S.4 Math Inequalities (3) Caltec S.4

1INEQUALITIESANDREGIONS Summary: 1.Thefollowingsymbolsareusedwhendealingwithinequalities<,,> and 2.Theinequalitysymbolreverseswhenyoumultiplyordivideaninequality byanegativenumber 3.Torepresentaninequalityonanumberline,useanopencirclefor<or> symbolandincaseofor,useaclosedcircle. 4.Integersintherangeofagiveninequalityarecalledintegralvalues 5.Theinequality2>xisthesameasx<2and4<xisthesameasx>4. EXAMPLES: 1.Representeachofthefollowinginequalitiesonanumberline: (i)x2(ii)x>3(iii)x2(iv)x<4(v)1x5 (vi)2<x<3(vii)1<x4(viii)3x<2 2.Giventhat and representPnQon anumberline.StatePnQ 3.Solvethefollowinginequalitiesandrepresenteachsolutiononanumber line: (i)5x+7<3(x+1)(ii)7(2x)+12(2x9)(iii)5x+3>11 2x (iv)3(x1)+2(x1)7x+7(v) (vi) (vii) (ix)743x>5(x)2x44>3x5 24.Usinganumberline,findtheintegralvaluesofxwhichsatisfythesets: 5.Findalltheintegralvaluesofxwhichsatisfytheinequalities: 6.Findthepositiveintegralvaluesofxwhichsatisfytheinequalities: 7.Findthegreatestintegralvalueofxwhichsatisfiestheinequality: 8.Giventhat1<x<4,findthevaluesofaandbforwhicha2x+3<b EER: 1.Solvetheinequality: 2.Solvetheinequality: 3.Solvetheinequality: 4.Solvetheinequality:3(x2)+42(2x3) 5.Solvetheinequality:62(x5)<4 6.Solvetheinequality: 7.Usinganumberline,findtheintegralvaluesofxwhichsatisfythesets: 38.Solvetheinequality: 9.Findtherangeofvaluesofxwhichsatisfytheinequalities: x43x+2<2(x+5) 10.Giventhat and representPnQ onanumberline.StatePnQ 11.Solvetheinequality: 12.Findalltheintegralvaluesofxwhichsatisfytheinequalities: 2x+35x3>8 13.Findalltheintegralvaluesofxwhichsatisfytheinequalities: 2x44>3x5 GRAPHINGLINEARINEQUALITIES Summary: 1.Inshadingouttheunwantedregion,weproceedasfollows: (i)Makeythesubjectinthegiveninequalityequation (ii)Rewritetheequationintheformy=mx+c (iii)Drawasolidlineiftheinequalityisorandincasetheinequalityis <or>,drawadottedline (iv)Iftheinequalityis>or,thewantedregionisabovethelineandIfthe inequalityis<or,thewantedregionisbelowtheline.Thusweshadeout theunwantedregion 2.Thepoints(x,y)fromthewantedregionarecalledanintegralsolution(x andyareintegers) 3.Themaximumandminimumvaluesofagivenexpressioninthewanted regionwillbefoundatoneofitsvertices 4EXAMPLES: 1.Giventhat and byshading theunwantedregion,showtheregionrepresentingPnQ 2.(i)Byshadingtheunwantedregion,showtheregionrepresenting (ii)Findtheintegralsolutionoftheinequalities (iii)Calculatetheareaofthewantedregion 3.(i)Byshadingtheunwantedregion,showtheregionwhichsatisfiesthe inequalities:3x+4y<12,y0andx0 (ii)Findtheintegralsolutionoftheinequalities (ii)Calculatetheareaofthewantedregion 4.(i)Byshadingtheunwantedregions,showclearlytheregionRwhich satisfiestheinequalities:yx<2,2y+5x25and6y+x5 (ii)GiventhatP(x,y)=50x+40y,determinethemaximumandminimum valuesofPintheregionR. (iii)DeterminetheareaoftheunshadedregionR 5.(i)Findtheinequalitiessatisfiedbytheunshadedregionbelow: 03 6yaxi xaxi2 2 5(ii)Calculatetheareaoftheunshadedregion 6.Byshadingtheunwantedregion,showtheregionrepresenting for 2x2 7.Byshadingtheunwantedregion,showtheregionrepresenting for2x2 EER: 1.Byshadingtheunwantedregion,showtheregionwhichsatisfiesthe inequality3x+4y<12 2.Byshadingtheunwantedregion,showtheregionrepresenting 3.Findtheinequalitythatsatisfiestheunshadedregionbelow: 4.(i)Byshadingtheunwantedregion,showtheregionwhichsatisfiesthe inequalities:x+y3,y>x4andy+7x4 3 xaxiyaxi 0 4 6(ii)Calculatetheareaofthewantedregion 5.(i)Byshadingtheunwantedregion,showtheregionrepresenting (ii)Calculatetheareaofthewantedregion 6.(i)Byshadingtheunwantedregion,showtheregionwhichsatisfiesthe inequalities:x4,2y+x4and4y3x8 (ii)Findtheintegralsolutionoftheinequalities (iii)FindthemaximumandminimumvaluesofP=x+yinthewanted region. 7.Findtheinequalitiessatisfiedbytheunshadedregionbelow: 8.Byshadingtheunwantedregion,showtheregionsatisfyingthe inequalities y2x+1andy3 9.(i)Onthesameaxes,drawthecurve for2x2andtheline y=1 (ii)Byshadingtheunwantedregion,showtheregionrepresented and y1 (iii)Statetheintegralcoordinatesofthepointswhichlieintheregion 04 6yaxi xaxi 710.(i)Byshadingtheunwantedregion,showtheregionrepresenting (ii)Calculatetheareaofthewantedregion QUADRATICINEQUALITIES Summary: 1.Solvingaquadraticinequalityisthesameasfindtherangeofxvalues wherethegraphintheequationwillbeaboveorbelowthexaxis 2.Thefollowingstepsapplywhensolvingaquadraticinequality: (i)Replacetheoriginalinequalitywithaquadraticequation (ii)Solvetheequationtogettheendpointsofthethreedifferentintervals (iii)Plotthesolutiononanumberlinetoidentifytheintervalsfor investigation (iv)Pickanumberfromeachintervalandworkoutthesignforeach interval (v)Thesymbolintheinequalitydeterminestherequiredrange.Inany intervalthegraphiseitheraboveorbelowthexaxis EXAMPLES: 1.Findtherangeofxforwhich Soln: Attheendpoints, 8Testingfornegativity(negativesign) Requiredrange=4x3 NOTE:Thefinalanswermusthavethesymbolusedintheoriginal inequality 2.Solveforxintheinequality: Soln: Attheendpoints, Testingforpositivity Requiredrange=x<2orx>3 3.Solveforxintheinequality: Soln: Attheendpoints, 4 3 + + 2 3 + + 9Testingfornegativity Requiredrange=6<x<6 EER: 1.Solveforxintheinequality: 2.Solveforxintheinequality: 3.Solveforxintheinequality: 4.Solveforxintheinequality: 5.Determinethesolutionsetoftheinequality: 6.Findtheintegralvaluesofxwhichsatisfytheinequality: 6 6 + + 10