5Inequalities A-level考试纯数学P1课件.ppt

InequalitiesChapter5

ContentsNotationforinequalitiesSolvinglinearinequalities Inquadraticinequalities

NotationforinequalitesThesefourexpressionsareequivalentabaisgreaterthanbbabislessthanaa≥baisgreaterthanorequaltobb≤abisnotgreaterthana0ab0ab0abThesymbolsandarecalledstrictinequalities,andthesymbols≤and≥arecalledweakinequalities.

SolvinglinearinequalitiesSolvetheinequality3x+1010x–11YoucanaddorsubtractanumberonbothsidesofaninequalityYoucanmultiplyordivideaninequalitybyapositivenumberYoucanmultiplyordivideaninequalitybyanegativenumber,butyoumustchangethedirectionoftheinequality

ExampleSolvetheinequality–3x21Solvetheinequality

QuadraticinequalitiesAquadraticfunctionmighttakeoneofthreeforms:f(x)=ax2+bx+ctheusualformf(x)=a(x–p)(x–q)thefactorformf(x)=a(x–r)2+sthecompletedsquareform

TheeasiestformtouseisthefactorformExampleSolvetheinequality(x–2)(x–4)0xy24

Findthevaluesofxforwhich(x–2)(x–4)=0.Thesevalues,x=2andx=4,arecalledthecriticalvaluesfortheinequality.Makeatableshowingthesignsofthefactorsintheproduct(x–2)(x–4).00+x=4+–0+(x–2)(x–4)+–––x–4++0–x–2x42x4x=2x2

ExampleSolvetheinequality(x+1)(5–x)≤0yx-15

ExampleSolvetheinequalityx2≤a2,wherea000+x=a+–0+(x+a)(x–a)+–––x–a++0–x+axa–axax=–ax–aIfa0,thenthesesta