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CHAPTER4116CHAPTERTABLEOFCONTENTS4-1SolvingEquationsUsingMoreThanOneOperation4-2SimplifyingEachSideofanEquation4-3SolvingEquationsThatHavetheVariableinBothSides4-4UsingFormulastoSolveProblems4-5SolvingforaVariableinTermsofAnotherVariable4-6TransformingFormulas4-7PropertiesofInequalities4-8FindingandGraphingtheSolutionSetofanInequality4-9UsingInequalitiestoSolveProblemsChapterSummaryVocabularyReviewExercisesCumulativeReviewFIRSTDEGREEEQUATIONSANDINEQUALITIESINONEVARIABLEAnequationisanimportantproblem-solvingtool.Asuccessfulbusinesspersonmustmakemanydeci-sionsaboutbusinesspractices.Someofthesedeci-sionsinvolveknownfacts,butothersrequiretheuseofinformationobtainedfromequationsbasedonexpectedtrends.Forexample,anequationcanbeusedtorepresentthefollowingsituation.Helgasewshand-madequiltsforsaleatalocalcraftshop.Sheknowsthatthemate-rialsforthelastquiltthatshemadecost$76andthatitrequired44hoursofworktocompletethequilt.IfHelgareceived$450forthequilt,howmuchdidsheearnforeachhourofwork,takingintoaccountthecostofthematerials?Mostoftheproblem-solvingequationsforbusinessarecomplex.Beforeyoucancopewithcomplexequa-tions,youmustlearnthebasicprinciplesinvolvedinsolvinganyequation.SomeTermsandDefinitionsAnequationisasentencethatstatesthattwoalgebraicexpressionsareequal.Forexample,x39isanequationinwhichx3iscalledtheleftside,orleftmember,and9istherightside,orrightmember.Anequationmaybeatruesentencesuchas527,afalsesentencesuchas634,oranopensentencesuchasx39.Thenumberthatcanreplacethevariableinanopensentencetomakethesentencetrueiscalledaroot,orasolution,oftheequation.Forexample,6isarootofx+39.AsdiscussedinChapter3,thereplacementsetordomainisthesetofpos-siblevaluesthatcanbeusedinplaceofthevariableinanopensentence.Ifnoreplacementsetisgiven,thereplacementsetisthesetofrealnumbers.Thesetconsistingofallelementsofthereplacementsetthataresolutionsoftheopensentenceiscalledthesolutionsetoftheopensentence.Forexample,ifthereplacementsetisthesetofrealnumbers,thesolutionsetofx39is{6}.Ifnoelementofthereplacementsetmakestheopensentencetrue,thesolutionsetistheemptyornullset,or{}.Ifeveryelementofthedomainsatisfiesanequation,theequationiscalledanidentity.Thus,5xx(5)isaniden-titywhenthedomainisthesetofrealnumbersbecauseeveryelementofthedomainmakesthesentencetrue.Twoequationsthathavethesamesolutionsetareequivalentequations.Tosolveanequationistofinditssolutionset.Thisisusuallydonebywritingsim-plerequivalentequations.Ifnoteveryelementofthedomainmakesthesentencetrue,theequationiscalledaconditionalequation,orsimplyanequation.Therefore,x39isaconditionalequation.PropertiesofEqualityWhentwonumericaloralgebraicexpressionsareequal,itisreasonabletoassumethatifwechangeeachinthesameway,theresultingexpressionswillbeequal.Forexample:5712(57)3123(57)81282(57)2(12)Theseexamplessuggestthefollowingpropertiesofequality:517351234-1SOLVINGEQUATIONSUSINGMORETHANONEOPERATIONSolvingEquationsUsingMoreThanOneOperation117PropertiesofEquality1.Theadditionpropertyofequality.Ifequalsareaddedtoequals,thesumsareequal.2.Thesubtractionpropertyofequality.Ifequalsaresubtractedfromequals,thedifferencesareequal.3.Themultiplicationpropertyofequality.Ifequalsaremultipliedbyequals,theproductsareequal.4.Thedivisionpropertyofequality.Ifequalsaredividedbynonzeroequals,thequotientsareequal.5.Thesubstitutionprinciple.Inastatementofequality,aquantitymaybesubstitutedforitsequal.Tosolveanequation,youneedtoworkbackwardor“undo”whathasbeendonebyusinginverseoperations.Toundotheadditionofanumber,additsopposite.Forexample,tosolvetheequationx719,usetheadditionprop-ertyofequality.Addtheoppositeof7tobothsides.Thevariablexisnowaloneononesideanditiseasytoreadthesolution,x12.Tosolveanequationinwhichthevariablehasbeenmultipliedbyanum-ber,eitherdividebythatnumberormultiplybyitsreciprocal.(Remembermultiplyingbythereciprocalisthesameasdividingbythenumber.)Tosolve6x24,dividebothsidesby6ormultiplybothsidesby.6x246x24orx4x4Tosolve,multiplyeachsidebythereciprocalofwhichis3.x15Intheequation2x315,therearetwooperationsintheleftside:mul-tiplicationandaddition.Informingtheleftsideoftheequation,xwasfirstmul-tipliedby2,andthen3wasaddedtotheproduct.Tosolvethisequation,wemustundotheseoperationsbyusingtheinverseelementsinthereverseorder.Sincethelastoperationwastoadd3,thefirststepinsolvingtheequationistoadditsopposite,3,tobothsidesoftheequationorsubtract3frombothsides(3)x35(3)5x35513x35516(6x)516(24)6x6524616x175192727x512118FirstDegreeEquationsandInequalitiesinOneVariableoftheequation.Hereweareusingeithertheadditionorthesubtractionprop-ertyofequality.orNowwehaveasimplerequationthathasthesamesolutionsetastheoriginalandincludesonlymultiplicationby2.Tosolvethissimplerequation,wemulti-plybothsidesoftheequationby,thereciprocalof2,ordividebothsidesoftheequationby2.Herewecanuseeitherthemultiplicationorthedivisionpropertyofequality.orAfteranequationhasbeensolved,wechecktheequation,thatis,weverifythatthesolutiondoesinfactmakethegivenequationtruebyreplacingthevari-ablewiththesolutionandperformi