College Algebra - 6Ed.pdf

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1、111.1LinearEquationsEQUATIONS,andProblemSolvingINEQUALITIES,AND1.2MoreEquationsandApplicationsPROBLEMSOLVING1.3QuadraticEquations1.4ApplicationsofLinearandQuadraticEquations1.5MiscellaneousEquations1.6Inequalities1.7InequalitiesInvolvingQuotientsandAbsoluteValueWecanusetheequation0.

2、2(8)20.2x1x50.9(8)todeterminethechangeofthemixtureinaradiatorfrom20%antifreezeto90%antifreeze.Wewoulddrain7litersofcoolantfromaradiatorthatholds8litersandreplacethemwithpureantifreeze,whichwillchangetheprotectionagainsttemperaturefrom12°Fto220°F.8212Equations,Inequalities,andProblem

3、Solving1.1LinearEquationsandProblemSolving83Acommonthreadthroughoutprecalculusalgebracoursesisoneofdevelop-ingalgebraicskills,thenusingtheskillstosolveequationsandinequalities,andthenusingequationsandinequalitiestosolveappliedproblems.Inthischapterweshallreviewandextendavarietyofcon

4、ceptsrelatedtothatthread.1.1LINEAREQUATIONSANDPROBLEMSOLVINGAnalgebraicequationsuchas5x12512isneithertruenorfalseasitstands;itissometimesreferredtoasanopensentence.Eachtimethatanumberissubstitutedforx,thealgebraicequation5x12512becomesanumericalstatementthatiseithertrueorfalse.Forex

5、ample,ifx55,then5x12512becomes5(5)12512,whichisafalsestatement.Ifx52,then5x12512becomes5(2)12512,whichisatruestatement.Solvinganequationreferstotheprocessoffindingthenumber(ornumbers)thatmake(s)analgebraicequationatruenumericalstatement.Suchnumbersarecalledthesolutionsorrootsoftheeq

6、uationandaresaidtosat-isfytheequation.Thesetofallsolutionsofanequationiscalleditssolutionset.Thus{2}isthesolutionsetof5x12512.Anequationthatissatisfiedbyallnumbersthatcanmeaningfullyreplacethevariableiscalledanidentity.Forexample,3(x12)53x16x2245(x12)(x22)and1121x}}1}}5}}x22xarealli

7、dentities.Inthelastidentity,xcannotequalzero;thusthestatement1121x}}1}}5}}x22xistrueforallrealnumbersexceptzero.Anequationthatistrueforsomebutnotallpermissiblevaluesofthevariableiscalledaconditionalequation.Thustheequa-tion5x12512isaconditionalequation.Equivalentequationsareequation

8、sthathavethesamesol