Chapter6:
SolvingandGraphingLinear
Inequalities
6.1/6.2SolvingandGraphingLinearInequalities
Alinearinequalityrepresentsasetofnumbersthatare
lessthan<
greaterthan>
lessthanorequalto≤
orgreaterthanorequalto≥
agivenvalue.
Theinequalitycanbeexpressedalgebraicallyorasagraph.
Solvelinearinequalitiesjustlikeyouwouldsolveanequationexceptwhenyoumultiplyordividebyanegative
numberyoumustreversetheinequalitysymbol.
Examples:Solveandgraphtheinequalitiesonanumberline.
1)
x+7>10
2)
x–10≤12 3)
x–23≥-40
________________________
________________________
________________________
$
'
4)
-4x>28
5)
≥− 6)
-6x≥48
%
(
________________________
________________________
________________________
Application:
WriteaninequalitytorepresentthetemperatureTofpeoplewithtemperaturesabove98.6degreesF.
Graphtheinequality.
________________________
Homework6.1/6.2
Sketchagraphoftheinequality.
1)
x≤ 3 2)
x>-2
____________________________________
____________________________________
Solvetheinequalityandgraphitssolutionsetonthenumberline.
3)
-3+x<-5
4)
x–4>-2
____________________________________
____________________________________
5)
-8x≤-24
6)
-4.2≥x+1.9
____________________________________
____________________________________
$
(
'
.
7)
> 8)
𝑥≤− *
*
,
%
____________________________________
____________________________________
Writeandgraphaninequalitytorepresentthefollowingscenario.
9)
Heliumistheelementthathasthelowestboilingpoint,-268.9degreesC.Writeaninequalitythat
describestheboilingpointbofanyotherelement.Graphtheinequality.
____________________________________
10) ThelowestelevationinCaliforniais282feetbelowsealevel.LetErepresenttheelevationofany
locationinCalifornia.WriteaninequalityforEandgraphtheinequality.
___________________________________
6.3 Multi-StepInequalities
Solveeachinequalityandgraphthesolutionsetonthenumberline.
Usethepropertiesofequalitytosolve,rememberingtoreversetheinequalitywhenevermultiplyingor
dividingbyanegativenumber.
1)
2x–5<7
________________________________________
2)
3(x+2)≥7 ________________________________________
3)
2x–3>4x–1
________________________________________
4)
12>-5x+2 ________________________________________
5)
-6(x+3)<4 ________________________________________
Homework6.3
Solveandgraphtheinequality.
1.
6x+1≤-2 2.
2x+3<6x–1
3.
2x–14>4x+4
4.
6x+3<3(x+2)
5.
-2(x+4)>6x–4
6.
7–3x≥x+9 %
7.
𝑥 − 8 > −4
'
8.
7–5x≤9–4x
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
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6.4 CompoundInequalities:ANDInequalities
Whenwewanttodescribeanintervalofnumberslocatedbetweentwovalues,wecanuseacompound
inequalityconnectedbytheword“AND.”Forexample,“Thenumberofstudentsintheclassroomcanbe
greaterthanorequalto12ANDlessthanorequalto25.”Wecouldwritemathematicalinequalitiesto
representthenumberofstudentsintheclassroom.
𝑥 ≥ 12𝑨𝑵𝑫𝑥 ≤ 25
Thiscouldalsobewritten: 12 ≤ 𝑥𝑨𝑵𝑫𝑥 ≤ 25
Moresimply: 𝑥 ≤ 12 ≤ 25 Eachofthesestatementsdescribesthenumberofstudentsinaclass.Thesearecalled“compound
inequalities.”
Examples:
Writeacompoundinequalitytorepresenteachstatementbelow.
1.
TheaveragedaytimetemperatureinJulyisgreaterthan70degreesFandlessthan95degreesF.
2.
Thepriceofgasolinelastweekrangedfrom$2.12pergallonto$2.35pergallon.
3.
Thenumberoftextmessagessentbyagroupofstudentsbetween4:00and5:00p.m.rangedfrom37
to85messages.
4.
Theaveragespeedofcarsonalocalhighwayrangefrom47mphto82mph.
5.
Thepriceofapairofjeansatalocalmallrangesfrom$25.99to$59.99.
Writeaninequalitythatrepresentsthestatementandgraphtheinequality.
6.
xislessthan4andgreaterthan0 ________________________________
7.
xislessthan4andgreaterthanorequalto-3
________________________________
8.
xisgreaterthanorequalto5andlessthanorequalto10
_________________________________
Solvetheinequalityandgraphitssolutiononthenumberline.
9.
-14<7x<21 _____________________________________
10.
-8<x–11<-6
_____________________________________
11.
3≤ 2𝑥 + 1 ≤ 5
_____________________________________
6.5CompoundInequalities:ORinequalities
Twoinequalitiesconnectedbytheword“OR”typicallydescribeasetofvaluesoutsideoftwoboundaries.
Examples:
1)
x<-7ORx>3
Graph:
_________________________________
2)
x≤ −10ORx>0 Graph:
_________________________________
3)
x<-5ORx≥ 5
Graph:
_________________________________
Solveandsketchthegraph.
4)
x–3≥ 2ORx+4≤6 Graph:
_________________________________
5)
18–x<5OR14–3x>5 Graph:
_________________________________
6)
-22>11xOR4+x>4
Graph:
_________________________________
Homework6.4/6.5
Solveandsketchagraphoftheinequalities.
1.
x≤-3orx>1
______________________________________________
2.
-5<xandx<12
______________________________________________
3.
-22>11xor4+x>4
______________________________________________
$
4.
< −3𝑜𝑟 − 5 > −6 − 𝑥 ______________________________________________
'
5.
1.5<xandx<3 ______________________________________________
6.
x>2andx<7
______________________________________________
7.
6+2x>20or8+x≤0
______________________________________________
8.
-3<x≤ 3
______________________________________________ Translatingverbalphrasesintoinequalities.
Translatetheverbalphraseusingthevariablexandasingleinequalityoracompoundinequality.
1.
Todonatebloodyoumustweighatleast110pounds.Writeaninequalityfortheweightofdonors.
2.
Youmustbeatleast16yearsoldtogetadrivingpermitintheU.S.Writeaninequalityforthepeople
eligibletogetadrivingpermit.
3.
Thespeedlimitontheexpresswayisaminimumof45mphandamaximumof60mph.Writean
inequalityforthespeedsthatyoucandriveontheexpressway.
4.
Thecostofgasolinethisyearrangedfrom$3.85pergallonto$1.95pergallon.Writeaninequalityfor
therangeofgasprices.
5.
Therangeofscoresonarecenttestwasfrom45%to90%.Writeaninequalityfortherangeoftest
scores.
6.
ThetypicaldaytimetemperatureinColoradointhewinterrangefrom22degreesto35degrees.Write
aninequalityforthetypicaldaytimetemperatures.
7.
Peoplewhoareunder4yearsofageorover65yearsofageareadmittedfreetotheamusementpark.
Writeaninequalityfortheagesofpeoplewhogetinfree.
8.
WriteaninequalitydescribingthegradelevelsofstudentsatPenncrestHighSchool.
9.
ThemostpointsthatJasonhasscoredinabasketballgameare27points.Hislowestscoringgamewas
whenhedidn’tscoreatall.WriteaninequalitytoexpressthepointsJasonscoresinagame.
10.
Theweightofatruckvariesfrom4200poundsto6300poundsdependingonhowmuchcargoit
carries.Writeaninequalityexpressingtheweightofthetruck.
CHAPTERREVIEW6.1-6.5 6.1SolvingInequalitiesUsingAdditionorSubtraction.
Solvetheinequality.Graphthesolution.
a + 6 > 28 x − 5 ≤ −3 1.
2.
3.
6.2SolvingInequalitiesUsingMultiplicationorDivision.
Solvetheinequality.Graphthesolution.
64 < 8x −6k > −30 5.
6.
7.
3
x
− n ≥ 9
3< 9.
10.
11.
2
5
6.3SolvingMulti-stepInequalities.
Solvetheinequality.
6x − 8 ≥ 4 10 − 3x < −5 13.
14.
−8 < −10 + x 4.
7 + z ≥ 20 −81 ≥ −3p 8.
−81 > 9r T
≤4
14
12.
1
− y ≥ 3
6
15.
4x − 9 ≥ 11 16.
5(x − 2) ≤ 10 17.
5 − 8x ≤ −3x 5x > 12 + x 19.
20.
6.4SolvingCompoundInequalitiesInvolving“and”
Solvetheinequality.Graphthesolution.
9 < x + 1 < 13 −3 ≤ 3x ≤ 15 22.
23.
25.
1 < 2x − 3 < 5 26.
18.
1
(x + 8) < 1 4
21.
3x − 9 ≤ 2x + 4 24.
−1 ≤ x − 2 < 3 27.
−7 < 3 −
−3(x − 1) > 4 0 < 4− x ≤ 5 1
x ≤1
4
6.5SolvingCompoundInequalitiesInvolving“or”
Solvetheinequality.Graphthesolution.
x > 4 or 3x ≤ −9 2x ≤ −10 or x + 3 > 1 28.
29.
31.
6x − 2 ≤ 4 or 3x > 21 32.
3x + 2 ≤ −7 or 2x + 1 ≥ 9 30.
x − 7 ≥ 0 or 3 + x < −2 33.
1
1
x<
or 3x − 6 > 24 4
2
6.6AbsoluteValueEquations
Absolutevalueisthedistancefrom0toagivennumberonthenumberline.
Forexample, 4 representsthedistancefrom0to4onthenumberline.Thatdistanceis4.
Anotherexample, −6 representsthedistancefrom0to-6onthenumberline.Thatdistanceis6.
Review:
3 =3 −3 =3
Itfollowsthat: If
𝑥 = 3,thenx=3orx=-3
Solveforx:
1)
𝑥 = 5
x=_______ or
x=_______
2)
𝑥 = 9
x=_______ or
x=_______
3)
𝑥 = 1.6
x=_______ or
x=_______
4)
𝑥 = −5
x=_______ or
x=_______
Whenthereisanexpressioninsideoftheabsolutevaluesymbolthesolutionisonestepmorecomplicated.
Solvetheabsolutevalueequationsbelowbyrewritingastwoequations.
5)
𝑥 − 2 = 3 6)
2𝑥 − 1 = 5
7)
3𝑥 − 6 = 21
8)
3𝑥 + 8 = 7
6.8 Inequalitiesinthecoordinateplane
Thegraphofaninequalityinthecoordinateplaneisashadedregionrepresentingallcoordinatepairsthat
satisfythegiveninequality.
Fortheinequality: x+y≥3
Circlethecoordinatepairsonthecoordinateplane
thatmaketheinequalityabovetrue.
Drawtheboundarylinethatdividesthepointsthatmaketheinequalitytruefromthepointsthatdonotmake
theinequalitytrue.Shadetheregionofpointsthatmaketheinequalitytrue.
1)
Sketchthegraphofaninequalityintwovariablesinthecoordinateplane.
y>2x–5
1)
Graphtherelatedequation:y=2x–5
2)
Drawthelineusingasolidlinefor≤ 𝑜𝑟 ≥
Drawthelineusingabrokenlinefor<or>
3)
Shadethehalf-planethatmakestheinequalitytrue.
2)
Sketchthegraphofaninequalityintwovariablesinthecoordinateplane.
𝑦 ≤ 2𝑥 + 6
1)
Graphtherelatedequation:y=2x+6
2)
Drawthelineusingasolidlinefor≤ 𝑜𝑟 ≥
Drawthelineusingabrokenlinefor<or>
3)
Shadethehalf-planethatmakestheinequalitytrue.
3)
Sketchthegraphofaninequalityintwovariablesinthecoordinateplane.
X>3
1)
Graphtherelatedequation:x=3 2)
Drawthelineusingasolidlinefor≤ 𝑜𝑟 ≥
Drawthelineusingabrokenlinefor<or>
3)
Shadethehalf-planethatmakestheinequalitytrue.
4)
Sketchthegraphofaninequalityintwovariablesinthecoordinateplane.
𝑦 + 3 ≤ −1
1)
Graphtherelatedequation:y+3=-1
2)
Drawthelineusingasolidlinefor≤ 𝑜𝑟 ≥
Drawthelineusingabrokenlinefor<or>
3)
Shadethehalf-planethatmakestheinequalitytrue.
Homework6.8
Iseachorderedpairasolutionoftheinequality?IndicatebywritingYESorNO.
1)
y–x>5
(3,2) (-1,4) 2)
y–2x≤6
(-3,-1)
3)
x>4 (3,2) (-1,4) 4)
y≥-2 (-4,0) Sketchthegraphoftheinequality.
5)
𝑦 − 𝑥 ≥ 3
6)
y+x≥2
7)
y+x<5
8)
y–x<-2
(2,7)
(3,7)
9)
11)
13)
y–2x<-4
10)
y+3x≤ 2
y<2 12)
x≤-2
-4y>-16
14)
5y≥ −25
ReviewSheet6.6,6.8
AbsoluteValueEquations
Solvetheabsolutevalueequation.Iftheequationhasnosolution,writenosolution.
6x − 4 = 2 3x + 5 = 22 2x + 5 = 3 1.
2.
3.
x − 7 = −5 5 − 4x − 3 = 4 2x − 4 − 8 = 10 4.
5.
6.
1
x =8
2x − 4 = 0 5x + 10 + 15 = 60 7.
8.
9.
2
GraphingLinearInequalities
Graphtheinequality.
x ≥ 4 1.
3.
y ≤ 3x − 5 2.
y < −2 4.
−2x + y ≥ 4 5.
7.
1
y>− x+4 3
y ≥ −3x + 7 6.
y ≤ 2x − 5 8.
y<
1
x−2
5