y +20 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 y +6 Exercise8.450 Solutiononp.649. 4 a a +2 )]TJ/F7 6.9738 Tf 14.271 3.922 Td [(3 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 a 2 + a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.451 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.452 Solutiononp.649. 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.453 3 x +4 + 5 x +4 = 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.454 Solutiononp.649. 2 y +2 + 8 y +2 = 9 y +3 Exercise8.455 4 a 2 +2 a = 3 a 2 + a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.456 Solutiononp.649. 2 b b +2 = 3 b 2 +6 b +8 Exercise8.457 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + 3 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 = 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 x +1 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x +4 Exercise8.458 Solutiononp.649. 4 x x +2 )]TJ/F10 6.9738 Tf 16.202 3.923 Td [(x x +1 = 3 x 2 +4 x +4 x 2 +3 x +2 Exercise8.459 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 )]TJ/F7 6.9738 Tf 18.32 3.922 Td [(4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +5 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.460 Solutiononp.649. )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +4 )]TJ/F7 6.9738 Tf 16.475 3.922 Td [(2 x +1 = 4 x +19 x 2 +5 x +4 Exercise8.461 2 x 2 + 1 x =1 Exercise8.462 Solutiononp.649. 6 y 2 )]TJ/F7 6.9738 Tf 11.318 3.923 Td [(5 y =1 Exercise8.463 12 a 2 )]TJ/F7 6.9738 Tf 11.333 3.923 Td [(4 a =1 Exercise8.464 Solutiononp.649. 20 x 2 )]TJ/F7 6.9738 Tf 11.432 3.923 Td [(1 x =1 Exercise8.465 12 y + 12 y 2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Exercise8.466 Solutiononp.649. 16 b 2 + 12 b =4

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599 Exercise8.467 1 x 2 =1 Exercise8.468 Solutiononp.649. 16 y 2 =1 Exercise8.469 25 a 2 =1 Exercise8.470 Solutiononp.649. 36 y 2 =1 Exercise8.471 2 x 2 + 3 x =2 Exercise8.472 Solutiononp.649. 2 a 2 )]TJ/F7 6.9738 Tf 11.333 3.923 Td [(5 a =3 Exercise8.473 2 x 2 + 7 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 Exercise8.474 Solutiononp.649. 4 a 2 + 9 a =9 Exercise8.475 2 x = 3 x +2 +1 Exercise8.476 Solutiononp.649. 1 x = 2 x +4 )]TJ/F7 6.9738 Tf 11.159 3.922 Td [(3 2 Exercise8.477 4 m )]TJ/F7 6.9738 Tf 17.806 3.922 Td [(5 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 =7 Exercise8.478 Solutiononp.649. 6 a +1 )]TJ/F7 6.9738 Tf 16.432 3.922 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 =5 Forthefollowingproblems,solveeachliteralequationforthedesignatedletter. Exercise8.479 V = GMm D for D: Exercise8.480 Solutiononp.649. PV = nrt for n: Exercise8.481 E = mc 2 for m: Exercise8.482 Solutiononp.649. P =2+ w for w: Exercise8.483 A = 1 2 h b + B for B: Exercise8.484 Solutiononp.649. A = P + rt for r: Exercise8.485 z = x )]TJETq1 0 0 1 122.22 177.109 cm[]0 d 0 J 0.339 w 0 0 m 4.518 0 l SQBT/F10 6.9738 Tf 122.22 172.921 Td [(x s for x: Exercise8.486 Solutiononp.649. F = S x 2 S y 2 for S y 2 : Exercise8.487 1 R = 1 E + 1 F for F: Exercise8.488 Solutiononp.649. K = 1 2 h s 1 + s 2 for s 2 :

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600 CHAPTER8.RATIONALEXPRESSIONS Exercise8.489 Q = 2 mn s + t for s: Exercise8.490 Solutiononp.649. V = 1 6 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 a 2 + h 2 for h 2 : Exercise8.491 I = E R + r for R: 8.7.11ExercisesForReview Exercise8.492 Solutiononp.649. Section3.7 Write )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 x 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 sothatonlypositiveexponentsappear. Exercise8.493 Section6.6 Factor x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 : Exercise8.494 Solutiononp.649. Section7.5 Supplythemissingword.Anslopeofalineisameasureofthe ofthe line. Exercise8.495 Section8.3 Findtheproduct. x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x 2 +6 x +9 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +8 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 : Exercise8.496 Solutiononp.649. Section8.6 Findthesum. 2 x x +1 + 1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 : 8.8Applications 8 8.8.1Overview TheFive-StepMethod 8.8.2TheFive-StepMethod Wearenowinapositiontostudysomeapplicationsofrationalequations.Someoftheseproblemswillhave practicalapplicationswhileothersareintendedaslogicdevelopers. Wewillapplytheve-stepmethodforsolvingwordproblems. Five-StepMethod 1.Representallunknownquantitiesintermsof x orsomeotherletter. 2.Translatetheverbalphrasestomathematicalsymbolsandformanequation. 3.Solvethisequation. 4.Checkthesolutionbysubstitutingtheresultintotheoriginalstatementoftheproblem. 5.Writetheconclusion. Remember,step1isveryimportant:always Introduceavariable. 8 Thiscontentisavailableonlineat.

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601 8.8.3SampleSetA Example8.103 Whenthesamenumberisaddedtothenumeratoranddenominatorofthefraction 3 5 ; theresult is 7 9 : Whatisthenumberthatisadded? Step1:Let x = thenumberbeingadded. Step2: 3+ x 5+ x = 7 9 : Step3: 3+ x 5+ x = 7 9 : Anexcludedvalueis )]TJ/F15 9.9626 Tf 9.963 0 Td [(5. TheLCDis9 + x Multiplyeachtermby9 + x : 9+ x 3+ x 5+ x =9+ x 7 9 9+ x =7+ x 27+9 x =35+7 x 2 x =8 x =4 Checkthispotentialsolution. Step4: 3+4 5+4 = 7 9 : Yes,thisiscorrect. Step5:Thenumberaddedis4. 8.8.4PracticeSetA Thesamenumberisaddedtothenumeratoranddenominatorofthefraction 4 9 : Theresultis 2 3 : Whatis thenumberthatisadded? Exercise8.497 Solutiononp.649. Step1:Let x = Step2: Step3: Step4: Step5:Thenumberaddedis .

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602 CHAPTER8.RATIONALEXPRESSIONS 8.8.5SampleSetB Example8.104 Twothirdsofanumberaddedtothereciprocalofthenumberyields 25 6 : Whatisthenumber? Step1:Let x =thenumber. Step2:Recallthatthereciprocalofanumber x isthenumber 1 x 2 3 x + 1 x = 25 6 Step3: 2 3 x + 1 x = 25 6 TheLCDis 6 x: Multiplyeachtermby 6 x: 6 x 2 3 x +6 x 1 x =6 x 25 6 4 x 2 +6=25 x Solvethisnonfractionalquadraticequationtoobtainthe potentialsolutions.Usethezero-factorproperty. 4 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 x +6=0 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6=0 x = 1 4 ; 6 Checkthesepotentialsolutions. Step4:Substitutingintotheoriginalequation,itcanbethatbothsolutionscheck. Step5:Therearetwosolutions: 1 4 and6. 8.8.6PracticeSetB Sevenhalvesofanumberaddedtothereciprocalofthenumberyields 23 6 : Whatisthenumber? Exercise8.498 Solutiononp.650. Step1:Let x = Step2: Step3: Step4: Step5:Thenumberis .

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603 8.8.7SampleSetC Example8.105 PersonA,workingalone,canpouraconcretewalkwayin6hours.PersonB,workingalone,can pourthesamewalkwayin4hours.Howlongwillittakebothpeopletopourtheconcretewalkway workingtogether? Step1:Let x =thenumberofhourstopourtheconcretewalkwayworkingtogethersincethis iswhatwe'relookingfor. Step2:IfpersonAcancompletethejobin6hours,Acancomplete 1 6 ofthejobin1hour. IfpersonBcancompletethejobin4hours,Bcancomplete 1 4 ofthejobin1hour. IfAandB,workingtogether,cancompletethejobin x hours,theycancomplete 1 x ofthejobin 1hour.Puttingthesethreefactsintoequationform,wehave 1 6 + 1 4 = 1 x Step3: 1 6 + 1 4 = 1 x : Anexcludedvalueis0. TheLCDis12 x .Multiplyeachtermby12 x 12 x 1 6 +12 x 1 4 =12 x 1 x 2 x +3 x =12 Solvethisnonfractional equationtoobtainthepotentialsolutions. 5 x =12 x = 12 5 or x =2 2 5 Checkthispotentialsolution. Step4: 1 6 + 1 4 = 1 x 1 6 + 1 4 = 1 12 5 : Isthiscorrect? 1 6 + 1 4 = 5 12 TheLCDis12.Isthiscorrect? 2 12 + 3 12 = 5 12 Isthiscorrect? 5 12 = 5 12 Isthiscorrect? Step5:Workingtogether,AandBcanpourtheconcretewalkwayin2 2 5 hours. 8.8.8PracticeSetC PersonA,workingalone,canpouraconcretewalkwayin9hours.PersonB,workingalone,canpourthe samewalkwayin6hours.Howlongwillittakebothpeopletopourtheconcretewalkwayworkingtogether? Exercise8.499 Solutiononp.650. Step1: Step2: Step3:

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604 CHAPTER8.RATIONALEXPRESSIONS Step4: Step5:Workingtogether,AandB 8.8.9SampleSetD Example8.106 Aninletpipecanllawatertankin12hours.Anoutletpipecandrainthetankin20hours.If bothpipesareopen,howlongwillittaketollthetank? Step1:Let x =thenumberofhoursrequiredtollthetank. Step2:Iftheinletpipecanllthetankin12hours,itcanll 1 12 ofthetankin1hour. Iftheoutletpipecandrainthetankin20hours,itcandrain 1 20 ofthetankin1hour. Ifbothpipesareopen,ittakes x hourstollthetank.So 1 x ofthetankwillbelledin1hour. Sincewaterisbeingaddedinletpipeandsubtractedoutletpipeweget 1 12 )]TJ/F7 6.9738 Tf 13.143 3.923 Td [(1 20 = 1 x Step3: 1 12 )]TJ/F7 6.9738 Tf 13.144 3.922 Td [(1 20 = 1 x : Anexcludedvalueis0.TheLCDis60 x Multiplyeachtermby60 x 60 x 1 12 )]TJ/F8 9.9626 Tf 9.962 0 Td [(60 x 1 20 =60 x 1 x 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x =60 Solvethisnonfractionalequationtoobtain thepotentialsolutions. 2 x =60 x =30 Checkthispotentialsolution. Step4: 1 12 )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(1 20 = 1 x 1 12 )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(1 20 = 1 30 : TheLCDis60.Isthiscorrect? 5 60 )]TJ/F7 6.9738 Tf 13.144 3.923 Td [(3 60 = 1 30 Isthiscorrect? 1 30 = 1 30 Yes,thisiscorrect. Step5:Withbothpipesopen,itwilltake30hourstollthewatertank. 8.8.10PracticeSetD Aninletpipecanllawatertankin8hoursandanoutletpipecandrainthetankin10hours.Ifboth pipesareopen,howlongwillittaketollthetank? Exercise8.500 Solutiononp.650. Step1: Step2: Step3:

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605 Step4: Step5: 8.8.11SampleSetE Example8.107 IttakespersonA3hourslongerthanpersonBtocompleteacertainjob.Workingtogether,both cancompletethejobin2hours.Howlongdoesittakeeachpersontocompletethejobworking alone? Step1:Let x =timerequiredforBtocompletethejobworkingalone.Then, x +3= time requiredforAtocompletethejobworkingalone. Step2: 1 x + 1 x +3 = 1 2 : Step3: 1 x + 1 x +3 = 1 2 : Thetwoexcludedvaluesare0and )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 : TheLCDis 2 x x +3 2 x x +3 1 x +2 x x +3 1 x +3 =2 x x +3 1 2 2 x +3+2 x = x x +3 2 x +6+2 x = x 2 +3 x Thisisaquadraticequationthatcan besolvedusingthezero-factorproperty. 4 x +6= x 2 +3 x x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6=0 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 x +2=0 x =3 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Checkthesepotentialsolutions. Step4:If x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; theequationchecks,butdoesnotevenmakephysicalsense. If x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 ,theequationchecks. x =3 and x +3=6 Step5:PersonBcandothejobin3hoursandpersonAcandothejobin6hours. 8.8.12PracticeSetE IttakespersonA4hourslessthanpersonBtocompleteacertaintask.Workingtogether,bothcancomplete thetaskin 8 3 hours.Howlongdoesittakeeachpersontocompletethetaskworkingalone? Exercise8.501 Solutiononp.650. Step1: Step2:

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606 CHAPTER8.RATIONALEXPRESSIONS Step3: Step4: Step5: 8.8.13SampleSetF Example8.108 Thewidthofarectangleis 1 3 itslength.Findthedimensionslengthandwidthiftheperimeter is16cm. Step1:Let x =length.Then, x 3 = width. Step2:Makeasketchoftherectangle. Theperimeterofagureisthetotallengtharoundthegure.

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607 x + x 3 + x + x 3 =16 2 x + 2 x 3 =16 Step3: 2 x + 2 x 3 =16 : TheLCDis3. 3 2 x +3 2 x 3 =3 16 6 x +2 x =48 8 x =48 x =6 Checkthispotentialsolution. Step4: 6+ 6 3 +6+ 6 3 =16 Isthiscorrect? 6+2+6+2=16 Isthiscorrect? 16=16 Yes,thisiscorrect. Since x =6 ; x 3 = 6 3 =2 Step5 : Thelength=6cmandthewidth=2cm. 8.8.14PracticeSetF Thewidthofarectangleis 1 12 itslength.Findthedimensionslengthandwidthiftheperimeteris78feet. Exercise8.502 Solutiononp.650. Step1: Step2: Step3: Step4: Step5: 8.8.15Exercises Forthefollowingproblems,solveusingtheve-stepmethod.

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608 CHAPTER8.RATIONALEXPRESSIONS Exercise8.503 Solutiononp.650. Whenthesamenumberisaddedtoboththenumeratoranddenominatorofthefraction 3 7 ; the resultis 2 3 : Whatisthenumber? Exercise8.504 Whenthesamenumberisaddedtoboththenumeratoranddenominatorofthefraction 5 8 ; the resultis 3 4 : Whatisthenumber? Exercise8.505 Solutiononp.650. Whenthesamenumberisaddedtoboththenumeratoranddenominatorofthefraction 3 8 ; the resultis 1 6 : Whatisthenumber? Exercise8.506 Whenthesamenumberisaddedtoboththenumeratoranddenominatorofthefraction 7 9 ; the resultis 2 3 : Whatisthenumber? Exercise8.507 Solutiononp.650. Whenthesamenumberissubtractedfromboththenumeratoranddenominatorof 1 10 ; theresult is 2 3 : Whatisthenumber? Exercise8.508 Whenthesamenumberissubtractedfromboththenumeratoranddenominatorof 3 4 ; theresultis 5 6 : Whatisthenumber? Exercise8.509 Solutiononp.650. Onethirdofanumberaddedtothereciprocalofnumberyields 13 6 : Whatisthenumber? Exercise8.510 Fourfthsofanumberaddedtothereciprocalofnumberyields 81 10 : Whatisthenumber? Exercise8.511 Solutiononp.650. Onehalfofanumberaddedtotwicethereciprocalofthenumberyields2.Whatisthenumber? Exercise8.512 Onefourthofanumberaddedtofourtimesthereciprocalofthenumberyields )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 3 : Whatisthe number? Exercise8.513 Solutiononp.650. Oneinletpipecanllatankin8hours.Anotherinletpipecanllthetankin5hours.Howlong doesittakebothpipesworkingtogethertollthetank? Exercise8.514 Onepipecandrainapoolin12hours.Anotherpipecandrainthepoolin15hours.Howlong doesittakebothpipesworkingtogethertodrainthepool? Exercise8.515 Solutiononp.650. Afaucetcanllabathroomsinkin1minute.Thedraincanemptythesinkin2minutes.Ifboth thefaucetanddrainareopen,howlongwillittaketollthesink? Exercise8.516 Afaucetcanllabathtubin 6 1 2 minutes.Thedraincanemptythetubin 8 1 3 minutes.Ifboththe faucetanddrainareopen,howlongwillittaketollthebathtub? Exercise8.517 Solutiononp.650. Aninletpipecanllatankin5hours.Anoutletpipecanemptythetankin4hours.Ifboth pipesareopen,canthetankbelled?Explain. Exercise8.518 Aninletpipecanllatankin a unitsoftime.Anoutletpipecanemptythetankin b unitsof time.Ifbothpipesareopen,howmanyunitsoftimearerequiredtollthetank?Arethereany restrictionson a and b ?

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609 Exercise8.519 Solutiononp.650. Adeliveryboy,workingalone,candeliverallhisgoodsin6hours.Anotherdeliveryboy,working alone,candeliverthesamegoodsin5hours.Howlongwillittaketheboystodeliverallthegoods workingtogether? Exercise8.520 ASpaceShuttleastronautcanperformacertainexperimentin2hours.AnotherSpaceShuttle astronautwhoisnotasfamiliarwiththeexperimentcanperformitin 2 1 2 hours.Workingtogether, howlongwillittakebothastronautstoperformtheexperiment? Exercise8.521 Solutiononp.650. Onepersoncancompleteatask8hourssoonerthananotherperson.Workingtogether,bothpeople canperformthetaskin3hours.Howmanyhoursdoesittakeeachpersontocompletethetask workingalone? Exercise8.522 Findtwoconsecutiveintegerssuchthattwothirdsofthesmallernumberaddedtotheotheryields 11. Exercise8.523 Solutiononp.650. Findtwoconsecutiveintegerssuchthatthreefourthsofthesmallernumberaddedtotheother yields29. Exercise8.524 Thewidthofarectangleis 2 5 itslength.Findthedimensionsiftheperimeteris42meters. Exercise8.525 Solutiononp.650. Thewidthofarectangleis 3 7 thelength.Findthedimensionsiftheperimeteris60feet. Exercise8.526 Twosidesofatrianglehavethesamelength.Thethirdsideistwiceaslongaseitheroftheother twosides.Theperimeterofthetriangleis56inches.Whatisthelengthofeachside? Exercise8.527 Solutiononp.650. Inatriangle,thesecondsideis3incheslongerthanrstside.Thethirdsideis 3 4 thelengthofthe secondside.Iftheperimeteris30inches,howlongiseachside? Exercise8.528 Thepressureduetosurfacetensioninasphericaldropofliquidisgivenby P = 2 T r ; where T is thesurfacetensionoftheliquidand r istheradiusofthedrop.Iftheliquidisabubble,ithastwo surfacesandthesurfacetensionisgivenby P = 2 T r + 2 T r = 4 T r aDeterminethepressureduetosurfacetensionwithinasoapbubbleofradius2inchesand surfacetension28. bDeterminetheradiusofabubbleifthepressureduetosurfacetensionis52andthesurface tensionis39. Exercise8.529 Solutiononp.650. Theequation 1 p + 1 q = 1 f relatesthedistance p ofanobjectfromalensandtheimagedistance q fromthelenstothefocallength f ofthelens.

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610 CHAPTER8.RATIONALEXPRESSIONS aDeterminethefocallengthofalensinwhichanobject10feetawayproducesanimage6 feetaway. bDeterminehowfaranobjectisfromalensifthefocallengthofthelensis6inchesandthe imagedistanceis10inches. cDeterminehowfaranimagewillbefromalensthathasafocallengthof 4 4 5 cmandtheobject is12cmawayfromthelens. Exercise8.530 PersonAcancompleteataskin4hours,personBcancompletethetaskin6hours,andpersonC cancompletethetaskin3hours.Ifallthreepeopleareworkingtogether,howlongwillittaketo completethetask? Exercise8.531 Solutiononp.650. Threeinletpipescanllastoragetankin4,6,and8hours,respectively.Howlongwillittakeall threepipestollthetank? Exercise8.532 Aninletpipecanllatankin10hours.Thetankhastwodrainpipes,eachofwhichcanempty thetankin30hours.Ifallthreepipesareopen,canthetankbelled?Ifso,howlongwillittake? Exercise8.533 Solutiononp.650. Aninletpipecanllatankin4hours.Thetankhasthreedrainpipes.Twoofthedrainpipes canemptythetankin12hours,andthethirdcanemptythetankin20hours.Ifallfourpipesare open,canthetankbelled?Ifso,howlongwillittake? 8.8.16ExercisesForReview Exercise8.534 Section6.8 Factor 12 a 2 +13 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 : Exercise8.535 Solutiononp.650. Section7.5 Findtheslopeofthelinepassingthroughthepoints ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 and ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 : Exercise8.536 Section8.4 Findthequotient: 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(24 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 : Exercise8.537 Solutiononp.650. Section8.6 Findthedierence: x +2 x 2 +5 x +6 )]TJ/F10 6.9738 Tf 20.405 3.923 Td [(x +1 x 2 +4 x +3 : Exercise8.538 Section8.7 Solvetheequation 9 2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 : 8.9ComplexRationalExpressions 9 8.9.1Overview SimpleAndComplexFractions TheCombine-DivideMethod TheLCD-Multiply-DivideMethod 9 Thiscontentisavailableonlineat.

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611 8.9.2SimpleAndComplexFractions SimpleFraction InSectionSection8.2wesawthata simplefraction wasafractionoftheform P Q ; where P and Q are polynomialsand Q 6 =0 ComplexFraction A complexfraction isafractioninwhichthenumeratorordenominator,orboth,isafraction.The fractions 8 15 2 3 and 1 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x 1 )]TJ/F6 4.9813 Tf 9.643 2.677 Td [(1 x 2 areexamplesofcomplexfractions,ormoregenerally,complexrationalexpressions. Therearetwomethodsforsimplifyingcomplexrationalexpressions:thecombine-dividemethodandthe LCD-multiply-dividemethod. 8.9.3TheCombine-DivideMethod 1.Ifnecessary,combinethetermsofthenumeratortogether. 2.Ifnecessary,combinethetermsofthedenominatortogether. 3.Dividethenumeratorbythedenominator. 8.9.4SampleSetA Simplifyeachcomplexrationalexpression. Example8.109 x 3 8 x 5 12 Steps1and2arenotnecessarysoweproceedwithstep3. x 3 8 x 5 12 = x 3 8 12 x 5 = x 3 2 3 x = 3 2 x 2

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612 CHAPTER8.RATIONALEXPRESSIONS Example8.110 1 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x 1 )]TJ/F6 4.9813 Tf 9.643 2.677 Td [(1 x 2 Step1:Combinethetermsofthenumerator:LCD = x 1 )]TJ/F7 6.9738 Tf 11.431 3.923 Td [(1 x = x x )]TJ/F7 6.9738 Tf 11.432 3.923 Td [(1 x = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x Step2:Combinethetermsofthedenominator:LCD = x 2 1 )]TJ/F7 6.9738 Tf 13.375 3.923 Td [(1 x 2 = x 2 x 2 )]TJ/F7 6.9738 Tf 13.375 3.923 Td [(1 x 2 = x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x 2 Step3:Dividethenumeratorbythedenominator. x )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 x x 2 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 x 2 = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x x 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x x x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = x x +1 Thus, 1 )]TJ/F6 4.9813 Tf 7.698 2.677 Td [(1 x 1 )]TJ/F6 4.9813 Tf 9.643 2.677 Td [(1 x 2 = x x +1

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613 Example8.111 2 )]TJ/F6 4.9813 Tf 7.423 2.677 Td [(13 m )]TJ/F6 4.9813 Tf 10.73 2.677 Td [(7 m 2 2+ 3 m + 1 m 2 Step1:Combinethetermsofthenumerator:LCD = m 2 2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(13 m )]TJ/F7 6.9738 Tf 14.651 3.923 Td [(7 m 2 = 2 m 2 m 2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(13 m m 2 )]TJ/F7 6.9738 Tf 14.652 3.923 Td [(7 m 2 = 2 m 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(13 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 m 2 Step2:Combinethetermsofthedenominator:LCD = m 2 2+ 3 m + 1 m 2 = 2 m 2 m 2 + 3 m m 2 + 1 m 2 = 2 m 2 +3 m +1 m 2 Step3:Dividethenumeratorbythedenominator. 2 m 2 )]TJ/F6 4.9813 Tf 5.396 0 Td [(13 m )]TJ/F6 4.9813 Tf 5.397 0 Td [(7 m 2 2 m 2 +3 m )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 m 2 = 2 m 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(13 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 m 2 m 2 2 m 2 +3 m +1 = m +1 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 m 2 m 2 m +1 m +1 = m )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 m +1 Thus, 2 )]TJ/F6 4.9813 Tf 7.423 2.677 Td [(13 m )]TJ/F6 4.9813 Tf 10.73 2.677 Td [(7 m 2 2+ 3 m + 1 m 2 = m )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 m +1 8.9.5PracticeSetA Usethecombine-dividemethodtosimplifyeachexpression. Exercise8.539 Solutiononp.650. 27 x 2 6 15 x 3 8 Exercise8.540 Solutiononp.650. 3 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x 3+ 1 x Exercise8.541 Solutiononp.651. 1+ x y x )]TJ/F9 4.9813 Tf 7.422 3.22 Td [(y 2 x Exercise8.542 Solutiononp.651. m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3+ 2 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(4+ 3 m Exercise8.543 Solutiononp.651. 1+ 1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 )]TJ/F6 4.9813 Tf 12.092 2.678 Td [(1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(1

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614 CHAPTER8.RATIONALEXPRESSIONS 8.9.6TheLCD-Multiply-DivideMethod 1.FindtheLCDofalltheterms. 2.MultiplythenumeratoranddenominatorbytheLCD. 3.Reduceifnecessary. 8.9.7SampleSetB Simplifyeachcomplexfraction. Example8.112 1 )]TJ/F6 4.9813 Tf 9.6 2.677 Td [(4 a 2 1+ 2 a Step1:TheLCD = a 2 Step2:Multiplyboththenumeratoranddenominatorby a 2 a 2 1 )]TJ/F6 4.9813 Tf 9.601 2.677 Td [(4 a 2 a 2 1+ 2 a = a 2 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 2 4 a 2 a 2 1+ a 2 2 a = a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 a 2 +2 a Step3:Reduce. a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a 2 +2 a = a +2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a a +2 = a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a Thus, 1 )]TJ/F6 4.9813 Tf 9.6 2.678 Td [(4 a 2 1+ 2 a = a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a

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615 Example8.113 1 )]TJ/F6 4.9813 Tf 7.699 2.678 Td [(5 x )]TJ/F6 4.9813 Tf 9.643 2.678 Td [(6 x 2 1+ 6 x + 5 x 2 Step 1: TheLCDis x 2 : Step 2: Multiplythenumeratoranddenominatorby x 2 : x 2 1 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(5 x )]TJ/F6 4.9813 Tf 9.643 2.677 Td [(6 x 2 x 2 1+ 6 x + 5 x 2 = x 2 1 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 5 x )]TJETq1 0 0 1 317.764 578.792 cm[]0 d 0 J 0.339 w 0 0 m 11.519 0 l SQBT/F7 6.9738 Tf 317.764 572.377 Td [( x 2 6 x 2 x 2 1+ x 6 x + x 2 5 x 2 = x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x 2 +6 x +5 Step 3: Reduce. x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x 2 +6 x +5 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +1 x +5 x +1 = x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x +5 Thus, 1 )]TJ/F6 4.9813 Tf 7.699 2.678 Td [(5 x )]TJ/F6 4.9813 Tf 9.643 2.678 Td [(6 x 2 1+ 6 x + 5 x 2 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +5 8.9.8PracticeSetB ThefollowingproblemsarethesameproblemsastheproblemsinPracticeSetA.Simplifytheseexpressions usingtheLCD-multiply-dividemethod.ComparetheanswerstotheanswersproducedinPracticeSetA. Exercise8.544 Solutiononp.651. 27 x 2 6 15 x 3 8 Exercise8.545 Solutiononp.651. 3 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x 3+ 1 x Exercise8.546 Solutiononp.651. 1+ x y x )]TJ/F9 4.9813 Tf 7.422 3.22 Td [(y 2 x Exercise8.547 Solutiononp.651. m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3+ 2 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(4+ 3 m Exercise8.548 Solutiononp.651. 1+ 1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 )]TJ/F6 4.9813 Tf 12.092 2.677 Td [(1 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 8.9.9Exercises Forthefollowingproblems,simplifyeachcomplexrationalexpression. Exercise8.549 Solutiononp.651. 1+ 1 4 1 )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 4 Exercise8.550 1 )]TJ/F6 4.9813 Tf 7.422 2.677 Td [(1 3 1+ 1 3

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616 CHAPTER8.RATIONALEXPRESSIONS Exercise8.551 Solutiononp.651. 1 )]TJ/F6 4.9813 Tf 7.636 2.678 Td [(1 y 1+ 1 y Exercise8.552 a + 1 x a )]TJ/F6 4.9813 Tf 7.699 2.678 Td [(1 x Exercise8.553 Solutiononp.651. a b + c b a b )]TJ/F9 4.9813 Tf 7.422 2.677 Td [(c b Exercise8.554 5 m + 4 m 5 m )]TJ/F6 4.9813 Tf 8.786 2.677 Td [(4 m Exercise8.555 Solutiononp.651. 3+ 1 x 3 x +1 x 2 Exercise8.556 1+ x x + y 1 )]TJ/F9 4.9813 Tf 11.891 2.678 Td [(x x + y Exercise8.557 Solutiononp.651. 2+ 5 a +1 2 )]TJ/F6 4.9813 Tf 11.911 2.678 Td [(5 a +1 Exercise8.558 1 )]TJ/F6 4.9813 Tf 12.049 2.677 Td [(1 a )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1+ 1 a )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 Exercise8.559 Solutiononp.651. 4 )]TJ/F6 4.9813 Tf 10.73 2.677 Td [(1 m 2 2+ 1 m Exercise8.560 9 )]TJ/F6 4.9813 Tf 9.643 2.677 Td [(1 x 2 3 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x Exercise8.561 Solutiononp.651. k )]TJ/F6 4.9813 Tf 7.656 2.677 Td [(1 k k +1 k Exercise8.562 m m +1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m +1 2 Exercise8.563 Solutiononp.651. 2 xy 2 x )]TJ/F9 4.9813 Tf 5.396 0 Td [(y )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 2 x )]TJ/F9 4.9813 Tf 5.396 0 Td [(y 3 Exercise8.564 1 a + b )]TJ/F6 4.9813 Tf 11.932 2.677 Td [(1 a )]TJ/F9 4.9813 Tf 5.396 0 Td [(b 1 a + b + 1 a )]TJ/F9 4.9813 Tf 5.396 0 Td [(b Exercise8.565 Solutiononp.651. 5 x +3 )]TJ/F6 4.9813 Tf 12.092 2.677 Td [(5 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(3 5 x +3 + 5 x )]TJ/F6 4.9813 Tf 5.396 0 Td [(3 Exercise8.566 2+ 1 y +1 1 y + 2 3 Exercise8.567 Solutiononp.651. 1 x 2 )]TJ/F6 4.9813 Tf 9.58 2.678 Td [(1 y 2 1 x + 1 y Exercise8.568 1+ 5 x + 6 x 2 1 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x )]TJ/F6 4.9813 Tf 7.948 2.677 Td [(12 x 2 Exercise8.569 Solutiononp.651. 1+ 1 y )]TJ/F6 4.9813 Tf 9.58 2.677 Td [(2 y 2 1+ 7 y + 10 y 2

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617 Exercise8.570 3 n m )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 )]TJ/F9 4.9813 Tf 7.423 2.677 Td [(m n 3 n m +4+ m n Exercise8.571 Solutiononp.651. x )]TJ/F6 4.9813 Tf 13.787 2.677 Td [(4 3 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 1 )]TJ/F6 4.9813 Tf 7.422 3.082 Td [(2 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 3 x )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 Exercise8.572 y x + y )]TJ/F9 4.9813 Tf 12.029 2.677 Td [(x x )]TJ/F9 4.9813 Tf 5.396 0 Td [(y x x + y + y x )]TJ/F9 4.9813 Tf 5.397 0 Td [(y Exercise8.573 Solutiononp.651. a a )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 )]TJ/F9 4.9813 Tf 11.677 2.677 Td [(a a +2 2 a a )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 + a 2 a +2 Exercise8.574 3 )]TJ/F7 6.9738 Tf 22.781 3.923 Td [(2 1 )]TJ/F6 4.9813 Tf 13.041 2.677 Td [(1 m +1 Exercise8.575 Solutiononp.651. x )]TJ/F6 4.9813 Tf 13.287 2.677 Td [(1 1 )]TJ/F6 4.9813 Tf 6.869 1.93 Td [(1 x x + 1 1+ 1 x Exercise8.576 Inelectricitytheory,whentworesistorsofresistance R 1 and R 2 ohmsareconnectedinparallel, thetotalresistance R is R = 1 1 R 1 + 1 R 2 Writethiscomplexfractionasasimplefraction. Exercise8.577 Solutiononp.651. AccordingtoEinstein'stheoryofrelativity,twovelocities v 1 and v 2 arenotaddedaccordingto v = v 1 + v 2 ,butratherby v = v 1 + v 2 1+ v 1 v 2 c 2 Writethiscomplexfractionasasimplefraction. Einstein'sformulaisreallyonlyapplicaleforvelocitiesnearthespeedoflight c =186 ; 000 milespersecond : Atverymuchlowervelocities,suchas500milesperhour, theformula v = v 1 + v 2 providesanextremelygoodapproximation. 8.9.10ExercisesForReview Exercise8.578 Section3.3 Supplythemissingword.Absolutevaluespeakstothequestionofhow andnotwhichway. Exercise8.579 Solutiononp.651. Section4.7 Findtheproduct. x +4 2 : Exercise8.580 Section6.6 Factor x 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 4 : Exercise8.581 Solutiononp.651. Section8.7 Solvetheequation 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 16.475 3.922 Td [(5 x +3 =0 : Exercise8.582 Section8.8 Oneinletpipecanllatankin10minutes.Anotherinletpipecanllthesame tankin4minutes.Howlongdoesittakebothpipesworkingtogethertollthetank?

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618 CHAPTER8.RATIONALEXPRESSIONS 8.10DividingPolynomials 10 8.10.1Overview DividingaPolynomialbyaMonomial TheProcessofDivision ReviewofSubtractionofPolynomials DividingaPolynomialbyaPolynomial 8.10.2DividingAPolynomialByAMonomial Thefollowingexamplesillustratehowtodivideapolynomialbyamonomial.Thedivisionprocessisquite simpleandisbasedonadditionofrationalexpressions. a c + b c = a + b c Turningthisequationaroundweget a + b c = a c + b c Nowwesimplydivide c into a ,and c into b .Thisshouldsuggestarule. DividingaPolynomialByaMonomial Todivideapolynomialbyamonomial,divideeverytermofthepolynomialbythemonomial. 8.10.3SampleSetA Example8.114 3 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 x : Divideeverytermof 3 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 by x: 3 x 2 x + x x )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(11 x =3 x +1 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(11 x Example8.115 8 a 3 +4 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 a +9 2 a 2 : Divideeverytermof 8 a 3 +4 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 a +9 by 2 a 2 : 8 a 3 2 a 2 + 4 a 2 2 a 2 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(16 a 2 a 2 + 9 2 a 2 =4 a +2 )]TJ/F7 6.9738 Tf 11.334 3.923 Td [(8 a + 9 2 a 2 Example8.116 4 b 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 b 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 b +5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 2 : Divideeverytermof 4 b 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 b 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b +5 by )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 b 2 : 4 b 6 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 2 )]TJ/F7 6.9738 Tf 14.272 3.923 Td [(9 b 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 2 )]TJ/F7 6.9738 Tf 16.216 3.923 Td [(2 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b 2 + 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 b 2 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(b 4 + 9 4 b 2 + 1 2 b )]TJ/F7 6.9738 Tf 14.854 3.923 Td [(5 4 b 2 8.10.4PracticeSetA Performthefollowingdivisions. Exercise8.583 Solutiononp.652. 2 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x Exercise8.584 Solutiononp.652. 3 x 3 +4 x 2 +10 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x 2 10 Thiscontentisavailableonlineat.

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619 Exercise8.585 Solutiononp.652. a 2 b +3 ab 2 +2 b ab Exercise8.586 Solutiononp.652. 14 x 2 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 xy 7 xy Exercise8.587 Solutiononp.652. 10 m 3 n 2 +15 m 2 n 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(20 mn )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 m 8.10.5TheProcessOfDivision InSectionSection8.3westudiedthemethodofreducingrationalexpressions.Forexample,weobserved howtoreduceanexpressionsuchas x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Ourmethodwastofactorboththenumeratoranddenominator,thendivideoutcommonfactors. x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x +2 x +1 Whenthenumeratoranddenominatorhavenofactorsincommon,thedivisionmaystilloccur,butthe processisalittlemoreinvolvedthanmerelyfactoring.Themethodofdividingonepolynomialbyanotheris muchthesameasthatofdividingonenumberbyanother.First,we'llreviewthestepsindividingnumbers. 1. 35 8 : Wearetodivide35by8. 2. Wetry4,since32dividedby8is4. 3. Multiply4and8. 4. Subtract32from35. 5. Sincetheremainder3islessthanthedivisor8,wearedonewiththe32division. 6. 4 3 8 : Thequotientisexpressedasamixednumber. Theprocesswastodivide,multiply,andsubtract. 8.10.6ReviewOfSubtractionOfPolynomials Averyimportantstepintheprocessofdividingonepolynomialbyanotherissubtractionofpolynomials. Let'sreviewtheprocessofsubtractionbyobservingafewexamples. 1.Subtract x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 from x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5; thatis,nd x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 )]TJ/F8 9.9626 Tf 9.963 0 Td [( x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : Since x )]TJ/F8 9.9626 Tf 9.21 0 Td [(2 isprecededbyaminussign,removetheparentheses,changethesignofeachterm,thenadd. x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 )]TJ/F8 9.9626 Tf 9.409 0 Td [( x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 )]TJ/F11 9.9626 Tf 7.748 0 Td [(x +2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3

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620 CHAPTER8.RATIONALEXPRESSIONS Theresultis )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 : 2.Subtract x 3 +3 x 2 from x 3 +4 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : Since x 3 +3 x 2 isprecededbyaminussign,removetheparentheses,changethesignofeachterm,then add. x 3 +4 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 3 +3 x 2 = x 3 +4 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 2 x 2 + x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Theresultisx 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : 3.Subtract x 2 +3 x from x 2 +1 : Wecanwrite x 2 +1 as x 2 +0 x +1 : x 2 +1 )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 +3 x = x 2 +0 x +1 )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x 2 +3 x = x 2 +0 x +1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +1 8.10.7DividingAPolynomialByAPolynomial Nowwe'llobservesomeexamplesofdividingonepolynomialbyanother.Theprocessisthesameasthe processusedwithwholenumbers:divide,multiply,subtract,divide,multiply,subtract,.... Thedivision,multiplication,andsubtractiontakeplaceonetermatatime.Theprocessisconcluded whenthepolynomialremainderisoflesserdegreethanthepolynomialdivisor. 8.10.8SampleSetB Performthedivision. Example8.117 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 : Wearetodivide x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 by x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : 1 )]TJ/F7 6.9738 Tf 16.53 3.923 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Thus, x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 =1 )]TJ/F7 6.9738 Tf 16.53 3.923 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2

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621 Example8.118 x 3 +4 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +3 : Wearetodivide x 3 +4 x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 by x +3 :

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622 CHAPTER8.RATIONALEXPRESSIONS

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623 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2+ 5 x +3 Thus, x 3 +4 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +3 = x 2 + x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+ 5 x +3 8.10.9PracticeSetB Performthefollowingdivisions. Exercise8.588 Solutiononp.652. x +6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.589 Solutiononp.652. x 2 +2 x +5 x +3 Exercise8.590 Solutiononp.652. x 3 + x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +8 Exercise8.591 Solutiononp.652. x 3 + x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +1 x 2 +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 8.10.10SampleSetC Example8.119 Divide 2 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x +1 by x +6 : 2 x 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +1 x +6 Noticethatthe x 2 terminthenumeratorismissing. Wecanavoidanyconfusionbywriting 2 x 3 +0 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x +6 Divide,multiply,andsubtract.

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624 CHAPTER8.RATIONALEXPRESSIONS 2 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 x +6 =2 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 x +68 )]TJ/F7 6.9738 Tf 12.503 3.922 Td [(407 x +6 8.10.11PracticeSetC Performthefollowingdivisions. Exercise8.592 Solutiononp.652. x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 Exercise8.593 Solutiononp.652. 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.594 Solutiononp.652. x 3 +2 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.595 Solutiononp.652. 6 x 3 +5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 x +3 8.10.12Exercises Forthefollowingproblems,performthedivisions. Exercise8.596 Solutiononp.652. 6 a +12 2

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625 Exercise8.597 12 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 3 Exercise8.598 Solutiononp.652. 8 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.599 21 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.600 Solutiononp.652. 3 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.601 4 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y 2 y Exercise8.602 Solutiononp.652. 9 a 2 +3 a 3 a Exercise8.603 20 x 2 +10 x 5 x Exercise8.604 Solutiononp.652. 6 x 3 +2 x 2 +8 x 2 x Exercise8.605 26 y 3 +13 y 2 +39 y 13 y Exercise8.606 Solutiononp.652. a 2 b 2 +4 a 2 b +6 ab 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 ab ab Exercise8.607 7 x 3 y +8 x 2 y 3 +3 xy 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 xy xy Exercise8.608 Solutiononp.652. 5 x 3 y 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 x 2 y 2 +20 xy )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 xy Exercise8.609 4 a 2 b 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 ab 4 +12 ab 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 ab 2 Exercise8.610 Solutiononp.652. 6 a 2 y 2 +12 a 2 y +18 a 2 24 a 2 Exercise8.611 3 c 3 y 3 +99 c 3 y 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 c 3 y 5 3 c 3 y 3 Exercise8.612 Solutiononp.652. 16 ax 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(20 ax 3 +24 ax 4 6 a 4 Exercise8.613 21 ay 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 ay 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 ay 6 ay 2 Exercise8.614 Solutiononp.652. )]TJ/F7 6.9738 Tf 6.226 0 Td [(14 b 2 c 2 +21 b 3 c 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(28 c 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 a 2 c 3 Exercise8.615 )]TJ/F7 6.9738 Tf 6.226 0 Td [(30 a 2 b 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(35 a 2 b 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(25 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 b 3 Exercise8.616 Solutiononp.652. x +6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.617 y +7 y +1 Exercise8.618 Solutiononp.652. x 2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x +4 x +2

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626 CHAPTER8.RATIONALEXPRESSIONS Exercise8.619 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +1 Exercise8.620 Solutiononp.653. x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +3 x +1 Exercise8.621 x 2 +5 x +5 x +5 Exercise8.622 Solutiononp.653. x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +1 Exercise8.623 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +2 Exercise8.624 Solutiononp.653. y 2 +4 y +2 Exercise8.625 x 2 +36 x +6 Exercise8.626 Solutiononp.653. x 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +1 Exercise8.627 a 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 a +2 Exercise8.628 Solutiononp.653. x 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.629 a 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.630 Solutiononp.653. x 3 +3 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.631 a 3 +2 a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a +1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.632 Solutiononp.653. a 3 + a +6 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.633 x 3 +2 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.634 Solutiononp.653. y 3 +3 y 2 +4 y +2 Exercise8.635 y 3 +5 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.636 Solutiononp.653. x 3 +3 x 2 x +3 Exercise8.637 a 2 +2 a a +2 Exercise8.638 Solutiononp.653. x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.639 a 2 +5 a +4 a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise8.640 Solutiononp.653. 2 y 2 +5 y +3 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4

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627 Exercise8.641 3 a 2 +4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a 2 +3 a +3 Exercise8.642 Solutiononp.653. 2 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +4 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.643 3 a 2 +4 a +2 3 a +4 Exercise8.644 Solutiononp.653. 6 x 2 +8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 x +4 Exercise8.645 20 y 2 +15 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 4 y +3 Exercise8.646 Solutiononp.653. 4 x 3 +4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.647 9 a 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 a 2 +8 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.648 Solutiononp.653. 4 x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 3 +2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.649 3 y 4 +9 y 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 y +4 y +3 Exercise8.650 Solutiononp.653. 3 y 2 +3 y +5 y 2 + y +1 Exercise8.651 2 a 2 +4 a +1 a 2 +2 a +3 Exercise8.652 Solutiononp.653. 8 z 6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 z 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 z 4 +8 z 3 +3 z 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 z 2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.653 9 a 7 +15 a 6 +4 a 5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a 4 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a 3 +12 a 2 + a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 3 a +1 Exercise8.654 Solutiononp.653. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 x 5 +5 x 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x +5 Exercise8.655 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(6 a 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 a 2 + a +4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 8.10.13ExercisesForReview Exercise8.656 Solutiononp.653. Section8.4 Findtheproduct. x 2 +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 2 x +6 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 : Exercise8.657 Section8.6 Findthesum. x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x +5 + x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 : Exercise8.658 Solutiononp.653. Section8.7 Solvetheequation 1 x +3 + 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 = 1 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 : Exercise8.659 Section8.8 Whenthesamenumberissubtractedfromboththenumeratoranddenominatorof 3 10 ,theresultis 1 8 .Whatisthenumberthatissubtracted? Exercise8.660 Solutiononp.653. Section8.9 Simplify 1 x +5 4 x 2 )]TJ/F6 4.9813 Tf 5.397 0 Td [(25 :

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628 CHAPTER8.RATIONALEXPRESSIONS 8.11SummaryofKeyConcepts 11 8.11.1SummaryOfKeyConcepts RationalExpressionSection8.2 A rationalexpression isanalgebraicexpressionthatcanbewrittenasthequotientoftwopolynomials. Anexampleofarationalexpressionis x 2 +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 DomainofaRationalExpressionSection8.2 The domain ofarationalexpressionisthecollectionofvaluesforwhichtheraticlnalexpressionisdened. Thesevaluescanbefoundbydeterminingthevaluesthatwillnotproducezerointhedenominatorofthe expression. Thedomainof x +6 x +8 isthecollectionofallnumbersexcept )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 EqualityPropertyofFractionSection8.2 1.If a b = c d ,then ad = bc 2.If ad = bc ,then a b = c d NegativePropertyofFractionsSection8.2 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a b = a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b = )]TJ/F10 6.9738 Tf 8.944 3.923 Td [(a b ReducingaRationalExpressionSection8.3 1.Factorthenumeratoranddenominatorcompletely. 2.Dividethenumeratoranddenominatorbyanyfactorstheyhaveincommon. CommonCancellingErrorSection8.3 x +4 x +7 6 = x +4 x +7 6 = 4 7 Since x isnotacommonfactor,itcannotbecancelled. MultiplyingRationalExpressionsSection8.4 1.Factorallnumeratorsanddenominators. 2.Reducetolowesttermsrstbydividingoutallcommonfactors. 3.Multiplynumeratorstogether. 4.Multiplydenominatorstogether. Itwillbemoreconvenienttoleavethedenominatorinfactoredform. DivisionofRationalExpressionsSection8.4 P Q R S = P Q S R = P S Q R BuildingRationalExpressionsSection8.5 P Q b b = Pb Qb Buildingrationalexpressionsisexactlytheoppositeofreducingrationalexpressions.Itisoftenusefulinaddingorsubtractingrationalexpressions. Thebuildingfactormaybedeterminedbydividingtheoriginaldenominatorintothenewdenominator. Thequotientwillbethebuildingfactor.Itisthisfactorthatwillmultiplytheoriginalnumerator. LeastCommonDenominatorLCDSection8.5 TheLCDisthepolynomialofleastdegreedivisiblebyeachdenominator.Itisfoundasfollows: 1.Factoreachdenominator.Useexponentsforrepeatedfactors. 11 Thiscontentisavailableonlineat.

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629 2.Writeeach dierent factorthatappears.Ifafactorappearsmorethanonce,useonlythefactorwith thehighestexponent. 3.TheLCDistheproductofthefactorswritteninstep2. FundamentalRuleforAddingorSubtractingRationalExpressionsSection8.6 Toaddorsubtractrationalexpressionsconveniently,theyshouldhavethesamedenominator. AddingandSubtractingRationalExpressionsSection8.6 a c + b c = a + b c and a c )]TJ/F10 6.9738 Tf 11.186 3.923 Td [(b c = a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b c Notethatwecombine only thenumerators. RationalEquationSection8.7 A rationalequation isastatementthattworationalexpressionsareequal. ClearinganEquationofFractionsSection8.7 Toclearanequationoffractions,multiplybothsidesoftheequationbytheLCD.Thisamountstomultiplying everytermbytheLCD. SolvingaRationalEquationSection8.7 1.Determineallvaluesthatmustbeexcludedassolutionsbyndingthevaluesthatproducezerointhe denominator. 2.CleartheequationoffractionsbymultiplyingeverytermbytheLCD. 3.Solvethisnonfractionalequationforthevariable.Checktoseeifanyofthesepotentialsolutionsare excludedvalues. 4.Checkthesolutionbysubstitution. ExtraneousSolutionSection8.7 Apotentialsolutionthathasbeenexcludedbecauseitcreatesanundenedexpressionperhaps,division byzeroiscalledan extraneoussolution. 8.12ExerciseSupplement 12 8.12.1ExerciseSupplement 8.12.1.1RationalExpressionsSection8.2 Forthefollowingproblems,ndthedomainofeachrationalexpression. Exercise8.661 Solutiononp.653. 9 x +4 Exercise8.662 10 x x +6 Exercise8.663 Solutiononp.653. x +1 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise8.664 2 a +3 7 a +5 Exercise8.665 Solutiononp.653. 3 m 2 m m )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.666 5 r +6 9 r r +1 12 Thiscontentisavailableonlineat.

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630 CHAPTER8.RATIONALEXPRESSIONS Exercise8.667 Solutiononp.653. s s s +8 s +7 Exercise8.668 )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x +18 Exercise8.669 Solutiononp.654. )]TJ/F10 6.9738 Tf 6.226 0 Td [(y +5 12 y 2 +28 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.670 16 12 a 3 +21 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a Forthefollowingproblems,showthatthefractionsareequivalent. Exercise8.671 Solutiononp.654. )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 5 ; )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(4 5 Exercise8.672 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 8 ; )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(3 8 Exercise8.673 Solutiononp.654. )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 10 ; )]TJ/F7 6.9738 Tf 10.93 3.923 Td [(7 10 Forthefollowingproblems,llinthemissingterm. Exercise8.674 )]TJ/F7 6.9738 Tf 14.203 3.922 Td [(3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.675 Solutiononp.654. )]TJ/F7 6.9738 Tf 13.988 3.922 Td [(6 a 2 a +1 = 2 a +1 Exercise8.676 )]TJ/F10 6.9738 Tf 9 3.922 Td [(x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.677 Solutiononp.654. )]TJ/F7 6.9738 Tf 17.277 3.923 Td [(9 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a +4 = a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.678 y +3 )]TJ/F10 6.9738 Tf 6.226 0 Td [(y )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 = y +5 Exercise8.679 Solutiononp.654. )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 6 m +7 Exercise8.680 )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(2 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 7 r +1 = 2 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 8.12.1.2ReducingRationalExpressionsSection8.3 Forthefollowingproblems,reducetherationalexpressionstolowestterms. Exercise8.681 Solutiononp.654. 12 6 x +24 Exercise8.682 16 4 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 Exercise8.683 Solutiononp.654. 5 m +25 10 m 2 +15 m Exercise8.684 7+21 r 7 r 2 +28 r Exercise8.685 Solutiononp.654. 3 a 2 +4 a 5 a 3 +6 a 2

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631 Exercise8.686 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.687 Solutiononp.654. 5 y +20 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 Exercise8.688 4 y 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 y 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise8.689 Solutiononp.654. 6 a 9 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 a 7 2 a 7 )]TJ/F7 6.9738 Tf 6.226 0 Td [(14 a 5 Exercise8.690 8 x 4 y 8 +24 x 3 y 9 4 x 2 y 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x 3 y 6 Exercise8.691 Solutiononp.654. 21 y 8 z 10 w 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 y 7 w 2 Exercise8.692 )]TJ/F7 6.9738 Tf 6.226 0 Td [(35 a 5 b 2 c 4 d 8 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 abc 3 d 6 Exercise8.693 Solutiononp.654. x 2 +9 x +18 x 3 +3 x 2 Exercise8.694 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 a +35 2 a 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 a 3 Exercise8.695 Solutiononp.654. y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 y +12 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 y +3 Exercise8.696 m 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 m 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(22 Exercise8.697 Solutiononp.654. 12 r 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 4 r 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 r +10 Exercise8.698 14 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 6 a 2 +9 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.699 Solutiononp.654. 4 a 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 a 3 4 a 2 Exercise8.700 5 m 2 10 m 3 +5 m 2 Exercise8.701 Solutiononp.654. )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.702 )]TJ/F10 6.9738 Tf 6.227 0 Td [(r )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1

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632 CHAPTER8.RATIONALEXPRESSIONS 8.12.1.3MultiplyingandDividingRationalExpressionsSection8.4-AddingandSubtracting RationalExpressionsSection8.6 Forthefollowingproblems,performtheindicatedoperations. Exercise8.703 Solutiononp.654. x 2 18 3 x 3 Exercise8.704 4 a 2 b 3 15 x 4 y 5 10 x 6 y 3 ab 2 Exercise8.705 Solutiononp.654. x +6 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +7 x +6 Exercise8.706 8 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 3 a +3 a +1 2 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.707 Solutiononp.654. 10 m 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 m 2 4 r 7 +20 r 3 m 16 r 8 +80 r 4 Exercise8.708 5 r +7 )]TJ/F7 6.9738 Tf 16.173 3.922 Td [(3 r +7 Exercise8.709 Solutiononp.654. 2 a 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 16.257 3.923 Td [(9 a 3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.710 9 x +7 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 + 3 x +2 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 Exercise8.711 Solutiononp.654. 15 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 8 y +1 )]TJ/F7 6.9738 Tf 11.158 4.445 Td [(2 y +1 8 y +1 Exercise8.712 4 a +3 + 6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 Exercise8.713 Solutiononp.654. 7 a a +6 + 5 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 Exercise8.714 x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 + x +7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.715 Solutiononp.654. 2 y +1 y +4 )]TJ/F10 6.9738 Tf 11.158 4.444 Td [(y +6 y +1 Exercise8.716 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 x +4 + 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +4 Exercise8.717 Solutiononp.654. 6 a +5 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 + 4 a +1 2 a +1 Exercise8.718 4 x 2 +3 x +2 + 9 x 2 +6 x +8 Exercise8.719 Solutiononp.655. 6 r r 2 +7 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 )]TJ/F13 6.9738 Tf 20.103 3.923 Td [()]TJ/F7 6.9738 Tf 6.226 0 Td [(3 r r 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 r +2 Exercise8.720 y +3 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(11 y +10 )]TJ/F10 6.9738 Tf 20.347 4.445 Td [(y +1 y 2 +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise8.721 Solutiononp.655. 2 a +5 16 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 24.333 3.923 Td [(6 a +7 16 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 a +2 Exercise8.722 7 y +4 6 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(32 y +32 + 6 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 2 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 y +40 Exercise8.723 Solutiononp.655. x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 x 2 +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(18

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633 Exercise8.724 y 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y 2 +9 y +20 y 2 +5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(16 Exercise8.725 Solutiononp.655. r +3 4 r +4 r +3 3 Exercise8.726 b +5 3 b +1 2 b +5 2 Exercise8.727 Solutiononp.655. x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 3 x +1 Exercise8.728 x +9 6 x +9 2 x +1 4 Exercise8.729 Solutiononp.655. 5 x + 2 x 2 +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 Exercise8.730 2 y + 4 y 2 +5 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.731 Solutiononp.655. y 2 +4 y +4 y 2 +10 y +21 y +2 Exercise8.732 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+ 4 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.733 Solutiononp.655. 3 x +1 x 2 +3 x +2 + 5 x +6 x 2 +6 x +5 )]TJ/F7 6.9738 Tf 20.46 3.923 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(35 Exercise8.734 5 a +3 b 8 a 2 +2 ab )]TJ/F10 6.9738 Tf 6.227 0 Td [(b 2 )]TJ/F7 6.9738 Tf 25.988 3.922 Td [(3 a )]TJ/F10 6.9738 Tf 6.227 0 Td [(b 4 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 ab +2 b 2 )]TJ/F10 6.9738 Tf 24.058 3.922 Td [(a +5 b 4 a 2 +3 ab )]TJ/F10 6.9738 Tf 6.227 0 Td [(b 2 Exercise8.735 Solutiononp.655. 3 x 2 +6 x +10 10 x 2 +11 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 + 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +15 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(11 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 8.12.1.4RationalEquationsSection8.7 Forthefollowingproblems,solvetherationalequations. Exercise8.736 4 x 5 + 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 15 = 29 25 Exercise8.737 Solutiononp.655. 6 a 7 + 2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 21 = 77 21 Exercise8.738 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 6 + 3 x +4 9 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 9 Exercise8.739 Solutiononp.655. 4 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 4 + 8 y +1 6 = )]TJ/F7 6.9738 Tf 6.226 0 Td [(69 12 Exercise8.740 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + 7 x +2 = 43 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Exercise8.741 Solutiononp.655. 5 a +3 + 6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 = 9 a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 Exercise8.742 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 + 2 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 = 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 Exercise8.743 Solutiononp.655. 2 m +5 m )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 + 9 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 = 30 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8

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634 CHAPTER8.RATIONALEXPRESSIONS Exercise8.744 r +6 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(3 r +2 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Exercise8.745 Solutiononp.655. 8 b +1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 )]TJ/F10 6.9738 Tf 11.214 3.923 Td [(b +5 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 = 45 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 Exercise8.746 Solve z = x )]TJETq1 0 0 1 148.227 622.44 cm[]0 d 0 J 0.339 w 0 0 m 4.518 0 l SQBT/F10 6.9738 Tf 148.227 618.251 Td [(x x for s: Exercise8.747 Solutiononp.655. Solve A = P + rt for t: Exercise8.748 Solve 1 R = 1 E + 1 F for E: Exercise8.749 Solutiononp.655. Solve Q = 2 mn s + t for t: Exercise8.750 Solve I = E R + r for r: 8.12.1.5ApplicationsSection8.8 Forthefollowingproblems,ndthesolution. Exercise8.751 Solutiononp.655. Whenthesamenumberissubtractedfrombothtermsofthefraction 7 12 ; theresultis 1 2 : Whatis thenumber? Exercise8.752 Whenthesamenumberisaddedtobothtermsofthefraction 13 15 ; theresultis 8 9 : Whatisthe number? Exercise8.753 Solutiononp.655. Whenthreefourthsofanumberisaddedtothereciprocalofthenumber,theresultis 173 16 : What isthenumber? Exercise8.754 Whenonethirdofanumberisaddedtothereciprocalofthenumber,theresultis )]TJ/F7 6.9738 Tf 6.227 0 Td [(127 90 : Whatis thenumber? Exercise8.755 Solutiononp.655. PersonAworkingalonecancompleteajobin9hours.PersonBworkingalonecancompletethe samejobin7hours.Howlongwillittakebothpeopletocompletethejobworkingtogether? Exercise8.756 Debbiecancompleteanalgebraassignmentin 3 4 ofanhour.Sandi,whoplaysherradiowhile working,cancompletethesameassignmentin 1 1 4 hours.IfDebbieandSandiworktogether,how longwillittakethemtocompletetheassignment? Exercise8.757 Solutiononp.655. Aninletpipecanllatankin6hoursandanoutletpipecandrainthetankin8hours.Ifboth pipesareopen,howlongwillittaketollthetank? Exercise8.758 Twopipescanllatankin4and5hours,respectively.Howlongwillittakebothpipestollthe tank? Exercise8.759 Solutiononp.655. Thepressureduetosurfacetensioninasphericalbubbleisgivenby P = 4 T r ; where T isthesurface tensionoftheliquid,and r istheradiusofthebubble. aDeterminethepressureduetosurfacetensionwithinasoapbubbleofradius 1 2 inchandsurface

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635 tension22. bDeterminetheradiusofabubbleifthepressureduetosurfacetensionis57.6andthesurface tensionis18. Exercise8.760 Theequation 1 p + 1 q = 1 f relatesanobjectsdistance p fromalensandtheimagedistance q from thelenstothefocallength f ofthelens. aDeterminethefocallengthofalensinwhichanobject8feetawayproducesanimage6feet away. bDeterminehowfaranobjectisfromalensifthefocallengthofthelensis10inchesandthe imagedistanceis10inches. cDeterminehowfaranobjectwillbefromalensthathasafocallengthof 1 7 8 cmandtheobject distanceis3cmawayfromthelens. 8.12.1.6DividingPolynomialsSection8.10 Forthefollowingproblems,dividethepolynomials. Exercise8.761 Solutiononp.655. a 2 +9 a +18 by a +3 Exercise8.762 c 2 +3 c )]TJ/F8 9.9626 Tf 9.963 0 Td [(88 by c )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise8.763 Solutiononp.655. x 3 +9 x 2 +18 x +28 by x +7 Exercise8.764 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(49 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 by y +6 Exercise8.765 Solutiononp.655. m 4 +2 m 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 m 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(m +2 by m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise8.766 3 r 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(17 r )]TJ/F8 9.9626 Tf 9.963 0 Td [(27 by r )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 Exercise8.767 Solutiononp.656. a 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(56 a +10 by a )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 Exercise8.768 x 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x +1 by x +3 Exercise8.769 Solutiononp.656. y 3 + y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y by y +4 Exercise8.770 5 x 6 +5 x 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 4 +5 x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x +6 by x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise8.771 Solutiononp.656. y 10 )]TJ/F11 9.9626 Tf 9.962 0 Td [(y 7 +3 y 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y by y 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y Exercise8.772 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 b 7 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(22 b 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(19 b 4 +12 b 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 b 2 + b +4 by b 2 +6 Exercise8.773 Solutiononp.656. x 3 +1 by x +1 Exercise8.774 a 4 +6 a 3 +4 a 2 +12 a +8 by a 2 +3 a +2 Exercise8.775 Solutiononp.656. y 10 +6 y 5 +9 by y 5 +3

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636 CHAPTER8.RATIONALEXPRESSIONS 8.13ProciencyExam 13 8.13.1ProciencyExam Exercise8.776 Solutiononp.656. Section8.2 Findthedomainof 5 a +1 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(24 : Exercise8.777 Solutiononp.656. Forthefollowingproblems,llinthemissingterm. Section8.2 )]TJ/F7 6.9738 Tf 14.261 3.922 Td [(3 x +4 = x +4 Exercise8.778 Solutiononp.656. Section8.2 2 x +5 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x +1 = x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.779 Solutiononp.656. Forthefollowingproblems,reducetolowestterms. Section8.3 30 x 6 y 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 x +5 2 6 xy 3 x +5 Exercise8.780 Solutiononp.656. Section8.3 x 2 +10 x +24 x 2 + x )]TJ/F7 6.9738 Tf 6.227 0 Td [(30 Exercise8.781 Solutiononp.656. Section8.3 8 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 4 x 2 +12 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise8.782 Solutiononp.656. Section8.5 Replace N withtheproperquantity. x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = N x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +3 Exercise8.783 Solutiononp.656. Section8.5 Assumethat a 2 + a )]TJ/F8 9.9626 Tf 9.994 0 Td [(6 ;a 2 )]TJ/F11 9.9626 Tf 9.994 0 Td [(a )]TJ/F8 9.9626 Tf 9.994 0 Td [(12 ; and a 2 )]TJ/F8 9.9626 Tf 9.994 0 Td [(2 a )]TJ/F8 9.9626 Tf 9.994 0 Td [(8 aredenominatorsofrational expressions.FindtheLCD. Exercise8.784 Solutiononp.656. Forthefollowingproblems,performtheoperations. Section8.6 3 a +4 a +6 )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a +6 Exercise8.785 Solutiononp.656. Section8.4 18 x 3 y 5 a 2 15 a 3 b 6 x 2 y Exercise8.786 Solutiononp.656. Section8.4 y 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 y 2 +3 y +2 y 2 +10 y +16 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 y +12 Exercise8.787 Solutiononp.656. Section8.6 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(11 y +24 + y +4 y 2 +3 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 Exercise8.788 Solutiononp.656. Section8.6 9 2 x +7 + 4 6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise8.789 Solutiononp.656. Section8.4 16 x 5 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 9 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x 3 Exercise8.790 Solutiononp.656. Section8.4 m +3 2 m +6 5 m +1 Exercise8.791 Solutiononp.656. Section8.6 3 y +10 8 y 2 +10 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )]TJ/F7 6.9738 Tf 22.277 4.444 Td [(5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 y 2 +23 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 13 Thiscontentisavailableonlineat.

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637 Exercise8.792 Solutiononp.656. Section8.7 Solve 1 x +3 + 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = x x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 : Exercise8.793 Solutiononp.656. Section8.7 Solve 12 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 +5= 3 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 : Exercise8.794 Solutiononp.656. Section8.8 Whenthesamenumberisaddedtoboththenumeratoranddenominatorofthe fraction 5 3 ; theresultis 6 5 : Whatisthenumberthatisadded? Exercise8.795 Solutiononp.656. Section8.8 PersonA,workingalone,cancompleteajobin20hours.PersonB,workingalone, cancompletethesamejobin30hours.Howlongwillittakebothpeople,workingtogether,to completethejob? Exercise8.796 Solutiononp.657. Section8.8 Thewidthofarectangleis1footlongerthanonehalfthelength.Findthe dimensionslenghandwidthoftherectangleiftheperimeteris44feet. Exercise8.797 Solutiononp.657. Section8.9 Simplifythecomplexfraction 4 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(3 x 4+ 3 x : Exercise8.798 Solutiononp.657. Section8.9 Simplifythecomplexfraction 1 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(5 x )]TJ/F6 4.9813 Tf 9.643 2.677 Td [(6 x 2 1+ 6 x + 5 x 2 : Exercise8.799 Solutiononp.657. Section8.10 Performthedivision: x 3 +10 x 2 +21 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 x +6 : Exercise8.800 Solutiononp.657. Section8.10 Performthedivision: 2 x 3 +5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 :

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638 CHAPTER8.RATIONALEXPRESSIONS SolutionstoExercisesinChapter8 SolutiontoExercise8.1p.549 7 SolutiontoExercise8.2p.549 0 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise8.3p.549 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 1 SolutiontoExercise8.4p.549 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise8.5p.549 )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(4 3 ; 2 SolutiontoExercise8.6p.549 Allrealnumberscomprisethedomain. SolutiontoExercise8.7p.549 Allrealnumberscomprisethedomain. SolutiontoExercise8.8p.551 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 SolutiontoExercise8.9p.551 a +2 SolutiontoExercise8.10p.551 8 SolutiontoExercise8.11p.551 x 6 =4 SolutiontoExercise8.13p.551 x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise8.15p.552 x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 2 SolutiontoExercise8.17p.552 x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; 6 SolutiontoExercise8.19p.552 b 6 =0 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 SolutiontoExercise8.21p.552 x 6 =0 ; 10 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.23p.552 x 6 =0 ; 5 ; 7 SolutiontoExercise8.25p.552 b 6 =1 ; 3 SolutiontoExercise8.27p.552 y 6 =5 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise8.29p.552 x 6 =0 ; 1 2 ; )]TJ/F7 6.9738 Tf 11.158 3.923 Td [(2 3 SolutiontoExercise8.31p.552 )]TJ/F8 9.9626 Tf 7.749 0 Td [(35= )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 ; )]TJ/F8 9.9626 Tf 9.963 0 Td [( 5= )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 SolutiontoExercise8.33p.552 )]TJ/F8 9.9626 Tf 9.409 0 Td [( 4= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 ; 4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1= )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 SolutiontoExercise8.35p.552 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 7.748 0 Td [(10=90 and =90 SolutiontoExercise8.37p.553 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise8.39p.553 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7

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639 SolutiontoExercise8.41p.553 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +4 SolutiontoExercise8.43p.553 a +1 SolutiontoExercise8.45p.553 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 SolutiontoExercise8.47p.553 1 x< 3 SolutiontoExercise8.49p.553 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 2 SolutiontoExercise8.51p.557 6 7 SolutiontoExercise8.52p.557 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 SolutiontoExercise8.53p.557 1 4 SolutiontoExercise8.54p.557 6 a 2 b 2 c 2 SolutiontoExercise8.55p.557 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a a +5 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.56p.557 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 SolutiontoExercise8.57p.557 )]TJ/F15 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise8.58p.557 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 SolutiontoExercise8.60p.557 3 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 SolutiontoExercise8.62p.558 1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise8.64p.558 1 4 y SolutiontoExercise8.66p.558 8 ab SolutiontoExercise8.68p.558 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +5 SolutiontoExercise8.70p.558 a +6 a +2 SolutiontoExercise8.72p.558 1 SolutiontoExercise8.74p.558 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x x +4 SolutiontoExercise8.76p.558 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 xy 4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 SolutiontoExercise8.78p.558 x +10 2 SolutiontoExercise8.80p.558 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x +6 3 SolutiontoExercise8.82p.558 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2

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640 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.84p.559 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 4 a +6 3 SolutiontoExercise8.86p.559 a +2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 a +1 SolutiontoExercise8.88p.559 4 x +2 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 4 SolutiontoExercise8.90p.559 x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise8.92p.559 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +1 SolutiontoExercise8.94p.559 x +2 x SolutiontoExercise8.96p.559 b +3 b +2 SolutiontoExercise8.98p.559 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.100p.559 x +3 x +4 SolutiontoExercise8.102p.559 a +2 SolutiontoExercise8.104p.560 )]TJ/F8 9.9626 Tf 9.409 0 Td [( a )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 or )]TJ/F11 9.9626 Tf 7.749 0 Td [(a +1 SolutiontoExercise8.106p.560 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise8.108p.560 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +10 10 SolutiontoExercise8.110p.560 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise8.112p.560 y 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.114p.560 a a +1 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.116p.560 2 a 2 +5 SolutiontoExercise8.118p.560 x 3 x 4 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 SolutiontoExercise8.120p.561 1 16 a 2 b 8 SolutiontoExercise8.122p.561 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise8.124p.561 x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 6 SolutiontoExercise8.125p.563 10 7 SolutiontoExercise8.126p.563 c 3 a 2 b 2 SolutiontoExercise8.127p.563 1 y 2 +1

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641 SolutiontoExercise8.128p.563 x +3 x +6 SolutiontoExercise8.129p.563 x +2 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 SolutiontoExercise8.130p.564 20 a 2 mn SolutiontoExercise8.131p.564 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.132p.564 3 a +4 3 a +2 SolutiontoExercise8.133p.564 6 a 2 5 SolutiontoExercise8.135p.564 1 SolutiontoExercise8.137p.565 16 a 2 5 SolutiontoExercise8.139p.565 9 x 4 14 SolutiontoExercise8.141p.565 20 x 5 y 2 3 SolutiontoExercise8.143p.565 14 3 x 4 y SolutiontoExercise8.145p.565 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 a 2 bpq SolutiontoExercise8.147p.565 3 a SolutiontoExercise8.149p.565 9 a 3 b 2 SolutiontoExercise8.151p.565 9 p 4 y 11 x SolutiontoExercise8.153p.565 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 b 3 x y 4 SolutiontoExercise8.155p.565 x +2 x +1 SolutiontoExercise8.157p.565 2 x +5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise8.159p.565 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.161p.566 3 4 SolutiontoExercise8.163p.566 4 abx 2 SolutiontoExercise8.165p.566 4 a 3 b 3 SolutiontoExercise8.167p.566 49 m 3 n SolutiontoExercise8.169p.566 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1

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642 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.171p.566 b +1 3 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 3 SolutiontoExercise8.173p.566 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 2 x +1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.175p.566 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.177p.566 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 b 2 a + b SolutiontoExercise8.179p.566 6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.181p.566 a +2 a +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a +1 SolutiontoExercise8.183p.567 1 SolutiontoExercise8.185p.567 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +1 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.187p.567 x x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(y SolutiontoExercise8.189p.567 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +1 SolutiontoExercise8.191p.567 a +3 b a + b SolutiontoExercise8.193p.567 2 ab 2 5 SolutiontoExercise8.195p.567 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 x 2 y 4 SolutiontoExercise8.197p.567 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +1 SolutiontoExercise8.199p.567 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 +4 x +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 SolutiontoExercise8.201p.567 )]TJ/F7 6.9738 Tf 6.226 0 Td [( b )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 3 b +2 SolutiontoExercise8.203p.567 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +3 x +1 SolutiontoExercise8.205p.568 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise8.207p.568 x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +3 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.209p.568 3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +2 2 x +3 SolutiontoExercise8.211p.568 binomial;2;4,2 SolutiontoExercise8.213p.568 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4= x +2 SolutiontoExercise8.215p.572 N =18 SolutiontoExercise8.216p.572 N =63 abx 3

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643 SolutiontoExercise8.217p.572 N = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y SolutiontoExercise8.218p.572 N = a 2 +9 a +14 SolutiontoExercise8.219p.572 N =24 a 4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.220p.572 N = )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 x 4 y 3 z 5 SolutiontoExercise8.221p.572 N =6 ab 2 +18 ab SolutiontoExercise8.222p.572 N = m 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(m )]TJ/F8 9.9626 Tf 9.962 0 Td [(30 SolutiontoExercise8.223p.572 N = r 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 r +12 SolutiontoExercise8.224p.572 N =8 ab 2 SolutiontoExercise8.225p.575 x 5 y SolutiontoExercise8.226p.575 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 2 x +1 SolutiontoExercise8.227p.575 m )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 m +1 2 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 3 SolutiontoExercise8.228p.575 x +1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 SolutiontoExercise8.229p.575 12 y 2 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 SolutiontoExercise8.230p.576 4 x 2 x 5 ; 7 x 5 SolutiontoExercise8.231p.576 2 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; x x +6 x +6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.232p.576 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b +1 b b )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 b +1 ; 4 b 2 b b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b +1 SolutiontoExercise8.233p.576 8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.234p.576 10 x x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +4 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 ; 5 x x +4 x +4 2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 SolutiontoExercise8.235p.576 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 2 b 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 a 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 ; 6 b a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 ; )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a 4 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a 3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 SolutiontoExercise8.236p.577 3 x 2 SolutiontoExercise8.238p.577 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y SolutiontoExercise8.240p.577 12 ab SolutiontoExercise8.242p.577 5 x 2 y 2 SolutiontoExercise8.244p.577 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 axy 2

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644 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.246p.577 40 x 4 y SolutiontoExercise8.248p.577 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise8.250p.577 4 ax a +2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise8.252p.577 4 x )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(b 4 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise8.254p.577 3 s s )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 SolutiontoExercise8.256p.578 a +2 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise8.258p.578 5 m m )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 SolutiontoExercise8.260p.578 9 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise8.262p.578 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x x +3 SolutiontoExercise8.264p.578 4 y y +8 SolutiontoExercise8.266p.578 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 SolutiontoExercise8.268p.578 z )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 SolutiontoExercise8.270p.578 b +4 SolutiontoExercise8.272p.578 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise8.274p.578 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x +1 SolutiontoExercise8.276p.578 y +6 y +2 SolutiontoExercise8.278p.579 a +3 a +5 SolutiontoExercise8.280p.579 x +9 x +5 SolutiontoExercise8.282p.579 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 a SolutiontoExercise8.284p.579 )]TJ/F11 9.9626 Tf 7.749 0 Td [(k )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 SolutiontoExercise8.286p.579 5 b b 3 ; 4 b 3 SolutiontoExercise8.288p.579 36 4 x 2 ; x 4 x 2 SolutiontoExercise8.290p.579 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x +5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 ; 4 x +5 x +5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 SolutiontoExercise8.292p.579 10 y +8 y +2 y +8 ; y +2 y +2 y +8 SolutiontoExercise8.294p.579 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b +5 b 2 b +5 ; b 4 b 2 b +5

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645 SolutiontoExercise8.296p.579 10 a 2 a a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 ; 2 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise8.298p.579 x +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 ; x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.300p.580 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 b +1 b +1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b +6 ; b +6 2 b +1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 b +6 SolutiontoExercise8.302p.580 x +7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 ; x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 SolutiontoExercise8.304p.580 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +6 ; 2 x x +1 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +6 SolutiontoExercise8.306p.580 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 ; 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 SolutiontoExercise8.308p.580 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 ; 5 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise8.310p.580 2 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 ; )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 SolutiontoExercise8.312p.580 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 SolutiontoExercise8.314p.580 y +2 SolutiontoExercise8.316p.583 2 3 SolutiontoExercise8.317p.583 5 b SolutiontoExercise8.318p.583 x y 2 SolutiontoExercise8.319p.583 3 x +4 y x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y SolutiontoExercise8.320p.583 3 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 x +10 SolutiontoExercise8.321p.583 4 x 2 +7 x x +3 SolutiontoExercise8.322p.583 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +7 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.323p.583 4 a 2 +2 a +1 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise8.324p.583 5 x 2 + x +3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.325p.585 9 ax +5 x 12 a 3 SolutiontoExercise8.326p.585 8 b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 b b +1 b )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise8.327p.586 2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(25 a +2 a +3 SolutiontoExercise8.328p.586 3 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(19 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(18 x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3

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646 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.329p.586 5 y 2 +6 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 y y +4 SolutiontoExercise8.330p.586 2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 a +26 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 SolutiontoExercise8.331p.586 4 b 2 +12 b +6 b +3 2 b +2 2 SolutiontoExercise8.332p.586 2 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x +8 3 x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise8.333p.586 2 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 SolutiontoExercise8.334p.586 8 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(45 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 SolutiontoExercise8.335p.586 a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 a +3 SolutiontoExercise8.336p.587 1 2 SolutiontoExercise8.338p.587 3 10 SolutiontoExercise8.340p.587 2 x SolutiontoExercise8.342p.587 14 y 5 x SolutiontoExercise8.344p.587 9 n 2 m SolutiontoExercise8.346p.587 2 y +12 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise8.348p.587 4 a +11 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.350p.587 2 x +4 5 x SolutiontoExercise8.352p.587 2 b +3 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.354p.587 8 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise8.356p.587 y +9 y +8 SolutiontoExercise8.358p.588 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 x +2 SolutiontoExercise8.360p.588 2 x +1 3 x 2 SolutiontoExercise8.362p.588 4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 10 a 3 SolutiontoExercise8.364p.588 5 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.366p.588 2 y y +13 y +4 y +3 SolutiontoExercise8.368p.588 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +9 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x +2

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647 SolutiontoExercise8.370p.588 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 y y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 SolutiontoExercise8.372p.588 y 2 y +1 y +6 SolutiontoExercise8.374p.588 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 a +9 a +4 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.376p.588 7 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 a +1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 SolutiontoExercise8.378p.588 2 x 2 + x +4 x +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +4 SolutiontoExercise8.380p.589 2 b 2 +3 b +29 b )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 b +4 b +5 SolutiontoExercise8.382p.589 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x +29 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +2 x +7 SolutiontoExercise8.384p.589 5 x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 x 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(34 x 2 +34 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(60 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x +3 x +4 SolutiontoExercise8.386p.589 a +5 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a +2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 SolutiontoExercise8.388p.589 )]TJ/F10 6.9738 Tf 6.226 0 Td [(a 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a 2 a +3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.390p.589 )]TJ/F10 6.9738 Tf 6.226 0 Td [(x 3 +2 x 2 +6 x +18 4 x 4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x +3 SolutiontoExercise8.392p.589 14 x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x 2 +9 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(36 8 x 3 x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.394p.589 8 x +50 x +6 SolutiontoExercise8.396p.589 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(13 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 SolutiontoExercise8.398p.589 2 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.400p.589 3 x 2 +2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 x +1 SolutiontoExercise8.402p.590 2 x 2 + x +2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise8.404p.590 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x +31 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 SolutiontoExercise8.406p.590 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( y 2 + y +1 y +4 SolutiontoExercise8.408p.590 7 a 3 +35 a 2 +85 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a +7 a +4 SolutiontoExercise8.410p.590 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 m m )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise8.412p.590 2 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 SolutiontoExercise8.414p.590 x 22 y 14 z 32

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648 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.416p.590 SolutiontoExercise8.418p.591 x +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise8.419p.594 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 SolutiontoExercise8.420p.594 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 SolutiontoExercise8.421p.595 y =3 isextraneous,sonosolution. SolutiontoExercise8.422p.596 a = 1 3 SolutiontoExercise8.423p.596 Thisequationhasnosolution. x =1 isextraneous. SolutiontoExercise8.424p.597 x =6 SolutiontoExercise8.426p.597 y =12 SolutiontoExercise8.428p.597 x =7 SolutiontoExercise8.430p.597 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 SolutiontoExercise8.432p.597 a =15 SolutiontoExercise8.434p.597 b = )]TJ/F8 9.9626 Tf 7.749 0 Td [(47 SolutiontoExercise8.436p.597 a = )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 SolutiontoExercise8.438p.597 y = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(1 2 SolutiontoExercise8.440p.597 m = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise8.442p.597 x =2 SolutiontoExercise8.444p.597 a =6 SolutiontoExercise8.446p.598 b = )]TJ/F7 6.9738 Tf 8.944 3.923 Td [(9 5

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649 SolutiontoExercise8.448p.598 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise8.450p.598 a =2 SolutiontoExercise8.452p.598 Nosolution;6isanexcludedvalue. SolutiontoExercise8.454p.598 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 SolutiontoExercise8.456p.598 b =8 SolutiontoExercise8.458p.598 nosolution SolutiontoExercise8.460p.598 Nosolution; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 isanexcludedvalue. SolutiontoExercise8.462p.598 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 ; 1 SolutiontoExercise8.464p.598 x =4 ; )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise8.466p.598 y =4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 SolutiontoExercise8.468p.599 y =4 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise8.470p.599 y =6 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 SolutiontoExercise8.472p.599 a = 1 3 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 SolutiontoExercise8.474p.599 a = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 3 ; 4 3 SolutiontoExercise8.476p.599 x = )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(4 3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise8.478p.599 a = 4 5 ; 1 SolutiontoExercise8.480p.599 n = PV rt SolutiontoExercise8.482p.599 W = P )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 SolutiontoExercise8.484p.599 r = A )]TJ/F10 6.9738 Tf 6.226 0 Td [(P Pt SolutiontoExercise8.486p.599 S y 2 = S x 2 F SolutiontoExercise8.488p.599 S 2 = 2 K h )]TJ/F11 9.9626 Tf 9.962 0 Td [(S 1 or 2 K )]TJ/F10 6.9738 Tf 6.227 0 Td [(hS 1 h SolutiontoExercise8.490p.600 h 2 = 6 V )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 a 2 SolutiontoExercise8.492p.600 y 8 16 x 6 SolutiontoExercise8.494p.600 steepness SolutiontoExercise8.496p.600 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x +1 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3

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650 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.497p.601 Thenumberaddedis6. SolutiontoExercise8.498p.602 Therearetwonumbers: 3 7 ; 2 3 : SolutiontoExercise8.499p.603 Workingtogether,AandBcanpourtheconcretewalkwayin 3 3 5 hr. SolutiontoExercise8.500p.604 Itwilltake40hrtollthetank. SolutiontoExercise8.501p.605 PersonA,4hrtocompletethetask;personB,8hrcompletethetask. SolutiontoExercise8.502p.607 length=36ft,width=3ft. SolutiontoExercise8.503p.607 Thenumberaddedis5. SolutiontoExercise8.505p.608 Thenumberaddedis )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 : SolutiontoExercise8.507p.608 Thenumbersubtractedis )]TJ/F8 9.9626 Tf 7.748 0 Td [(17 : SolutiontoExercise8.509p.608 x = 1 2 ; 6 SolutiontoExercise8.511p.608 2 SolutiontoExercise8.513p.608 3 1 13 hours SolutiontoExercise8.515p.608 twominutes SolutiontoExercise8.517p.608 No. x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 hours. SolutiontoExercise8.519p.609 2 8 11 hours SolutiontoExercise8.521p.609 Firstperson:12hours;secondperson:4hours SolutiontoExercise8.523p.609 16 ; 17 SolutiontoExercise8.525p.609 width =9 ft;length =21 ft SolutiontoExercise8.527p.609 side 1=9 inches ; side 2=12 inches ; side 3=9 inches SolutiontoExercise8.529p.609 a f = 15 4 ftb p =15 inchesc q =8 cm SolutiontoExercise8.531p.610 1 11 13 hours SolutiontoExercise8.533p.610 30hours SolutiontoExercise8.535p.610 m =1 SolutiontoExercise8.537p.610 0 SolutiontoExercise8.539p.613 12 5 x

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651 SolutiontoExercise8.540p.613 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 3 x +1 SolutiontoExercise8.541p.613 x y x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y SolutiontoExercise8.542p.613 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.543p.613 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.544p.615 12 5 x SolutiontoExercise8.545p.615 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 3 x +1 SolutiontoExercise8.546p.615 x y x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y SolutiontoExercise8.547p.615 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.548p.615 x x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.549p.615 5 3 SolutiontoExercise8.551p.615 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y +1 SolutiontoExercise8.553p.616 a + c a )]TJ/F10 6.9738 Tf 6.227 0 Td [(c SolutiontoExercise8.555p.616 x SolutiontoExercise8.557p.616 2 a +7 2 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.559p.616 2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 m SolutiontoExercise8.561p.616 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.563p.616 3 y 2 x )]TJ/F10 6.9738 Tf 6.227 0 Td [(y 2 SolutiontoExercise8.565p.616 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 x SolutiontoExercise8.567p.616 y )]TJ/F10 6.9738 Tf 6.227 0 Td [(x xy SolutiontoExercise8.569p.616 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 y +5 SolutiontoExercise8.571p.617 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise8.573p.617 4 a 2 +4 SolutiontoExercise8.575p.617 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x +1 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x +2 SolutiontoExercise8.577p.617 c 2 V 1 + V 2 c 2 + V 1 V 2 SolutiontoExercise8.579p.617 9 x 2 +24 x +16

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652 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.581p.617 x =7 SolutiontoExercise8.583p.618 2 x +1 )]TJ/F7 6.9738 Tf 11.431 3.922 Td [(1 x SolutiontoExercise8.584p.618 3 x +4+ 10 x )]TJ/F7 6.9738 Tf 13.375 3.923 Td [(4 x 2 SolutiontoExercise8.585p.619 a +3 b + 2 a SolutiontoExercise8.586p.619 2 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.587p.619 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 m 2 n 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 mn 3 +4 n SolutiontoExercise8.588p.623 1+ 7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.589p.623 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1+ 8 x +3 SolutiontoExercise8.590p.623 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +55 )]TJ/F7 6.9738 Tf 12.504 3.923 Td [(442 x +8 SolutiontoExercise8.591p.623 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3+ 14 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(14 x 2 +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3+ 14 x +5 SolutiontoExercise8.592p.624 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+ 1 x +2 SolutiontoExercise8.593p.624 4 x +12+ 35 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.594p.624 x 2 +2 x +6+ 14 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 SolutiontoExercise8.595p.624 3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +3 )]TJ/F7 6.9738 Tf 16.475 3.922 Td [(10 2 x +3 SolutiontoExercise8.596p.624 3 a +6 SolutiontoExercise8.598p.625 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y +1 SolutiontoExercise8.600p.625 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise8.602p.625 3 a +1 SolutiontoExercise8.604p.625 3 x 2 + x +4 SolutiontoExercise8.606p.625 ab +4 a +6 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 SolutiontoExercise8.608p.625 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 2 y 2 +3 xy )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 SolutiontoExercise8.610p.625 1 4 y 2 + 1 2 y + 3 4 SolutiontoExercise8.612p.625 8 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 x 3 +12 x 4 3 a 3 or 12 x 4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 x 3 +8 x 2 3 a 3 SolutiontoExercise8.614p.625 2 b 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 b 3 c +4 c a 2 c SolutiontoExercise8.616p.625 1+ 8 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2

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653 SolutiontoExercise8.618p.625 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3+ 10 x +2 SolutiontoExercise8.620p.626 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+ 5 x +1 SolutiontoExercise8.622p.626 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 )]TJ/F7 6.9738 Tf 16.475 3.923 Td [(1 x +1 SolutiontoExercise8.624p.626 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+ 8 y +2 SolutiontoExercise8.626p.626 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x +1 )]TJ/F7 6.9738 Tf 16.475 3.923 Td [(2 x +1 SolutiontoExercise8.628p.626 x 2 + x +1 SolutiontoExercise8.630p.626 x 2 +5 x +11+ 20 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.632p.626 a 2 + a +2+ 8 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.634p.626 y 2 + y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2+ 8 y +2 SolutiontoExercise8.636p.626 x 2 SolutiontoExercise8.638p.626 1+ 1 x +1 SolutiontoExercise8.640p.626 2+ 11 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 SolutiontoExercise8.642p.627 x + 4 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.644p.627 2 x )]TJ/F7 6.9738 Tf 18.461 3.923 Td [(1 3 x +4 SolutiontoExercise8.646p.627 2 x 2 +3 x )]TJ/F7 6.9738 Tf 18.516 3.923 Td [(2 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.648p.627 4 x 3 +2 x )]TJ/F7 6.9738 Tf 16.531 3.922 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.650p.627 3+ 2 y 2 + y +1 SolutiontoExercise8.652p.627 4 z 5 +4 z 4 +2 z 3 +7 z 2 +12 z +11+ 33 2 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 SolutiontoExercise8.654p.627 x 4 )]TJ/F7 6.9738 Tf 18.461 3.922 Td [(1 2 x +5 SolutiontoExercise8.656p.627 x +4 2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.658p.627 x = 1 2 SolutiontoExercise8.660p.627 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 4 SolutiontoExercise8.661p.629 x 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 SolutiontoExercise8.663p.629 x 6 = 5 2 SolutiontoExercise8.665p.629 m 6 =0 ; 1

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654 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.667p.629 s 6 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 ; )]TJ/F7 6.9738 Tf 8.945 3.923 Td [(7 4 ; 0 SolutiontoExercise8.669p.630 y 6 = 1 6 ; )]TJ/F7 6.9738 Tf 8.945 3.922 Td [(5 2 SolutiontoExercise8.671p.630 )]TJ/F8 9.9626 Tf 9.409 0 Td [( 5=20 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(4= )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 SolutiontoExercise8.673p.630 )]TJ/F8 9.9626 Tf 9.409 0 Td [( 10= )]TJ/F8 9.9626 Tf 7.749 0 Td [(70 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(7= )]TJ/F8 9.9626 Tf 7.749 0 Td [(70 SolutiontoExercise8.675p.630 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 a SolutiontoExercise8.677p.630 9 SolutiontoExercise8.679p.630 5 m +1 SolutiontoExercise8.681p.630 2 x +4 SolutiontoExercise8.683p.630 m +5 m m +3 SolutiontoExercise8.685p.630 3 a +4 a a +6 SolutiontoExercise8.687p.631 5 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 SolutiontoExercise8.689p.631 3 a 2 a 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 SolutiontoExercise8.691p.631 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 yz 10 SolutiontoExercise8.693p.631 x +6 x 2 SolutiontoExercise8.695p.631 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.697p.631 3 r +2 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.699p.631 a a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 SolutiontoExercise8.701p.631 6 a +1 5 a +2 SolutiontoExercise8.703p.632 1 6 x SolutiontoExercise8.705p.632 x +7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.707p.632 20 mr )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.709p.632 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 a 3 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 SolutiontoExercise8.711p.632 13 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 8 y +1 SolutiontoExercise8.713p.632 2 a a )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 a +6 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 SolutiontoExercise8.715p.632 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(23 y +4 y +1

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655 SolutiontoExercise8.717p.632 2 8 a 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(a +1 a +1 a )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.719p.632 3 r r +7 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 r )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 r +9 SolutiontoExercise8.721p.632 )]TJ/F7 6.9738 Tf 6.227 0 Td [(16 a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(18 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(17 2 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a +1 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise8.723p.632 x +4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 SolutiontoExercise8.725p.633 r +3 r +4 SolutiontoExercise8.727p.633 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 x +1 SolutiontoExercise8.729p.633 7 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(20 x +1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 SolutiontoExercise8.731p.633 y +2 y +3 y +7 SolutiontoExercise8.733p.633 5 x 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(26 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(192 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(105 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(35 x +1 x +2 SolutiontoExercise8.735p.633 13 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(39 x 2 +51 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(100 x +3 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 SolutiontoExercise8.737p.633 a =4 SolutiontoExercise8.739p.633 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise8.741p.633 a =1 SolutiontoExercise8.743p.633 Nosolution; m =8 isexcluded. SolutiontoExercise8.745p.634 Nosolution; b =7 isexcluded. SolutiontoExercise8.747p.634 t = A )]TJ/F10 6.9738 Tf 6.227 0 Td [(P Pr SolutiontoExercise8.749p.634 t = 2 mn )]TJ/F10 6.9738 Tf 6.227 0 Td [(Qs Q SolutiontoExercise8.751p.634 2 SolutiontoExercise8.753p.634 Norationalsolution. SolutiontoExercise8.755p.634 3 15 16 hrs SolutiontoExercise8.757p.634 24hrs SolutiontoExercise8.759p.634 a176unitsofpressure;b 5 4 unitsoflength SolutiontoExercise8.761p.635 a +6 SolutiontoExercise8.763p.635 x 2 +2 x +4

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656 CHAPTER8.RATIONALEXPRESSIONS SolutiontoExercise8.765p.635 m 3 +4 m 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 SolutiontoExercise8.767p.635 a 2 +6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F7 6.9738 Tf 16.432 3.922 Td [(8 a )]TJ/F7 6.9738 Tf 6.226 0 Td [(9 SolutiontoExercise8.769p.635 y 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 y +11 )]TJ/F7 6.9738 Tf 14.376 3.923 Td [(44 y +4 SolutiontoExercise8.771p.635 y 6 +3 SolutiontoExercise8.773p.635 x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x +1 SolutiontoExercise8.775p.635 y 5 +3 SolutiontoExercise8.776p.636 a 6 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 ; 8 SolutiontoExercise8.777p.636 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 SolutiontoExercise8.778p.636 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 SolutiontoExercise8.779p.636 5 x 5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 2 x +5 SolutiontoExercise8.780p.636 x +4 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 SolutiontoExercise8.781p.636 4 x +3 2 x +7 SolutiontoExercise8.782p.636 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +2 SolutiontoExercise8.783p.636 a +2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 a +3 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 SolutiontoExercise8.784p.636 a +5 a +6 SolutiontoExercise8.785p.636 9 abx SolutiontoExercise8.786p.636 y +3 y +8 y +1 y )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 SolutiontoExercise8.787p.636 2 y 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(22 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(8 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 y +6 SolutiontoExercise8.788p.636 62 x +19 x +7 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise8.789p.636 8 x 4 x +1 3 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 SolutiontoExercise8.790p.636 5 m +1 2 SolutiontoExercise8.791p.636 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 y 2 +15 y +63 y )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 y +3 y +6 SolutiontoExercise8.792p.636 x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 SolutiontoExercise8.793p.637 Nosolution; m =4 isexcluded. SolutiontoExercise8.794p.637 7

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657 SolutiontoExercise8.795p.637 12hours SolutiontoExercise8.796p.637 8ftby14ft SolutiontoExercise8.797p.637 4 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 4 x +3 SolutiontoExercise8.798p.637 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 x +5 SolutiontoExercise8.799p.637 x 2 +4 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 SolutiontoExercise8.800p.637 2 x 2 +4 x +13+ 25 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2

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658 CHAPTER8.RATIONALEXPRESSIONS

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Chapter9 Roots,Radicals,andSquareRoot Equations 9.1Objectives 1 Aftercompletingthischapter,youshould SquareRootExpressionsSection9.2 understandtheconceptofsquareroot beabletodistinguishbetweentheprincipalandsecondarysquarerootsofanumber beabletorelatesquarerootsandmeaningfulexpressionsandtosimplifyasquarerootexpression SimplifyingSquareRootExpressionsSection9.3 beabletoidentifyaperfectsquare befamiliarwiththeproductandquotientpropertiesofsquareroots beabletosimplifysquarerootsinvolvingandnotinvolvingfractions MultiplicationofSquareRootExpressionsSection9.4 beabletousetheproductpropertyofsquarerootstomultiplysquareroots DivisionofSquareRootExpressionsSection9.5 beabletousethedivisionpropertyofsquareroots,themethodofrationalizingthedenominator,and conjugatestodividesquareroots AdditionandSubtractionofSquareRootExpressionsSection9.6 understandtheprocessusedinaddingandsubtractingsquareroots beabletoaddandsubtractsquareroote SquareRootEquationswithApplicationsSection9.7 beabletorecognizesquarerootequationsandextraneoussolutions beabletosolvesquarerootequations 1 Thiscontentisavailableonlineat. 659

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660 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.2SquareRootExpressions 2 9.2.1Overview SquareRoots PrincipalandSecondarySquareRoots MeaningfulExpressions SimplifyingSquareRoots 9.2.2SquareRoots WhenwestudiedexponentsinSectionSection2.5,wenotedthat 4 2 =16 and )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 2 =16 : Wecanseethat16 isthesquareofboth4and )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 .Since16comesfromsquaring4or )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 ,4and )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 arecalledthe squareroots of16.Thus16hastwosquareroots,4and )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 .Noticethatthesetwosquarerootsareoppositesofeachother. Wecansaythat SquareRoot Thesquarerootofapositivenumber x isanumbersuchthatwhenitissquaredthenumber x results. Everypositivenumberhastwosquareroots,onepositivesquarerootandonenegativesquareroot. Furthermore,thetwosquarerootsofapositivenumberareoppositesofeachother.Thesquarerootof0is 0. 9.2.3SampleSetA Example9.1 Thetwosquarerootsof49are7and )]TJ/F15 9.9626 Tf 7.749 0 Td [(7since 7 2 =49 and )]TJ/F8 9.9626 Tf 7.748 0 Td [(7 2 =49 Example9.2 Thetwosquarerootsof 49 64 are 7 8 and )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 8 since )]TJ/F7 6.9738 Tf 5.761 -4.147 Td [(7 8 2 = 7 8 7 8 = 49 64 and )]TJ/F13 6.9738 Tf 5.761 -4.147 Td [()]TJ/F7 6.9738 Tf 6.227 0 Td [(7 8 2 = )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 8 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 8 = 49 64 9.2.4PracticeSetA Namebothsquarerootsofeachofthefollowingnumbers. Exercise9.1 Solutiononp.713. 36 Exercise9.2 Solutiononp.713. 25 Exercise9.3 Solutiononp.713. 100 Exercise9.4 Solutiononp.713. 64 Exercise9.5 Solutiononp.713. 1 2 Thiscontentisavailableonlineat.

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661 Exercise9.6 Solutiononp.713. 1 4 Exercise9.7 Solutiononp.713. 9 16 Exercise9.8 Solutiononp.713. 0 : 1 Exercise9.9 Solutiononp.713. 0 : 09 9.2.5PrincipalandSecondarySquareRoots Thereisanotationfordistinguishingthepositivesquarerootofanumber x fromthenegativesquareroot of x PrincipalSquareRoot: p x If x isapositiverealnumber,then p x representsthepositivesquarerootof x .Thepositivesquarerootofanumberiscalledthe principalsquareroot ofthenumber. SecondarySquareRoot: )]TJ 7.749 7.175 Td [(p x )]TJ 7.749 7.176 Td [(p x representsthenegativesquarerootof x .Thenegativesquarerootofanumberiscalledthe secondarysquareroot ofthenumber. )]TJ 7.749 7.176 Td [(p x indicatesthesecondarysquarerootof x RadicalSign,Radicand,andRadical Intheexpression p x; q iscalleda radicalsign x iscalledthe radicand p x iscalleda radical Thehorizontalbarthatappearsattachedtotheradicalsign, q ,isagroupingsymbolthatspecies theradicand. Because p x and )]TJ 7.749 7.175 Td [(p x arethetwosquarerootof x p x p x = x and )]TJ 7.749 7.176 Td [(p x )]TJ 7.749 7.176 Td [(p x = x 9.2.6SampleSetB Writetheprincipalandsecondarysquarerootsofeachnumber. Example9.3 9 : Principalsquarerootis p 9=3 : Secondarysquarerootis )]TJ 9.962 8.241 Td [(p 9= )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 :

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662 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Example9.4 15 : Principalsquarerootis p 15 : Secondarysquarerootis )]TJ 9.963 8.241 Td [(p 15 : Example9.5 Useacalculatortoobtainadecimalapproximationforthetwosquarerootsof34.Roundtotwo decimalplaces. OntheCalculater Type 34 Press p x Displayreads: 5 : 8309519 Roundto5.83. Noticethatthesquarerootsymbolonthecalculatoris q .Thismeans,ofcourse,thata calculatorwillproduceonlythepositivesquareroot.Wemustsupplythenegativesquareroot ourselves. p 34 5 : 83 and )]TJ 7.749 8.241 Td [(p 34 )]TJ/F8 9.9626 Tf 18.265 0 Td [(5 : 83 Note: Thesymbol means"approximatelyequalto." Example9.6 Thenumber p 50 isbetweenwhattwowholenumbers? Since 7 2 =49 ; p 49=7 : Since 8 2 =64 ; p 64=8 : Thus, 7 < p 50 < 8 Thus, p 50 isanumberbetween7and8. 9.2.7PracticeSetB Writetheprincipalandsecondarysquarerootsofeachnumber. Exercise9.10 Solutiononp.713. 100 Exercise9.11 Solutiononp.713. 121

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663 Exercise9.12 Solutiononp.713. 35 Exercise9.13 Solutiononp.713. Useacalculatortoobtainadecimalapproximationforthetwosquarerootsof35.Roundtotwo decimalplaces. 9.2.8MeaningfulExpressions Sinceweknowthatthesquareofanyrealnumberisapositivenumberorzero,wecanseethatexpressions suchas p )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 donotdescriberealnumbers.Thereisnorealnumberthatcanbesquaredthatwillproduce )]TJ/F15 9.9626 Tf 7.749 0 Td [(16.For p x tobearealnumber,wemusthave x 0 : Inourstudyofalgebra,wewillassumethatall variablesandallexpressionsinradicandsrepresentnonnegativenumbersnumbersgreaterthanorequalto zero. 9.2.9SampleSetC Writetheproperrestrictionsthatmustbeplacedonthevariablesothateachexpressionrepresentsareal number. Example9.7 For p x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 tobearealnumber,wemusthave x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 0 or x 3 Example9.8 For p 2 m +7 tobearealnumber,wemusthave 2 m +7 0 or 2 m )]TJ/F8 9.9626 Tf 18.265 0 Td [(7 or m )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 2 9.2.10PracticeSetC Writetheproperrestrictionsthatmustbeplacedonthevariablesothateachexpressionrepresentsareal number. Exercise9.14 Solutiononp.713. p x +5 Exercise9.15 Solutiononp.713. p y )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise9.16 Solutiononp.713. p 3 a +2 Exercise9.17 Solutiononp.713. p 5 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(6

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664 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.2.11SimplifyingSquareRoots Whenvariablesoccurintheradicand,wecanoftensimplifytheexpressionbyremovingtheradicalsign. Wecandosobykeepinginmindthattheradicandisthesquareofsomeotherexpression.Wecansimplify aradicalbyseekinganexpressionwhosesquareistheradicand.Thefollowingobservationswillhelpusnd thesquarerootofavariablequantity. Example9.9 Since )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 3 2 = x 3 2 = x 6 ;x 3 isasquarerootof x 6 : Also Example9.10 Since )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x 4 2 = x 4 2 = x 8 ;x 4 isasquarerootof x 8 : Also Example9.11 Since )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(x 6 2 = x 6 2 = x 12 ;x 6 isasquarerootof x 12 : Also Theseexamplessuggestthefollowingrule: Ifavariablehasanevenexponent,itssquarerootcanbefoundbydividingthatexponentby2. TheexamplesofSampleSetBillustratetheuseofthisrule.

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665 9.2.12SampleSetD Simplifyeachexpressionbyremovingtheradicalsign.Assumeeachvariableis nonnegative Example9.12 p a 2 : Weseekanexpressionwhosesquareis a 2 : Since a 2 = a 2 ; p a 2 = a Noticethat 2 2=1 : Example9.13 p y 8 : Weseekanexpressionwhosesquareis y 8 : Since )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(y 4 2 = y 8 ; p y 8 = y 4 Noticethat 8 2=4 : Example9.14 p 25 m 2 n 6 : Weseekanexpressionwhosesquareis 25 m 2 n 6 : Since )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(5 mn 3 2 =25 m 2 n 6 ; p 25 m 2 n 6 =5 mn 3 Noticethat 2 2=1 and 6 2=3 : Example9.15 )]TJ/F1 9.9626 Tf 7.749 12.522 Td [(q 121 a 10 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4 : Weseekanexpressionwhosesquareis 121 a 10 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4 : Since h 11 a 5 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 i 2 =121 a 10 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4 ; q 121 a 10 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4 =11 a 5 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 Then, )]TJ/F1 9.9626 Tf 9.963 12.523 Td [(q 121 a 10 b )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 4 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 a 5 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 Noticethat 10 2=5 and 4 2=2 : 9.2.13PracticeSetD Simplifyeachexpressionbyremovingtheradicalsign.Assumeeachvariableisnonnegative. Exercise9.18 Solutiononp.713. p y 8 Exercise9.19 Solutiononp.713. p 16 a 4 Exercise9.20 Solutiononp.713. p 49 x 4 y 6 Exercise9.21 Solutiononp.713. )]TJ/F1 9.9626 Tf 7.749 8.745 Td [(p 100 x 8 y 12 z 2 Exercise9.22 Solutiononp.713. )]TJ/F1 9.9626 Tf 7.749 12.522 Td [(q 36 a +5 4 Exercise9.23 Solutiononp.713. q 225 w 4 z 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 2 Exercise9.24 Solutiononp.713. p 0 : 25 y 6 z 14 Exercise9.25 Solutiononp.713. p x 2 n ; where n isanaturalnumber. Exercise9.26 Solutiononp.714. p x 4 n ; where n isanaturalnumber.

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666 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.2.14Exercises Exercise9.27 Solutiononp.714. Howmanysquarerootsdoeseverypositiverealnumberhave? Exercise9.28 Thesymbol q representswhichsquarerootofanumber? Exercise9.29 Solutiononp.714. Thesymbol q representswhichsquarerootofanumber? Forthefollowingproblems,ndthetwosquarerootsofthegivennumber. Exercise9.30 64 Exercise9.31 Solutiononp.714. 81 Exercise9.32 25 Exercise9.33 Solutiononp.714. 121 Exercise9.34 144 Exercise9.35 Solutiononp.714. 225 Exercise9.36 10,000 Exercise9.37 Solutiononp.714. 1 16 Exercise9.38 1 49 Exercise9.39 Solutiononp.714. 25 36 Exercise9.40 121 225 Exercise9.41 Solutiononp.714. 0 : 04 Exercise9.42 0 : 16 Exercise9.43 Solutiononp.714. 1 : 21 Forthefollowingproblems,evaluateeachexpression.Iftheexpressiondoesnotrepresentarealnumber, write"notarealnumber." Exercise9.44 p 49 Exercise9.45 Solutiononp.714. p 64 Exercise9.46 )]TJ 7.749 8.241 Td [(p 36

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667 Exercise9.47 Solutiononp.714. )]TJ 7.749 8.241 Td [(p 100 Exercise9.48 )]TJ 7.749 8.241 Td [(p 169 Exercise9.49 Solutiononp.714. )]TJ/F1 9.9626 Tf 7.749 11.507 Td [(q 36 81 Exercise9.50 )]TJ/F1 9.9626 Tf 7.749 11.507 Td [(q 121 169 Exercise9.51 Solutiononp.714. p )]TJ/F8 9.9626 Tf 7.749 0 Td [(225 Exercise9.52 p )]TJ/F8 9.9626 Tf 7.749 0 Td [(36 Exercise9.53 Solutiononp.714. )]TJ 7.749 7.827 Td [(p )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Exercise9.54 )]TJ 7.749 7.826 Td [(p )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Exercise9.55 Solutiononp.714. )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F14 9.9626 Tf 4.567 -8.07 Td [()]TJ 7.748 8.241 Td [(p 9 Exercise9.56 )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F14 9.9626 Tf 4.567 -8.07 Td [()]TJ 7.748 8.241 Td [(p 0 : 81 Forthefollowingproblems,writetheproperrestrictionsthatmustbeplacedonthevariablesothatthe expressionrepresentsarealnumber. Exercise9.57 Solutiononp.714. p y +10 Exercise9.58 p x +4 Exercise9.59 Solutiononp.714. p a )]TJ/F8 9.9626 Tf 9.963 0 Td [(16 Exercise9.60 p h )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 Exercise9.61 Solutiononp.714. p 2 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Exercise9.62 p 7 x +8 Exercise9.63 Solutiononp.714. p )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 Exercise9.64 p )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y +15 Forthefollowingproblems,simplifyeachexpressionbyremovingtheradicalsign. Exercise9.65 Solutiononp.714. p m 6 Exercise9.66 p k 10 Exercise9.67 Solutiononp.714. p a 8

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668 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.68 p h 16 Exercise9.69 Solutiononp.714. p x 4 y 10 Exercise9.70 p a 6 b 20 Exercise9.71 Solutiononp.714. p a 4 b 6 Exercise9.72 p x 8 y 14 Exercise9.73 Solutiononp.714. p 81 a 2 b 2 Exercise9.74 p 49 x 6 y 4 Exercise9.75 Solutiononp.715. p 100 m 8 n 2 Exercise9.76 p 225 p 14 r 16 Exercise9.77 Solutiononp.715. p 36 x 22 y 44 Exercise9.78 q 169 w 4 z 6 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 Exercise9.79 Solutiononp.715. q 25 x 12 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 4 Exercise9.80 q 64 a 10 a +4 14 Exercise9.81 Solutiononp.715. q 9 m 6 n 4 m + n 18 Exercise9.82 p 25 m 26 n 42 r 66 s 84 Exercise9.83 Solutiononp.715. q f )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 g +6 4 Exercise9.84 q c )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 6 + c +1 2 Exercise9.85 Solutiononp.715. )]TJ 7.749 8.717 Td [(p 64 r 4 s 22 Exercise9.86 )]TJ/F1 9.9626 Tf 7.749 12.523 Td [(q 121 a 6 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 8 Exercise9.87 Solutiononp.715. )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( )]TJ/F1 9.9626 Tf 7.748 12.522 Td [(q w +6 2 Exercise9.88 )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( )]TJ/F1 9.9626 Tf 7.749 12.522 Td [(q 4 a 2 b 2 c 2 +8 2

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669 Exercise9.89 Solutiononp.715. p 1 : 21 h 4 k 4 Exercise9.90 p 2 : 25 m 6 p 6 Exercise9.91 Solutiononp.715. )]TJ/F1 9.9626 Tf 7.749 11.183 Td [(q 169 a 2 b 4 c 6 196 x 4 y 6 z 8 Exercise9.92 )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( q 81 y 4 z )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2 225 x 8 z 4 w 6 9.2.15ExercisedforReview Exercise9.93 Solutiononp.715. Section8.4 Findthequotient. x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 x 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2 x +1 : Exercise9.94 Section8.6 Findthesum. 1 x +1 + 3 x +1 + 2 x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : Exercise9.95 Solutiononp.715. Section8.7 Solvetheequation,ifpossible: 1 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 = 3 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 )]TJ/F7 6.9738 Tf 16.475 3.923 Td [(3 x +1 : Exercise9.96 Section8.10 Performthedivision: 15 x 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x 2 +10 x 5 x : Exercise9.97 Solutiononp.715. Section8.10 Performthedivision: x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x 2 +13 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(21 x )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 : 9.3SimplifyingSquareRootExpressions 3 9.3.1Overview PerfectSquares TheProductPropertyofSquareRoots TheQuotientPropertyofSquareRoots SquareRootsNotInvolvingFractions SquareRootsInvolvingFractions Tobeginourstudyoftheprocessofsimplifyingasquarerootexpression,wemustnotethreefacts:onefact concerningperfectsquaresandtwoconcerningpropertiesofsquareroots. 9.3.2PerfectSquares PerfectSquares Realnumbersthataresquaresofrationalnumbersarecalled perfectsquares. Thenumbers25and 1 4 are examplesofperfectsquaressince 25=5 2 and 1 4 = )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 2 ; and5and 1 2 arerationalnumbers.Thenumber2 is not aperfectsquaresince 2= )]TJ/F14 9.9626 Tf 4.567 0.172 Td [(p 2 2 and p 2 isnotarationalnumber. Althoughwewillnotmakeadetailedstudyofirrationalnumbers,wewillmakethefollowingobservation: Anyindicatedsquarerootwhoseradicandisnotaperfectsquareisanirrationalnumber. Thenumbers p 6 ; p 15 ; and q 3 4 areeachirrationalsinceeachradicand )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6 ; 15 ; 3 4 isnotaperfectsquare. 3 Thiscontentisavailableonlineat.

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670 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.3.3TheProductPropertyofSquareRoots Noticethat p 9 4= p 36=6 and p 9 p 4=3 2=6 Sinceboth p 9 4 and p 9 p 4 equal6,itmustbethat p 9 4= p 9 p 4 TheProductProperty p xy = p x p y Thissuggeststhatingeneral,if x and y arepositiverealnumbers, p xy = p x p y Thesquarerootoftheproductistheproductofthesquareroots. 9.3.4TheQuotientPropertyofSquareRoots Wecansuggestasimilarruleforquotients.Noticethat q 36 4 = p 9=3 and p 36 p 4 = 6 2 =3 Sinceboth 36 4 and p 36 p 4 equal3,itmustbethat q 36 4 = p 36 p 4 TheQuotientProperty q x y = p x p y Thissuggeststhatingeneral,if x and y arepositiverealnumbers, q x y = p x p y ;y 6 =0 Thesquarerootofthequotientisthequotientofthesquareroots. CAUTION Itisextremelyimportanttorememberthat p x + y 6 = p x + p y or p x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 6 = p x )]TJ 9.962 6.208 Td [(p y Forexample,noticethat p 16+9= p 25=5 ; but p 16+ p 9=4+3=7 : Weshallstudytheprocessofsimplifyingasquarerootexpressionbydistinguishingbetweentwotypes ofsquareroots:squarerootsnotinvolvingafractionandsquarerootsinvolvingafraction. 9.3.5SquareRootsNotInvolvingFractions Asquarerootthatdoesnotinvolvefractionsisin simpliedform iftherearenoperfectsquareinthe radicand. Thesquareroots p x; p ab; p 5 mn; p 2 a +5 areinsimpliedformsincenoneoftheradicandscontains aperfectsquare. Thesquareroots p x 2 ; p a 3 = p a 2 a are not insimpliedformsinceeachradicandcontainsaperfect square. Tosimplifyasquarerootexpressionthatdoesnotinvolveafraction,wecanusethefollowingtworules: SIMPLIFYINGSQUAREROOTSWITHOUTFRACTIONS

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671 1.Ifafactoroftheradicandcontainsavariablewithan even exponent,thesquarerootisobtainedby dividingtheexponentby2. 2.Ifafactoroftheradicandcontainsavariablewithan odd exponent,thesquarerootisobtainedby rstfactoringthevariablefactorintotwofactorssothatonehasanevenexponentandtheotherhas anexponentof1,thenusingtheproductpropertyofsquareroots. 9.3.6SampleSetA Simplifyeachsquareroot. Example9.16 p a 4 : Theexponentiseven: 4 2 =2 : Theexponentonthesquarerootis2. p a 4 = a 2 Example9.17 p a 6 b 10 : Bothexponentsareeven: 6 2 =3 and 10 2 =5 : Theexponentonthesquarerootof a 6 is3. Theexponentonthesquarerootif b 10 is5. p a 6 b 10 = a 3 b 5 Example9.18 p y 5 : Theexponentisodd: y 5 = y 4 y: Then p y 5 = p y 4 y = p y 4 p y = y 2 p y Example9.19 p 36 a 7 b 11 c 20 = p 6 2 a 6 ab 10 bc 20 a 7 = a 6 a;b 11 = b 10 b = p 6 2 a 6 b 10 c 20 ab bythecommutativepropertyofmultiplication. = p 6 2 a 6 b 10 c 20 p ab bytheproductpropertyofsquareroots. =6 a 3 b 5 c 10 p ab Example9.20 q 49 x 8 y 3 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 6 = q 7 2 x 8 y 2 y a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 6 = q 7 2 x 8 y 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 6 p y =7 x 4 y a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 3 p y Example9.21 p 75= p 25 3= p 5 2 3= p 5 2 p 3=5 p 3 9.3.7PracticeSetA Simplifyeachsquareroot. Exercise9.98 Solutiononp.715. p m 8 Exercise9.99 Solutiononp.715. p h 14 k 22 Exercise9.100 Solutiononp.715. p 81 a 12 b 6 c 38

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672 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.101 Solutiononp.715. q 144 x 4 y 80 b +5 16 Exercise9.102 Solutiononp.715. p w 5 Exercise9.103 Solutiononp.715. p w 7 z 3 k 13 Exercise9.104 Solutiononp.715. p 27 a 3 b 4 c 5 d 6 Exercise9.105 Solutiononp.715. q 180 m 4 n 15 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 15 9.3.8SquareRootsInvolvingFractions Asquarerootexpressionisinsimpliedformifthereare 1.noperfectsquaresintheradicand, 2.nofractionsintheradicand,or 3.3.nosquarerootexpressionsinthedenominator. Thesquarerootexpressions p 5 a; 4 p 3 xy 5 ; and 11 m 2 n p a )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 2 x 2 areinsimpliedform. Thesquarerootexpressions q 3 x 8 ; q 4 a 4 b 3 5 ; and 2 y p 3 x are not insimpliedform. SIMPLIFYINGSQUAREROOTSWITHFRACTIONS Tosimplifythesquarerootexpression q x y ; 1.Writetheexpressionas p x p y usingtherule q x y = p x p y : 2.Multiplythefractionby1intheformof p y p y : 3.Simplifytheremainingfraction, p xy y : RationalizingtheDenominator Theprocessinvolvedinstep2iscalled rationalizingthedenominator. Thisprocessremovessquareroot expressionsfromthedenominatorusingthefactthat )]TJ/F14 9.9626 Tf 4.566 -1.862 Td [(p y )]TJ/F14 9.9626 Tf 10.793 -1.862 Td [(p y = y: 9.3.9SampleSetB Simplifyeachsquareroot. Example9.22 r 9 25 = p 9 p 25 = 3 5 Example9.23 r 3 5 = p 3 p 5 = p 3 p 5 p 5 p 5 = p 15 5

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673 Example9.24 r 9 8 = p 9 p 8 = p 9 p 8 p 8 p 8 = 3 p 8 8 = 3 p 4 2 8 = 3 p 4 p 2 8 = 3 2 p 2 8 = 3 p 2 4 Example9.25 r k 2 m 3 = p k 2 p m 3 = k p m 3 = k p m 2 m = k p m 2 p m = k m p m = k m p m p m p m = k p m m p m p m = k p m m m = k p m m 2 Example9.26 p x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 x +16= q x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 2 = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 9.3.10PracticeSetB Simplifyeachsquareroot. Exercise9.106 Solutiononp.715. q 81 25 Exercise9.107 Solutiononp.715. q 2 7 Exercise9.108 Solutiononp.715. q 4 5 Exercise9.109 Solutiononp.715. q 10 4 Exercise9.110 Solutiononp.715. q 9 4 Exercise9.111 Solutiononp.716. q a 3 6 Exercise9.112 Solutiononp.716. q y 4 x 3 Exercise9.113 Solutiononp.716. q 32 a 5 b 7 Exercise9.114 Solutiononp.716. q x +9 2 Exercise9.115 Solutiononp.716. p x 2 +14 x +49

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674 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.3.11Exercises Forthefollowingproblems,simplifyeachoftheradicalexpressions. Exercise9.116 Solutiononp.716. p 8 b 2 Exercise9.117 p 20 a 2 Exercise9.118 Solutiononp.716. p 24 x 4 Exercise9.119 p 27 y 6 Exercise9.120 Solutiononp.716. p a 5 Exercise9.121 p m 7 Exercise9.122 Solutiononp.716. p x 11 Exercise9.123 p y 17 Exercise9.124 Solutiononp.716. p 36 n 9 Exercise9.125 p 49 x 13 Exercise9.126 Solutiononp.716. p 100 x 5 y 11 Exercise9.127 p 64 a 7 b 3 Exercise9.128 Solutiononp.716. 5 p 16 m 6 n 7 Exercise9.129 8 p 9 a 4 b 11 Exercise9.130 Solutiononp.716. 3 p 16 x 3 Exercise9.131 8 p 25 y 3 Exercise9.132 Solutiononp.716. p 12 a 4 Exercise9.133 p 32 m 8 Exercise9.134 Solutiononp.716. p 32 x 7 Exercise9.135 p 12 y 13 Exercise9.136 Solutiononp.716. p 50 a 3 b 9

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675 Exercise9.137 p 48 p 11 q 5 Exercise9.138 Solutiononp.716. 4 p 18 a 5 b 17 Exercise9.139 8 p 108 x 21 y 3 Exercise9.140 Solutiononp.716. )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 p 75 a 4 b 6 Exercise9.141 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 p 72 x 2 y 4 z 10 Exercise9.142 Solutiononp.716. )]TJ 7.749 8.718 Td [(p b 12 Exercise9.143 )]TJ 7.749 8.718 Td [(p c 18 Exercise9.144 Solutiononp.716. p a 2 b 2 c 2 Exercise9.145 p 4 x 2 y 2 z 2 Exercise9.146 Solutiononp.716. )]TJ 7.749 8.717 Td [(p 9 a 2 b 3 Exercise9.147 )]TJ/F1 9.9626 Tf 7.749 8.745 Td [(p 16 x 4 y 5 Exercise9.148 Solutiononp.716. p m 6 n 8 p 12 q 20 Exercise9.149 p r 2 Exercise9.150 Solutiononp.716. p p 2 Exercise9.151 q 1 4 Exercise9.152 Solutiononp.716. q 1 16 Exercise9.153 q 4 25 Exercise9.154 Solutiononp.716. q 9 49 Exercise9.155 5 p 8 p 3 Exercise9.156 Solutiononp.717. 2 p 32 p 3 Exercise9.157 q 5 6 Exercise9.158 Solutiononp.717. q 2 7

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676 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.159 q 3 10 Exercise9.160 Solutiononp.717. q 4 3 Exercise9.161 )]TJ/F1 9.9626 Tf 7.749 11.507 Td [(q 2 5 Exercise9.162 Solutiononp.717. )]TJ/F1 9.9626 Tf 7.749 11.507 Td [(q 3 10 Exercise9.163 q 16 a 2 5 Exercise9.164 Solutiononp.717. q 24 a 5 7 Exercise9.165 q 72 x 2 y 3 5 Exercise9.166 Solutiononp.717. q 2 a Exercise9.167 q 5 b Exercise9.168 Solutiononp.717. q 6 x 3 Exercise9.169 q 12 y 5 Exercise9.170 Solutiononp.717. q 49 x 2 y 5 z 9 25 a 3 b 11 Exercise9.171 q 27 x 6 y 15 3 3 x 3 y 5 Exercise9.172 Solutiononp.717. q b +2 4 Exercise9.173 q a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 8 Exercise9.174 Solutiononp.717. q x +2 6 Exercise9.175 q x +2 2 x +1 2 Exercise9.176 Solutiononp.717. q a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 4 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 2 Exercise9.177 q b +7 8 b )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 6 Exercise9.178 Solutiononp.717. p a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(10 a +25

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677 Exercise9.179 p b 2 +6 b +9 Exercise9.180 Solutiononp.717. q a 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a +1 4 Exercise9.181 q x 2 +2 x +1 12 9.3.12ExercisesForReview Exercise9.182 Solutiononp.717. Section5.7 Solvetheinequality 3 a +2 2 a +4 Exercise9.183 Section7.2 Graphtheinequality 6 x 5 x +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 : Exercise9.184 Solutiononp.717. Section7.5 Supplythemissingwords.Whenlookingatagraphfromleft-to-right,lineswith _______sloperise,whilelineswith__________slopefall. Exercise9.185 Section8.9 Simplifythecomplexfraction 5+ 1 x 5 )]TJ/F6 4.9813 Tf 7.699 2.677 Td [(1 x : Exercise9.186 Solutiononp.717. Section9.2 Simplify p 121 x 4 w 6 z 8 byremovingtheradicalsign. 9.4MultiplicationofSquareRootExpressions 4 9.4.1Overview TheProductPropertyofSquareRoots MultiplicationRuleforSquareRootExpressions 9.4.2TheProductPropertyofSquareRoots Inourworkwithsimplifyingsquarerootexpressions,wenotedthat p xy = p x p y Sincethisisanequation,wemaywriteitas p x p y = p xy 4 Thiscontentisavailableonlineat.

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678 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Tomultiplytwosquarerootexpressions,weusetheproductpropertyofsquareroots. TheProductProperty p x p y = p xy p x p y = p xy Theproductofthesquarerootsisthesquarerootoftheproduct. Inpractice,itisusuallyeasiertosimplifythesquarerootexpressionsbeforeactuallyperformingthe multiplication.Toseethis,considerthefollowingproduct: p 8 p 48 Wecanmultiplythesesquarerootsin either oftwoways: Example9.27 Simplifythenmultiply. p 4 2 p 16 3= 2 p 2 4 p 3 =2 4 p 2 3=8 p 6 Example9.28 Multiplythensimplify. p 8 p 48= p 8 48= p 384= p 64 6=8 p 6 Noticethatinthesecondmethod,theexpandedtermthethirdexpression, p 384 maybediculttofactor intoaperfectsquareandsomeothernumber. 9.4.3MultiplicationRuleforSquareRootExpressions Theprecedingexamplesuggeststhatthefollowingruleformultiplyingtwosquarerootexpressions. RuleforMultiplyingSquareRootExpressions 1.Simplifyeachsquarerootexpression,ifnecessary. 2.Performthemultiplecation. 3.Simplify,ifnecessary. 9.4.4SampleSetA Findeachofthefollowingproducts. Example9.29 p 3 p 6= p 3 6= p 18= p 9 2=3 p 2 Example9.30 p 8 p 2=2 p 2 p 2=2 p 2 2=2 p 4=2 2=4

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679 Thisproductmightbeeasierifweweretomultiplyrstandthensimplify. p 8 p 2= p 8 2= p 16=4 Example9.31 p 20 p 7= p 4 p 5 p 7=2 p 5 7=2 p 35 Example9.32 p 5 a 3 p 27 a 5 = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(a p 5 a )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 a 2 p 3 a =3 a 3 p 15 a 2 =3 a 3 a p 15 =3 a 4 p 15 Example9.33 q x +2 7 p x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1= q x +2 6 x +2 p x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1= x +2 3 p x +2 p x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = x +2 3 p x +2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 or = x +2 3 p x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Example9.34 Example9.35 Example9.36

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680 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.4.5PracticeSetA Findeachofthefollowingproducts. Exercise9.187 Solutiononp.717. p 5 p 6 Exercise9.188 Solutiononp.717. p 32 p 2 Exercise9.189 Solutiononp.717. p x +4 p x +3 Exercise9.190 Solutiononp.717. p 8 m 5 n p 20 m 2 n Exercise9.191 Solutiononp.717. q 9 k )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 3 p k 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 k +36 Exercise9.192 Solutiononp.717. p 3 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 2+ p 5 Exercise9.193 Solutiononp.717. p 2 a p 5 a )]TJ 9.962 8.717 Td [(p 8 a 3 Exercise9.194 Solutiononp.717. p 32 m 5 n 8 p 2 mn 2 )]TJ 9.962 8.717 Td [(p 10 n 7 9.4.6Exercises Exercise9.195 Solutiononp.717. p 2 p 10 Exercise9.196 p 3 p 15 Exercise9.197 Solutiononp.718. p 7 p 8 Exercise9.198 p 20 p 3 Exercise9.199 Solutiononp.718. p 32 p 27 Exercise9.200 p 45 p 50 Exercise9.201 Solutiononp.718. p 5 p 5 Exercise9.202 p 7 p 7 Exercise9.203 Solutiononp.718. p 8 p 8 Exercise9.204 p 15 p 15 Exercise9.205 Solutiononp.718. p 48 p 27

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681 Exercise9.206 p 80 p 20 Exercise9.207 Solutiononp.718. p 5 p m Exercise9.208 p 7 p a Exercise9.209 Solutiononp.718. p 6 p m Exercise9.210 p 10 p h Exercise9.211 Solutiononp.718. p 20 p a Exercise9.212 p 48 p x Exercise9.213 Solutiononp.718. p 75 p y Exercise9.214 p 200 p m Exercise9.215 Solutiononp.718. p a p a Exercise9.216 p x p x Exercise9.217 Solutiononp.718. p y p y Exercise9.218 p h p h Exercise9.219 Solutiononp.718. p 3 p 3 Exercise9.220 p 6 p 6 Exercise9.221 Solutiononp.718. p k p k Exercise9.222 p m p m Exercise9.223 Solutiononp.718. p m 2 p m Exercise9.224 p a 2 p a Exercise9.225 Solutiononp.718. p x 3 p x Exercise9.226 p y 3 p y Exercise9.227 Solutiononp.718. p y p y 4 Exercise9.228 p k p k 6

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682 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.229 Solutiononp.718. p a 3 p a 5 Exercise9.230 p x 3 p x 7 Exercise9.231 Solutiononp.718. p x 9 p x 3 Exercise9.232 p y 7 p y 9 Exercise9.233 Solutiononp.718. p y 3 p y 4 Exercise9.234 p x 8 p x 5 Exercise9.235 Solutiononp.718. p x +2 p x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 Exercise9.236 p a )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 p a +1 Exercise9.237 Solutiononp.718. p y +3 p y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 Exercise9.238 p h +1 p h )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 Exercise9.239 Solutiononp.718. p x +9 q x +9 2 Exercise9.240 p y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 q y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 5 Exercise9.241 Solutiononp.718. p 3 a 2 p 15 a 3 Exercise9.242 p 2 m 4 n 3 p 14 m 5 n Exercise9.243 Solutiononp.718. q 12 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(q 3 q 3 p )]TJ/F11 9.9626 Tf 9.963 0 Td [(q 5 Exercise9.244 q 15 a 2 b +4 4 q 21 a 3 b +4 5 Exercise9.245 Solutiononp.718. p 125 m 5 n 4 r 8 p 8 m 6 r Exercise9.246 q 7 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 11 k +1 3 q 14 k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 10 Exercise9.247 Solutiononp.719. p y 3 p y 5 p y 2 Exercise9.248 p x 6 p x 2 p x 9 Exercise9.249 Solutiononp.719. p 2 a 4 p 5 a 3 p 2 a 7 Exercise9.250 p x n p x n

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683 Exercise9.251 Solutiononp.719. p y 2 n p y 4 n Exercise9.252 p a 2 n +5 p a 3 Exercise9.253 Solutiononp.719. p 2 m 3 n +1 p 10 m n +3 Exercise9.254 q 75 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 7 p 48 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(96 Exercise9.255 Solutiononp.719. p 2 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 8+ p 6 Exercise9.256 p 5 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 3+ p 7 Exercise9.257 Solutiononp.719. p 3 )]TJ/F14 9.9626 Tf 4.566 -0.894 Td [(p x + p 2 Exercise9.258 p 11 )]TJ/F14 9.9626 Tf 4.566 -1.862 Td [(p y + p 3 Exercise9.259 Solutiononp.719. p 8 )]TJ/F14 9.9626 Tf 4.566 -0.894 Td [(p a )]TJ 9.962 8.242 Td [(p 3 a Exercise9.260 p x p x 3 )]TJ 9.963 8.718 Td [(p 2 x 4 Exercise9.261 Solutiononp.719. p y p y 5 + p 3 y 3 Exercise9.262 p 8 a 5 p 2 a )]TJ 9.963 8.718 Td [(p 6 a 11 Exercise9.263 Solutiononp.719. p 12 m 3 p 6 m 7 )]TJ 9.963 8.241 Td [(p 3 m Exercise9.264 p 5 x 4 y 3 )]TJ/F14 9.9626 Tf 4.566 -0.797 Td [(p 8 xy )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 p 7 x 9.4.7ExercisesforReview Exercise9.265 Solutiononp.719. Section6.6 Factor a 4 y 4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 w 2 : Exercise9.266 Section7.5 Findtheslopeofthelinethatpassesthroughthepoints )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 ; 4 and )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; 4 : Exercise9.267 Solutiononp.719. Section8.4 Performtheindicatedoperations: 15 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(20 x 6 x 2 + x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 8 x +12 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(15 5 x 2 +15 x x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 Exercise9.268 Section9.2 Simplify p x 4 y 2 z 6 byremovingtheradicalsign.

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684 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.269 Solutiononp.719. Section9.3 Simplify p 12 x 3 y 5 z 8 : 9.5DivisionofSquareRootExpressions 5 9.5.1Overview TheDivisionPropertyofSquareRoots RationalizingtheDenominator ConjugatesandRationalizingtheDenominator 9.5.2TheDivisionPropertyofSquareRoots Inourworkwithsimplifyingsquarerootexpressions,wenotedthat r x y = p x p y Sincethisisanequation,wemaywriteitas p x p y = r x y Todividetwosquarerootexpressions,weusethedivisionpropertyofsquareroots. TheDivisionProperty p x p y = q x y p x p y = r x y Thequotientofthesquarerootsisthesquarerootofthequotient. 9.5.3RationalizingtheDenominator Aswecanseebyobservingtherightsideoftheequationgoverningthedivisionofsquareroots,theprocess mayproduceafractionintheradicand.Thismeans,ofcourse,thatthesquarerootexpressionisnotin simpliedform.Itissometimesmoreusefultorationalizethedenominatorofasquarerootexpressionbefore actuallyperformingthedivision. 5 Thiscontentisavailableonlineat.

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685 9.5.4SampleSetA Simplifythesquarerootexpressions. Example9.37 q 3 7 : Thisradicalexpressionisnotinsimpliedformsincethereisafractionundertheradical sign.Wecaneliminatethisproblemusingthedivisionpropertyofsquareroots. r 3 7 = p 3 p 7 = p 3 p 7 p 7 p 7 = p 3 p 7 7 = p 21 7 Example9.38 p 5 p 3 : Adirectapplicationoftheruleproduces q 5 3 ; whichmustbesimplied.Letusrationalize thedenominatorbeforeweperformthedivision. p 5 p 3 = p 5 p 3 p 3 p 3 = p 5 p 3 3 = p 15 3 Example9.39 p 21 p 7 = q 21 7 = p 3 : Theruleproducesthequotientquickly.Wecouldalsorationalizethedenominatorrstand producethesameresult. p 21 p 7 = p 21 7 p 7 p 7 = p 21 7 7 = p 3 7 7 7 = p 3 7 2 7 = 7 p 3 7 = p 3 Example9.40 p 80 x 9 p 5 x 4 = r 80 x 9 5 x 4 = p 16 x 5 = p 16 p x 4 x =4 x 2 p x Example9.41 p 50 a 3 b 7 p 5 ab 5 = r 50 a 3 b 7 5 ab 5 = p 10 a 2 b 2 = ab p 10 Example9.42 p 5 a p b : Someobservationshowsthatadirectdivisionoftheradicandswillproduceafraction.This suggeststhatwerationalizethedenominatorrst. p 5 a p b = p 5 a p b p b p b = p 5 a p b b = p 5 ab b

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686 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Example9.43 p m )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 p m +2 = p m )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 p m +2 p m +2 p m +2 = p m 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 m )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 m +2 Example9.44 p y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 p y +3 = s y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 y +3 = s y +3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y +3 = s y +3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 y +3 = p y )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 9.5.5PracticeSetA Simplifythesquarerootexpressions. Exercise9.270 Solutiononp.719. p 26 p 13 Exercise9.271 Solutiononp.719. p 7 p 3 Exercise9.272 Solutiononp.719. p 80 m 5 n 8 p 5 m 2 n Exercise9.273 Solutiononp.719. p 196 x +7 8 p 2 x +7 3 Exercise9.274 Solutiononp.719. p n +4 p n )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 Exercise9.275 Solutiononp.719. p a 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 a +8 p a )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 Exercise9.276 Solutiononp.719. p x 3 n p x n Exercise9.277 Solutiononp.719. p a 3 m )]TJ/F6 4.9813 Tf 5.397 0 Td [(5 p a m )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 9.5.6ConjugatesandRationalizingtheDenominator Toperformadivisionthatcontainsabinomialinthedenominator,suchas 3 4+ p 6 ; wemultiplythenumerator anddenominatorbya conjugate ofthedenominator. Conjugate Aconjugateofthebinomial a + b is a )]TJ/F11 9.9626 Tf 9.963 0 Td [(b .Similarly,aconjugateof a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b is a + b Noticethatwhentheconjugates a + b and a )]TJ/F11 9.9626 Tf 10.384 0 Td [(b aremultipliedtogether,theyproduceadierenceof twosquares. a + b a )]TJ/F11 9.9626 Tf 9.962 0 Td [(b = a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(ab + ab )]TJ/F11 9.9626 Tf 9.962 0 Td [(b 2 = a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(b 2 Thisprinciplehelpsuseliminatesquarerootradicals,asshownintheseexamplesthatillustrate ndingtheproductofconjugates.

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687 Example9.45 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5+ p 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(5 )]TJ 9.963 8.242 Td [(p 2 =5 2 )]TJ/F1 9.9626 Tf 9.963 8.07 Td [()]TJ/F14 9.9626 Tf 4.566 0.172 Td [(p 2 2 =25 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 =23 Example9.46 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 6 )]TJ 9.963 8.241 Td [(p 7 )]TJ/F14 9.9626 Tf 10.793 0.171 Td [(p 6+ p 7 = )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 6 2 )]TJ/F1 9.9626 Tf 9.963 8.07 Td [()]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 7 2 =6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 9.5.7SampleSetB Simplifythefollowingexpressions. Example9.47 3 4+ p 6 Theconjugateofthedenominatoris 4 )]TJ 12.086 8.241 Td [(p 6 : Multiplythefractionby1intheformof 4 )]TJ 6.227 5.776 Td [(p 6 4 )]TJ 6.227 5.776 Td [(p 6 3 4+ p 6 4 )]TJ 6.226 5.776 Td [(p 6 4 )]TJ 6.226 5.776 Td [(p 6 = 3 4 )]TJ 6.227 5.776 Td [(p 6 4 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( p 6 2 = 12 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 p 6 16 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 = 12 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 p 6 10 Example9.48 p 2 x p 3 )]TJ 6.227 5.777 Td [(p 5 x : Theconjugateofthedenominatoris p 3+ p 5 x: Multiplythefractionby1intheformof p 3+ p 5 x p 3+ p 5 x : p 2 x p 3 )]TJ 6.227 5.777 Td [(p 5 x p 3+ p 5 x p 3+ p 5 x = p 2 x p 3+ p 5 x p 3 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( p 5 x 2 = p 2 x p 3+ p 2 x p 5 x 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x = p 6 x + p 10 x 2 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 x = p 6 x + x p 10 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 x 9.5.8PracticeSetB Simplifythefollowingexpressions. Exercise9.278 Solutiononp.719. 5 9+ p 7 Exercise9.279 Solutiononp.719. )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 )]TJ 6.227 5.777 Td [(p 3 x Exercise9.280 Solutiononp.719. p 8 p 3 x + p 2 x

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688 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.281 Solutiononp.719. p 2 m m )]TJ 6.227 5.776 Td [(p 3 m 9.5.9Exercises Forthefollowingproblems,simplifyeachexpressions. Exercise9.282 Solutiononp.719. p 28 p 2 Exercise9.283 p 200 p 10 Exercise9.284 Solutiononp.720. p 28 p 7 Exercise9.285 p 96 p 24 Exercise9.286 Solutiononp.720. p 180 p 5 Exercise9.287 p 336 p 21 Exercise9.288 Solutiononp.720. p 162 p 18 Exercise9.289 q 25 9 Exercise9.290 Solutiononp.720. q 36 35 Exercise9.291 q 225 16 Exercise9.292 Solutiononp.720. q 49 225 Exercise9.293 q 3 5 Exercise9.294 Solutiononp.720. q 3 7 Exercise9.295 q 1 2 Exercise9.296 Solutiononp.720. q 5 2 Exercise9.297 q 11 25 Exercise9.298 Solutiononp.720. q 15 36

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689 Exercise9.299 q 5 16 Exercise9.300 Solutiononp.720. q 7 25 Exercise9.301 q 32 49 Exercise9.302 Solutiononp.720. q 50 81 Exercise9.303 p 125 x 5 p 5 x 3 Exercise9.304 Solutiononp.720. p 72 m 7 p 2 m 3 Exercise9.305 p 162 a 11 p 2 a 5 Exercise9.306 Solutiononp.720. p 75 y 10 p 3 y 4 Exercise9.307 p 48 x 9 p 3 x 2 Exercise9.308 Solutiononp.720. p 125 a 14 p 5 a 5 Exercise9.309 p 27 a 10 p 3 a 5 Exercise9.310 Solutiononp.720. p 108 x 21 p 3 x 4 Exercise9.311 p 48 x 6 y 7 p 3 xy Exercise9.312 Solutiononp.720. p 45 a 3 b 8 c 2 p 5 ab 2 c Exercise9.313 p 66 m 12 n 15 p 11 mn 8 Exercise9.314 Solutiononp.720. p 30 p 5 q 14 p 5 q 7 Exercise9.315 p b p 5 Exercise9.316 Solutiononp.720. p 5 x p 2 Exercise9.317 p 2 a 3 b p 14 a Exercise9.318 Solutiononp.720. p 3 m 4 n 3 p 6 mn 5

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690 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.319 p 5 p )]TJ/F10 6.9738 Tf 6.226 0 Td [(q 6 r + s 4 p 25 r + s 3 Exercise9.320 Solutiononp.720. p m m )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 )]TJ/F10 6.9738 Tf 6.226 0 Td [(m 2 +6 m p 3 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 Exercise9.321 p r +1 p r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise9.322 Solutiononp.720. p s +3 p s )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise9.323 p a 2 +3 a +2 p a +1 Exercise9.324 Solutiononp.720. p x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 x +24 p x )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 Exercise9.325 p x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 p x +2 Exercise9.326 Solutiononp.720. p x 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 x +3 p x )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Exercise9.327 p 2 x 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p x )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise9.328 Solutiononp.720. )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 4+ p 5 Exercise9.329 1 1+ p x Exercise9.330 Solutiononp.720. 2 1 )]TJ 6.227 5.031 Td [(p a Exercise9.331 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 p 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise9.332 Solutiononp.721. )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 p 7+2 Exercise9.333 3 p 3 )]TJ 6.227 5.776 Td [(p 2 Exercise9.334 Solutiononp.721. 4 p 6+ p 2 Exercise9.335 p 5 p 8 )]TJ 6.227 5.776 Td [(p 6 Exercise9.336 Solutiononp.721. p 12 p 12 )]TJ 6.227 5.777 Td [(p 8 Exercise9.337 p 7 x 2 )]TJ 6.227 5.777 Td [(p 5 x Exercise9.338 Solutiononp.721. p 6 y 1+ p 3 y Exercise9.339 p 2 p 3 )]TJ 6.227 5.776 Td [(p 2

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691 Exercise9.340 Solutiononp.721. p a p a + p b Exercise9.341 p 8 3 b 5 4 )]TJ 6.227 5.951 Td [(p 2 ab Exercise9.342 Solutiononp.721. p 7 x p 5 x + p x Exercise9.343 p 3 y p 2 y )]TJ 6.226 4.352 Td [(p y 9.5.10ExercisesforReview Exercise9.344 Solutiononp.721. Section2.6 Simplify x 8 y 7 x 4 y 8 x 3 y 4 : Exercise9.345 Section5.7 Solvethecompoundinequality )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 x )]TJ/F8 9.9626 Tf 18.265 0 Td [(23 : Exercise9.346 Solutiononp.721. Section7.6 Constructthegraphof y = 2 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 : Exercise9.347 Section9.2 Thesymbol p x representswhichsquarerootofthenumber x;x 0 ? Exercise9.348 Solutiononp.721. Section9.4 Simplify p a 2 +8 a +16 9.6AdditionandSubtractionofSquareRootExpressions 6 9.6.1Overview TheLogicBehindTheProcess TheProcess 6 Thiscontentisavailableonlineat.

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692 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.6.2TheLogicBehindTheProcess Nowwewillstudymethodsofsimplifyingradicalexpressionssuchas 4 p 3+8 p 3 or 5 p 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 p 2 x +4 )]TJ/F14 9.9626 Tf 4.566 0.172 Td [(p 2 x +1 Theprocedureforaddingandsubtractingsquarerootexpressionswillbecomeapparentifwethinkback totheprocedureweusedforsimplifyingpolynomialexpressionssuchas 4 x +8 x or 5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 a +4 a +1 Thevariables x and a arelettersrepresentingsomeunknownquantitiesperhaps x represents p 3 and a represents p 2 x .Combiningliketermsgivesus 4 x +8 x =12 x or 4 p 3+8 p 3=12 p 3 and 5 a )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 a +4 a +1 or 5 p 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 p 2 x +4 )]TJ/F14 9.9626 Tf 4.567 0.172 Td [(p 2 x +1 5 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 a +4 a +45 p 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(11 p 2 x +4 p 2 x +4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a +4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 p 2 x +4 9.6.3TheProcess Let'sconsidertheexpression 4 p 3+8 p 3 : Therearetwowaystolookatthesimplicationprocess: 1.Weareasking,Howmanysquarerootsof3dowehave? 4 p 3 meanswehave4squarerootsof3 8 p 3 meanswehave8squarerootsof3 Thus,altogetherwehave12squarerootsof3. 2.Wecanalsousetheideaofcombiningliketerms.Ifwerecall,theprocessofcombiningliketermsis basedonthedistributiveproperty 4 x +8 x =12 x because 4 x +8 x =+8 x =12 x Wecouldsimplify 4 p 3+8 p 3 usingthedistributiveproperty. 4 p 3+8 p 3=+8 p 3=12 p 3 Bothmethodswillgiveusthesameresult.Therstmethodisprobablyabitquicker,butkeepinmind, however,thattheprocessworksbecauseitisbasedononeofthebasicrulesofalgebra,thedistributive propertyofrealnumbers. 9.6.4SampleSetA Simplifythefollowingradicalexpressions. Example9.49 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 p 10+11 p 10=5 p 10

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693 Example9.50 4 p 32+5 p 2 : Simplify p 32 : 4 p 16 2+5 p 2=4 p 16 p 2+5 p 2 =4 4 p 2+5 p 2 =16 p 2+5 p 2 =21 p 2 Example9.51 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 x p 75+2 x p 48 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x p 27 : Simpleeachofthethreeradicals. = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 x p 25 3+2 x p 16 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x p 9 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 x p 3+8 x p 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x p 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 x +8 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x p 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 x p 3 Example9.52 5 a p 24 a 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 p 54 a 5 + a 2 p 6 a +6 a: Simplifyeachradical. =5 a p 4 6 a 2 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 p 9 6 a 4 a + a 2 p 6 a +6 a =10 a 2 p 6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 a 2 p 6 a + a 2 p 6 a +6 a = )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(10 a 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(21 a 2 + a 2 p 6 a +6 a = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 a 2 p 6 a +6 a Factorout )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 a Thisstepisoptional. = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 a )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5 a p 6 a )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 9.6.5PracticeSetA Findeachsumordierence. Exercise9.349 Solutiononp.721. 4 p 18 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 p 8 Exercise9.350 Solutiononp.721. 6 x p 48+8 x p 75 Exercise9.351 Solutiononp.721. )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 p 84 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 p 189 x +2 p 21 x Exercise9.352 Solutiononp.721. 9 p 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(8 p 6+3 Exercise9.353 Solutiononp.721. p a 3 +4 a p a Exercise9.354 Solutiononp.721. 4 x p 54 x 3 + p 36 x 2 +3 p 24 x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x

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694 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS 9.6.6SampleSetB Example9.53 Example9.54 Example9.55

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695 Example9.56 3+ p 8 3 )]TJ 6.226 5.777 Td [(p 8 : We'llrationalizethedenominatorbymultiplyingthisfraction by1intheform 3+ p 8 3+ p 8 : 3+ p 8 3 )]TJ 6.227 5.776 Td [(p 8 3+ p 8 3+ p 8 = 3+ p 8 3+ p 8 3 2 )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( p 8 2 = 9+3 p 8+3 p 8+ p 8 p 8 9 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 = 9+6 p 8+8 1 =17+6 p 8 =17+6 p 4 2 =17+12 p 2 Example9.57 2+ p 7 4 )]TJ 6.226 5.776 Td [(p 3 : Rationalizethedenominatorbymultiplyingthisfractionby 1inthefrom 4+ p 3 4+ p 3 : 2+ p 7 4 )]TJ 6.227 5.777 Td [(p 3 4+ p 3 4+ p 3 = 2+ p 7 4+ p 3 4 2 )]TJ/F8 9.9626 Tf 6.226 -0.747 Td [( p 3 2 = 8+2 p 3+4 p 7+ p 21 16 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = 8+2 p 3+4 p 7+ p 21 13 9.6.7PracticeSetB Simplifyeachbyperformingtheindicatedoperation. Exercise9.355 Solutiononp.721. p 5 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 6 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 Exercise9.356 Solutiononp.721. )]TJ/F14 9.9626 Tf 4.567 0.171 Td [(p 5+ p 7 )]TJ/F14 9.9626 Tf 10.793 0.171 Td [(p 2+ p 8 Exercise9.357 Solutiononp.721. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 p 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 p 3+ p 8 Exercise9.358 Solutiononp.721. 4+ p 5 3 )]TJ 6.227 5.776 Td [(p 8 9.6.8Exercises Forthefollowingproblems,simplifyeachexpressionbyperformingtheindicatedoperation. Exercise9.359 Solutiononp.722. 4 p 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 5 Exercise9.360 10 p 2+8 p 2 Exercise9.361 Solutiononp.722. )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 p 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(12 p 6 Exercise9.362 )]TJ 7.749 8.241 Td [(p 10 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p 10 Exercise9.363 Solutiononp.722. 3 p 7 x +2 p 7 x

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696 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.364 6 p 3 a + p 3 a Exercise9.365 Solutiononp.722. 2 p 18+5 p 32 Exercise9.366 4 p 27 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 p 48 Exercise9.367 Solutiononp.722. p 200 )]TJ 9.963 8.241 Td [(p 128 Exercise9.368 4 p 300+2 p 500 Exercise9.369 Solutiononp.722. 6 p 40+8 p 80 Exercise9.370 2 p 120 )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 p 30 Exercise9.371 Solutiononp.722. 8 p 60 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 p 15 Exercise9.372 p a 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 a p a Exercise9.373 Solutiononp.722. p 4 x 3 + x p x Exercise9.374 2 b p a 3 b 5 +6 a p ab 7 Exercise9.375 Solutiononp.722. 5 xy p 2 xy 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y 2 p 2 x 3 y Exercise9.376 5 p 20+3 p 45 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 p 40 Exercise9.377 Solutiononp.722. p 24 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 54 )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 p 12 Exercise9.378 6 p 18+5 p 32+4 p 50 Exercise9.379 Solutiononp.722. )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 p 20 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9 p 125+10 p 180 Exercise9.380 2 p 27+4 p 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 p 12 Exercise9.381 Solutiononp.722. p 14+2 p 56 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 p 136 Exercise9.382 3 p 2+2 p 63+5 p 7 Exercise9.383 Solutiononp.722. 4 ax p 3 x +2 p 3 a 2 x 3 +7 p 3 a 2 x 3 Exercise9.384 3 by p 5 y +4 p 5 b 2 y 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p 5 b 2 y 3 Exercise9.385 Solutiononp.722. p 2 )]TJ/F14 9.9626 Tf 4.566 0.172 Td [(p 3+1 Exercise9.386 p 3 )]TJ/F14 9.9626 Tf 4.566 0.172 Td [(p 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3

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697 Exercise9.387 Solutiononp.722. p 5 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 3 )]TJ 9.963 8.241 Td [(p 2 Exercise9.388 p 7 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 6 )]TJ 9.963 8.241 Td [(p 3 Exercise9.389 Solutiononp.722. p 8 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 3+ p 2 Exercise9.390 p 10 )]TJ/F14 9.9626 Tf 4.566 0.172 Td [(p 10 )]TJ 9.962 8.241 Td [(p 5 Exercise9.391 Solutiononp.722. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ p 3 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(2 )]TJ 9.963 8.241 Td [(p 3 Exercise9.392 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5+ p 6 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 )]TJ 9.963 8.242 Td [(p 6 Exercise9.393 Solutiononp.722. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 )]TJ 9.962 8.242 Td [(p 2 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 )]TJ 9.963 8.242 Td [(p 2 Exercise9.394 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(5+ p 7 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(4 )]TJ 9.963 8.241 Td [(p 7 Exercise9.395 Solutiononp.722. )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 2+ p 5 )]TJ/F14 9.9626 Tf 10.793 0.171 Td [(p 2+3 p 5 Exercise9.396 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 p 6 )]TJ 9.963 8.241 Td [(p 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 p 6+2 p 3 Exercise9.397 Solutiononp.722. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(4 p 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 p 5+ p 3 Exercise9.398 )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(3 p 8 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 2 )]TJ/F8 9.9626 Tf 10.793 -8.069 Td [(4 p 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 p 8 Exercise9.399 Solutiononp.722. )]TJ/F14 9.9626 Tf 4.567 0.172 Td [(p 12+5 p 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 p 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 12 Exercise9.400 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ p 3 2 Exercise9.401 Solutiononp.722. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3+ p 5 2 Exercise9.402 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(2 )]TJ 9.962 8.242 Td [(p 6 2 Exercise9.403 Solutiononp.722. )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2 )]TJ 9.962 8.241 Td [(p 7 2 Exercise9.404 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ p 3 x 2 Exercise9.405 Solutiononp.722. )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(2+ p 5 x 2 Exercise9.406 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3 )]TJ 9.962 8.242 Td [(p 3 x 2 Exercise9.407 Solutiononp.722. 8 )]TJ 9.963 8.49 Td [(p 6 b 2 Exercise9.408 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 a + p 5 a 2

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698 CHAPTER9.ROOTS,RADICALS,ANDSQUAREROOTEQUATIONS Exercise9.409 Solutiononp.723. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(3 y )]TJ 9.962 7.273 Td [(p 7 y 2 Exercise9.410 )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(3+ p 3 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(3 )]TJ 9.963 8.241 Td [(p 3 Exercise9.411 Solutiononp.723. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2+ p 5 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(2 )]TJ 9.963 8.241 Td [(p 5 Exercise9.412 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8+ p 10 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(8 )]TJ 9.963 8.241 Td [(p 10 Exercise9.413 Solutiononp.723. )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(6+ p 7 )]TJ/F8 9.9626 Tf 10.793 -8.07 Td [(6 )]TJ 9.963 8.241 Td [(p 7 Exercise9.414 )]TJ/F14 9.9626 Tf 4.566 0.171 Td [(p 2+ p 3 )]TJ/F14 9.9626 Tf 10.793 0.172 Td [(p 2 )]TJ 9.963 8.241 Td [(p 3 Exercise9.415 Solutiononp.723. )]TJ/F14 9.9626 Tf 4.566 0.172 Td [(p 5+ p 2 )]TJ/F14 9.9626 Tf 10.793 0.172 Td [(p 5 )]TJ 9.963 8.241 Td [(p 2 Exercise9.416 p a + p b p a )]TJ 9.963 8.491 Td [(p b Exercise9.417 Solutiononp.723. )]TJ/F14 9.9626 Tf 4.566 -0.894 Td [(p x + p y )]TJ/F14 9.9626 Tf 10.793 -0.894 Td [(p x )]TJ 9.963 6.207 Td [(p y Exercise9.418 2 5+ p 3 Exercise9.419 Solutiononp.723. 4 6+ p 2 Exercise9.420 1 3 )]TJ 6.227 5.776 Td [(p 2 Exercise9.421 Solutiononp.723. 1 4 )]TJ 6.227 5.776 Td [(p 3 Exercise9.422 8 2 )]TJ 6.227 5.777 Td [(p 6 Exercise9.423 Solutiononp.723. 2 3 )]TJ 6.227 5.777 Td [(p 7 Exercise9.424