Chapter 8 More than how many binoculars must be produced and sold for

Ch.8 SystemsofEquationsandInequalities

8.1 SystemsofLinearEquationsinTwoVariables

1 DecideWhetheranOrderedPairisaSolutionofaLinearSystem

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Determinewhetherthegivenorderedpairisasolutionofthesystem.

1) (1

,

–1)

x+y=0

x–y=2

A) solution B) notasolution

2) (–6

,

–3)

x+y=3

x–y=9

A) notasolution B) solution

3) (2

,

3)

2x+y=7

3x+2y=12

A) solution B) notasolution

4) (6

,

–6)

4x–y=18

2x+4y=–12

A) notasolution B) solution

5) (2

,

3)

2x–y=7

3x–2y=12

A) notasolution B) solution

6) (–4

,

3)

4x+y=–19

2x+4y=–20

A) notasolution B) solution

7) (–3

,

5)

2x=–1–y

3x=1–2y

A) solution B) notasolution

8) (–5

,

1)

3x=16–y

2x=13–3y

A) notasolution B) solution

Page1

2 SolveLinearSystemsbySubstitution

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequationsbythesubstitutionmethod.

1)

x+y=-

3

y=-

2x

A) {(3

,

–6)} B) {(3

,

6)} C) {(–3

,

–6)} D) {(–3

,

6)}

2)

–5x–3y =-

168

x=5y

A) {(30

,

6)} B) {(6

,

30)} C) {(30

,

–6)} D) {(31

,

6)}

3)

y=4x–3

2y+8x =26

A) {(2

,

5)} B) {(5

,

2)} C) {(–2

,

5)} D) ∅

4)

2y =x+4

3x+6y =0

A) {(–2

,

1)} B) {(1

,

–2)} C) {(–2

,

–1)} D) {(2

,

0)}

5)

y=4x+7

y=9x+6

A) 1

5,39

5B) 39

5,1

5C) {(1,11)} D) ∅

6)

4x+2y =162

x–4y =0

A) {(36

,

9)} B) {(9

,

36)} C) {(36

,

–9)} D) {(37

,

9)}

7) x–7y=40

–2x–8y=30

A) {(5

,

–5)} B) {(4

,

–4)} C) {(–5

,

–4)} D) ∅

8) 9x–6y=–12

4x–2y=–2

A) {(2

,

5)} B) {(1

,

6)} C) {(2

,

6)} D) ∅

9) 4x+y=11

–4x+5y=–5

A) 5

2,1 B) {(1,–9)} C) 5

,

–1 D) 7

2,–3

10) 3x–2y=2–3x

3x+7y=x+6y+4

A) {(1

,

2)} B) {(1

,

–2)} C) {(0,2)} D) 1

3,0

Page2

11) y=1

5x–5

y=1

3x–7

A) {(15

,

–2)} B) {(15

,

2)} C) {(25

,

0)} D) 16

5,–13

2

3 SolveLinearSystemsbyAddition

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystembytheadditionmethod.

1) x+y=–9

x–y=16

A) {(3.5

,

–12.5)} B) {(3.5

,

12.5)} C) {(9

,

–12.5)} D) {(9

,

3.5)}

2) 4x +3y =-

2

–4x –9y =14

A) {(1,–2)} B) {(–1,2)} C) {(–1,–2)} D) ∅

3) 2x+7y=–10

2x+2y=20

A) {(16

,

–6)} B) {(–16

,

2)} C) {(–16

,

7)} D) {(–6

,

16)}

4) x+4y=–24

5x+3y=–35

A) {(–4

,

–5)} B) {(–5

,

–4)} C) {(4

,

–4)} D) ∅

5) 9x+y=–15

–3x–2y=10

A) –4

3,–3 B) {(1,9)} C) –2

,

–2 D) –1

3,–12

6) 8x+16y=16

3x–8y=–8

A) {(0,1)} B) {(1,0)} C) {(1,1)} D) {(0,0)}

7) 2x+20y=–162

8x+5y=27

A) {(9

,

–9)} B) {(–9

,

9)} C) {(8

,

–8)} D) {(–5

,

9)}

8) 7x+8y=–19

4x–3y=–26

A) {(–5

,

2)} B) {(–6

,

3)} C) {(–5

,

3)} D) ∅

9) 3x+7y=8

–7x+4y=–39

A) {(5

,

–1)} B) {(–5

,

1)} C) {(5

,

1)} D) {(–5

,

–1)}

10) 3y=24–6x

2x=40–3y

A) {(–4

,

16)} B) {(3

,

–16)} C) {(6

,

–16)} D) ∅

Page3

4 IdentifySystemsThatDoNotHaveExactlyOneOrdered –PairSolution

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystembythemethodofyourchoice.Identifysystemswithnosolutionandsystemswithinfinitelymany

solutions,usingsetnotationtoexpresstheirsolutionsets.

1) x+y=6

x+y=–8

A) {(0,–2)} B) {(6

,

–8)} C) {(x,y) x+y=6} D) ∅

2) y=18–6x

6x+y=54

A) {(10

,

12)} B) {(12

,

6)}

C) {(x,y)6x+y=18}D)

∅

3) y=10–3x

9x+3y=30

A) {(0,10)} B) {(5,–5)}

C) {(x,y)3x+y=10}D)

∅

4) 3x +y=9

12x +4y =36

A) {(0,9)} B) {(5,–6)} C) {(x,y)3x+y=9}D)

∅

5) x+y=–5

x–y=11

A) {(3

,

–8)} B) {(3

,

8)} C) {(x,y)|x+y= –5} D) ∅

6) 4x–8y=8

–12x+24y=–32

A) {(3

,

4)} B) 1

3,–2

3

C) {(x,y) 4x–8y=8}D)

∅

7) x+5y=–21

7x+4y=–54

A) {(–6

,

–3)} B) {(–7

,

–2)}

C) {(x,y) x+5y=–21} D) ∅

8) 4y=43–5x

2x=58–4y

A) {(–5

,

17)} B) {(4

,

–17)}

C) {(x,y) 5x+4y=43}D)

∅

9) x–7y=8

2x–14y=16

A) {(8

,

0)} B) {(0,0)} C) {(x,y)|x–7y=8} D) ∅

10) y=–6x–5

18x+3y=–15

A) {(0,–5)} B) {(0,0)}

C) {(x,y)|6x+y=–5} D) ∅

Page4

11) 4x+y=7

2y=14–8x

A) 7

4,0 B) {(0,7)} C) {(x,y)|4x+y=7} D) ∅

12) y=1

3x+2

x–3y=–6

A) {(–6

,

0)} B) {(0,2)} C) {(x,y)|x–3y= –6} D) ∅

13) 9x–8y=4

–18x+16y=–16

A) 3

4,–2

3B) {(2

,

4)}

C) {(x,y)|9x–8y=4} D) ∅

14) x

2+y

2=–2

x–y=8

A) {(2

,

–6)} B) {(1

,

–5)} C) {(x,y) x–y=8}D)

∅

15) x

5+y

15 =1

x

4–y

12 =0

A) 5

2,15

2B) 15

2,5

2

C) (x,y) x

5+y

15 =1 D) ∅

5 SolveProblemsUsingSystemsofLinearEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Thesumoftwonumbersis–9.Ifonenumberissubtractedfromtheother,theirdifferenceis5.Findthe

numbers.

A) –2

,

–7B)

–2

,

7C)2

,

–7D)

–10

,

1

2) Onenumberis1lessthanasecondnumber.Twicethesecondnumberis12 lessthan3timesthefirst.Findthe

twonumbers.

A) 14and15 B) 13and14 C) 15 and16 D) –15 and–14

3) Onenumberis2lessthanasecondnumber.Twicethesecondnumberis22 morethan5timesthefirst.Find

thetwonumbers.

A) –6and–4B)

–7and–5C)

–5 and–3D)4and6

4) Twocarsleaveacityandheadinthesamedirection.After3 hours,thefastercaris24milesaheadofthe

slowercar.Theslowercarhastraveled159miles.Findthespeedsofthetwocars.

A) 53mphand61mph B) 55 mphand63 mph

C) 80mphand88mph D) 45 mphand53 mph

Page5

Thefigureshowsthegraphsofthecostandrevenuefunctionsforacompanythatmanufacturesandsellsbinoculars.

Usetheinformationinthefiguretoanswerthequestion.

5) Howmanybinocularsmustbeproducedandsoldforthecompanytobreakeven?

A) 750binoculars B) 2250binoculars C) 1500binoculars D) 2700binoculars

6) Atthebreak–evenpointbothcostandrevenuearewhat?

A) $2250 B) $750 C) $1500 D) $2700

7) Morethanhowmanybinocularsmustbeproducedandsoldforthecompanytohaveaprofitgain?

A) 750binoculars B) 2250binoculars C) 1500binoculars D) 2700binoculars

8) Fewerthanhowmanybinocularsmustbeproducedandsoldforthecompanytohaveaprofitloss?

A) 750binoculars B) 2250binoculars C) 1500binoculars D) 2700binoculars

9) Usetherevenueandcostfunctionstowritetheprofitfunctionfromproducingandsellingxbinoculars.

A) P(x)=2x–1500 B) P(x)=2x+1500 C) P(x)=4x+1500 D) P(x)=4x–1500

10) Isthereaprofitwhen518binocularsareproduced?

A) No B) Yes

11) Isthereaprofitwhen889binocularsareproduced?

A) Yes B) No

12) Whatistheprofitwhen980binocularsareproduced?

A) $460 B) $3460 C) $2420 D) $5420

Solvetheproblem.

13) Steveinvestsinacircusproduction.Thecostincludesanoverheadof$24,000

,

plusproductioncostsof$5000

perperformance.Asold–outperformancebringsin$7000.Letxrepresentthenumberofsold–out

performancesandwritethecostfunction,Candrevenuefunction,R.

A) C(x)=24,000+5000x

R(x)=7000x

B) C(x)=24,000 +7000x

R(x)=5000x

C) C(x)=5000x

R(x)=24,000+7000x

D) C(x)=24,000x+5000

R(x)=7000x

14) Steveinvestsinacircusproduction.Thecostincludesanoverheadof$20,000

,

plusproductioncostsof$2000

perperformance.Asold–outperformancebringsin$4000.Determinethenumberofsold–outperformances,x,

neededtobreakeven.

A) 10performances B) 11performances C) 12 performances D) 4performances

Page6

15) Steveinvestsinacircusproduction.Thecostincludesanoverheadof$36,000

,

plusproductioncostsof$4000

perperformance.Asold–outperformancebringsin$7000.Determinethedollaramountcominginandgoing

outatthebreak–evenpoint.

A) $84,000 B) $48,000 C) $12 D) $22,909

16) An–Meiownsabusinessmakingandsellingjackets.Shehasafixedcostof$1960.Itcosts$95toproduceeach

jacket.Thesellingpriceis$123perjacket.Letxrepresentthenumberofjacketsproducedandsoldandwrite

thecostfunction,C,andrevenuefunction,R.

A) C(x)=95x+1960

R(x)=123x

B) C(x)=95 +123x

R(x)=1960x

C) C(x)=95 +1960x

R(x)=123

D) C(x)=95x+1960x

R(x)=123x

17) An–Meiownsabusinessmakingandsellingjackets.Shehasafixedcostof$900.Itcosts$92toproduceeach

jacket.Thesellingpriceis$107perjacket.Determinethenumberofjacketsthatmustbemadeandsoldin

ordertobreakeven.

A) 60units B) 61units C) 62 units D) 8units

18) Youinvested$41,800andstartedabusinesssellingvases.Suppliescost$19 pervaseandyouaresellingeach

vasefor$38.Letxrepresentthenumberofvasesproducedandsoldandwritethecostfunction,C,and

revenuefunction,R.

A) C(x)=19x+41,800

R(x)=38x

B) C(x)=19x+38

R(x)=41,800x

C) C(x)=19x+41,800

R(x)=38

D) C(x)=19x+41,800x

R(x)=38x

19) Youinvested$17,000andstartedabusinesssellingvases.Suppliescost$16 pervaseandyouaresellingeach

vasefor$33.Determinethenumberofvases,x,thatmustbeproducedandsoldtobreakeven.

A) 1000units B) 1001 units C) 1002 units D) 341 units

20) Asthepriceofaproductincreases,thedemandforthatproductdecreases.However,athigherprices,

suppliersarewillingtoproducegreaterquantitiesoftheproduct.Theweeklysupplyanddemandmodelsfor

acertaintypeoftelevisionareasfollows:

Demand:N=–3p+650

Supply:N=2.5p

wherepisthepriceindollarspertelevision.

Howmanyofthesetelevisionscanbesoldandsuppliedat$120pertelevision?

A) sold:290;supplied:300 B) sold:1010;supplied:300

C) sold:290;supplied:950 D) sold:300;supplied:290

21) Asthepriceofaproductincreases,thedemandforthatproductdecreases.However,athigherprices,

suppliersarewillingtoproducegreaterquantitiesoftheproduct.Theweeklysupplyanddemandmodelsfor

acertaintypeoftelevisionareasfollows:

Demand: N=–6p+680

Supply: N=2.3p

wherepisthepriceindollarspertelevision.

Findthepriceatwhichsupplyanddemandareequal.Atthisprice,howmanytelevisionscanbesuppliedand

soldeachweek?

A) $81.93;188 B) $183.78;423 C) $293.04;674 D) $81.93;205

Page7

22) InMiguelʹshometown,thepercentageofwomenwhosmokeisincreasingwhilethepercentageofmenwho

smokeisdecreasing.Thefunctiony=0.26x+16.7modelsthepercentage,y,ofwomeninthiscitywhosmokex

yearsafter1990.Thefunction0.20x+y=27.1modelsthepercentage,y,ofmeninthiscitywhosmokexyears

after1990.Usethesemodelstodeterminewhenthepercentageofwomenwhosmokewillbethesameasthe

percentageofmenwhosmoke.Roundtothenearestyear.Whatpercentageofwomenandwhatpercentageof

men(tothenearestwholepercent)willsmokeatthattime?

A) 2013;23% B) 2014;22% C) 2012;23% D) 2015;22%

23) In1985,inthetownofAppleby,20.1%ofHispanicswereoverweight,increasingbyanaverageof0.48%per

year.In1985,inthetownofAppleby,0.17%ofwhiteswereoverweight,increasingbyanaverageof27.7%per

year.Writeafunctionthatmodelsthepercentage,y,ofHispanicswhoareoverweightxyearsafter1985.Write

afunctionthatmodelsthepercentage,y,ofwhiteswhoareoverweightxyearsafter1985.

A) Hispanics:y=0.48x+20.1

Whites:y=27.7x+0.17

B) Hispanics:y=20.1x+0.48

Whites:y=0.17x+27.7

C) Hispanics:y=20.58x

Whites:y=27.87x

D) Hispanics:y+0.48x=20.1

Whites:y+27.7x=0.17

24) In1985,inthetownofAppleby,22.0%ofHispanicswereoverweight,increasingbyanaverageof0.41%per

year.In1985,inthetownofAppleby,0.18%ofwhiteswereoverweight,increasingbyanaverageof32.8%per

year.Ifthesetrendscontinue,inwhichyearwillthepercentageofHispanicswhoareoverweightbethesame

asthepercentageofwhiteswhoareoverweight?Roundtothenearestyear.WhatpercentageofHispanics(to

thenearestwholepercent)willbeoverweightatthattime?

A) 1986;22% B) 1988;23% C) 1984;21% D) 1982;21%

25) InthetownofMiltonLake,thepercentageofwomenwhosmokeisincreasingwhilethepercentageofmen

whosmokeisdecreasing.

Letxrepresentthenumberofyearssince1990andyrepresentthepercentageofwomeninMiltonLakewho

smoke.Thegraphofyagainstxincludesthedatapoints(0,17.5)and(12,20.86).Letxrepresentthenumberof

yearssince1990andyrepresentthepercentageofmeninMiltonLakewhosmoke.Thegraphofyagainstx

includesthedatapoints(0,28.9)and(14,26.1).

Determinewhenthepercentageofwomenwhosmokewillbethesameasthepercentageofmenwhosmoke.

Roundtothenearestyear.Whatpercentageofwomenandwhatpercentageofmen(tothenearestwhole

percent)willsmokeatthattime?

[Hint:firstfindtheslope–interceptequationofthelinethatmodelsthepercentage,y,ofwomenwhosmokex

yearsafter1990andtheslope–interceptequationofthelinethatmodelsthepercentage,y,ofmenwhosmoke

xyearsafter1990]

A) 2014;24% B) 2018;23% C) 2012;25% D) 2016;24%

26)

J

ohnnyʹscafeservesdesserts.Oneservingoficecreamandtwoservingsofblueberrypieprovides790 calories.

Threeservingsoficecreamandtwoservingsofblueberrypieprovides1290calories.Findthecaloriccontentof

eachitem.

A) Servingoficecream:250calories

Servingofblueberrypie:270calories

B) Servingoficecream:270calories

Servingofblueberrypie:250calories

C) Servingoficecream:232calories

Servingofblueberrypie:279calories

D) Servingoficecream:270calories

Servingofblueberrypie:260calories

27) Avendorsellshotdogsandbagsofpotatochips.Acustomerbuys3 hotdogsand2bagsofpotatochipsfor

$9.00.Anothercustomerbuys5hotdogsand4bagsofpotatochipsfor$16.00.Findthecostofeachitem.

A) $2.00forahotdog;$1.50forabagofpotatochips

B) $1.50forahotdog;$2.00forabagofpotatochips

C) $2.25forahotdog;$1.75forabagofpotatochips

D) $2.00forahotdog;$1.75forabagofpotatochips

Page8

28) TheFamilyFineArtsCentercharges$21 peradultand$15 perseniorcitizenforitsperformances.Onarecent

weekendeveningwhen525peoplepaidadmission,thetotalreceiptswere$8973.Howmanywhopaidwere

seniorcitizens?

A) 342seniorcitizens B) 183seniorcitizens C) 252 seniorcitizens D) 273 seniorcitizens

29) Abanktellerhas43$10and$5billsinhercashdrawer.Thevalueofthebillsis$280.Howmany$10 billsare

there?

A) 13$10bills B) 30$10 bills C) 15 $10 bills D) 28$10 bills

30) Arectangularlotwhoseperimeteris460 feetisfencedalongthreesides.Anexpensivefencingalongthelotʹs

lengthcosts$19perfoot,andaninexpensivefencingalongthetwosidewidthscostsonly$8perfoot.Thetotal

costofthefencingalongthethreesidescomesto$4130.Whatarethelotʹsdimensions?

A) Length150ft;Width80ft B) Length152 ft;Width78ft

C) Length146ft;Width84ft D) Length300 ft;Width160ft

31) Atwin–enginedaircraftcanfly1024milesfromcityAtocityBin4 hourswiththewindandmakethereturn

tripin8hoursagainstthewind.Whatisthespeedofthewind?

A) 64mph B) 48mph C) 80 mph D) 96mph

32)

J

ulieandEricrowtheirboat(ataconstantspeed)27 milesdownstreamfor3 hours,helpedbythecurrent.

Rowingatthesamerate,thetripbackagainstthecurrenttakes9hours.Findtherateofthecurrent.

A) 3mph B) 6mph C) 4 mph D) 2.5 mph

33)

J

amilalwaysthrowsloosechangeintoapencilholderonhisdeskandtakesitouteverytwoweeks.Thistimeit

isallnickelsanddimes.Thereare2timesasmanydimesasnickels,andthevalueofthedimesis$0.90more

thanthevalueofthenickels.HowmanynickelsanddimesdoesJamilhave?

A) 6nickelsand12dimes B) 7 nickelsand14 dimes

C) 5nickelsand10dimes D) 12 nickelsand6 dimes

34) Aflatrectangularpieceofaluminumhasaperimeterof66 inches.Thelengthis9incheslongerthanthewidth.

Findthewidth.

A) 12inches B) 21inches C) 30 inches D) 33inches

35)

J

arodishavingaproblemwithrabbitsgettingintohisvegetablegarden,sohedecidestofenceitin.Thelength

ofthegardenis5feetmorethan3timesthewidth.Heneeds74feetoffencingtodothejob.Findthelength

andwidthofthegarden.

A) length:29feet;width:8feet B) length:56 3

4feet;width:17 1

4feet

C) length:32feet;width:9feet D) length:26 feet;width:7feet

8.2 SystemsofLinearEquationsinThreeVariables

1 VerifytheSolutionofaSystemofLinearEquationsinThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Determineifthegivenorderedtripleisasolutionofthesystem.

1) (4

,

–2

,

–3)

x+y+z=–1

x–y+5z=–9

5x+y+z=15

A) solution B) notasolution

Page9

2) (–3

,

–5

,

4)

x+y+z=–4

x–y+4z=–3

2x+y+z=0

A) notasolution B) solution

3) (0

,

5

,

–4)

x–y+5z=–25

5x+z=–4

x+4y+z=16

A) solution B) notasolution

4) (1

,

3

,

0)

x–y+5z=2

5x +z=1

x+3y+z=10

A) notasolution B) solution

5) (1

,

–3

,

–4)

x–y+z=0

x+y+z=–6

x+y–z=2

A) solution B) notasolution

6) (–2

,

1

,

–3)

x–y+z=2

x+y+z=–4

x+y–z=0

A) notasolution B) solution

7) (–2

,

–1

,

4)

x+3y+5z=15

2y+2z=6

z=4

A) solution B) notasolution

8) (5

,

1

,

3)

x+5y+3z=23

3y+5z=28

z=5

A) notasolution B) solution

9) (–5

,

–1

,

3)

5x+2y+z=–24

2x–2y–z=–11

2x+y+3z=–2

A) solution B) notasolution

10) (–2

,

4

,

–3)

2x+2y+z=0

4x–4y–z=–26

2x+y+4z=–10

A) notasolution B) solution

Page10

2 SolveSystemsofLinearEquationsinThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequations.

1) x+y+z=2

x–y+2z=–3

2x+y+z=0

A) {(–2

,

3

,

1)} B) {(3

,

–2

,

1)} C) {(1

,

3

,

–2)} D) {(1

,

–2

,

3)}

2) x–y+2z=4

3x +z=3

x+4y+z=11

A) {(0

,

2

,

3)} B) {(2

,

0

,

3)} C) {(3

,

2

,

0)} D) {(3

,

0

,

2)}

3) x–y+z=3

x+y+z=–5

x+y–z=–1

A) {(1

,

–4

,

–2)} B) {(1

,

–2

,

–4)} C) {(–4

,

–2

,

1)} D) {(–2

,

1

,

–4)}

4) x+2y+5z=–12

5y+5z=–30

z=–1

A) {(3

,

–5

,

–1)} B) {(3

,

–1

,

–5)} C) {(–1

,

–5

,

3)} D) {(–5

,

3

,

–1)}

5) 4x+5y+z=–7

4x–4y–z=19

3x+y+4z=22

A) {(2

,

–4

,

5)} B) {(2

,

5

,

–4)} C) {(5

,

–4

,

2)} D) {(–4

,

2

,

5)}

6) x–y+2z=4

5x +z=0

x+5y+z=–20

A) {(0

,

–4

,

0)} B) {(0

,

0

,

–4)} C) {(–4

,

0

,

0)} D) {(0

,

–4

,

4)}

7)

x+y=7

5x +2y +5z =51

x–z=-

1

A) {(4

,

3

,

5)} B) {(4

,

5

,

3)} C) {(5

,

3

,

4)} D) {(5

,

4

,

3)}

8)

y–5z =12

–2x +y–3z =0

–2x +3z =-

14

A) {(4

,

2

,

–2)} B) {(4

,

7

,

–1)} C) {(7

,

3

,

0)} D) {(7

,

14

,

0)}

9) x+y+z=9

x–y+2z=4

2x+2y+2z=11

A) {(2

,

4

,

3)} B) {(3

,

4

,

2)} C) {(3

,

2

,

4)} D) ∅

Page11

10) –32+4z=4(x–3y)

2(x–4y–z)=–18

–4(2x+y)+3z=16

A) {(–1

,

1

,

4)} B) {(–1

,

4

,

4)} C) {(–1

,

2

,

8)} D) {(–1

,

–4

,

4)}

3 SolveProblemsUsingSystemsinThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Thesumofthreenumbersis–6.Ifthesecondnumberissubtractedfromthesumofthefirstandthird

numbers,theresultis4.Ifthethirdnumberissubtractedfromthesumofthefirstandsecondnumbers,the

resultis–8.Findthethreenumbers.

[Hint:letxrepresentthefirstnumber,ythesecondnumber,andzthethirdnumber.Usethegivenconditions

towriteandsolveasystemofequations.]

A) x=–2

,

y=–5

,

z=1B)x= –1

,

y= –5

,

z=0

C) x=–1

,

y=–4

,

z=–1D)x= –4

,

y= –3

,

z=1

2) Thefollowingisknownaboutthreenumbers:Ifthesecondnumberissubtractedfromthesumofthefirst

numberand2timesthethirdnumber,theresultis2.Thethirdnumberplus2timesthefirstnumberis3.The

firstnumberplus4timesthesecondnumberplusthethirdnumberis19.Findthethreenumbers.

[Hint:letxrepresentthefirstnumber,ythesecondnumber,andzthethirdnumber.Usethegivenconditions

towriteandsolveasystemofequations.]

A) x=0

,

y=4

,

z=3B)x=1

,

y=5

,

z=3C)x= –1

,

y=4

,

z=5D)x=0

,

y=3

,

z=7

3) Findthevaluesofa,b,andcsuchthatthegraphofthequadraticequationy=ax2+bx+cpassesthroughthe

points(–1,2),(3,14),and(4,12).

A) a=–1;b=5;c=8B)a=–1;b=8;c=5C)a=1;b=5;c=8D)a=1;b=8;c=5

4) Youthrowaballstraightupfromarooftop.Theballmissestherooftoponitswaydownandeventuallystrikes

theground.Amathematicalmodelcanbeusedtodescribetherelationshipfortheballʹsheightabovethe

ground,y,afterxseconds.Considerthefollowingdata:



x,secondsafter

ballisthrown

y,ballʹsheight,infeet,

abovetheground

194

2 126

494

Findthequadraticfunctiony=ax2+bx+cwhosegraphpassesthroughthegivenpoints.

A) y=–16x2+80x+30 B) y=–12x2+80x+26

C) y=–16x2+60x+50 D) y=–10x2+100x+4

Page12

5) ThetableshowsthepercentageofpeoplelivingbelowthepovertylineinoneU.S.cityintheyears2000

through2003.



Year

Percentageofpeopleliving

belowpovertyline

2000 11.2

2001 12.6

2002 13.2

2003 13

Thedatainthetablecanbewrittenasorderedpairs(x,y)wherexisthenumberofyearsafter2000andyisthe

percentageofpeoplelivingbelowthepovertylineinthatyear.Usethedatafor2000,2002,and2003tofind

thequadraticfunctiony=ax2+bx+cthatmodelsthepercentage,y,ofpeopleinthiscitylivingbelowthe

povertylinexyearsafter2000.

[Hint:Finda,b,andcbysubstitutingeachofthreeorderedpairsintothefunctionandwritingandsolvinga

systemoflinearequationsinthreevariables.]

A) y=–0.4x2+1.8x+11.2 B) y=–0.2x2+1.4x+11.2

C) y=–0.6x2+2.2x+11.2 D) y=–0.2x2+1.2x+11.2

6) Abasketballplayerscored26pointsinagame.Thenumberofthree–pointfieldgoalstheplayermadewas2

lessthanthreetimesthenumberoffreethrows(eachworth1point).Twicethenumberoftwo–pointfieldgoals

theplayermadewas8morethanthenumberofthree–pointfieldgoalsmade.Findthenumberoffree–throws,

two–pointfieldgoals,andthree–pointfieldgoalsthattheplayermadeinthegame.

A) 2freethrows;6two–pointfieldgoals;4 three–pointfieldgoals

B) 3freethrows;6two–pointfieldgoals;7 three–pointfieldgoals

C) 2freethrows;7two–pointfieldgoals;6 three–pointfieldgoals

D) 2freethrows;4two–pointfieldgoals;6three–pointfieldgoals

7) Avendorsellshotdogs,bagsofpotatochips,andsoftdrinks.Acustomerbuys5hotdogs,4bagsofpotato

chips,and5softdrinksfor$17.00.Thepriceofahotdogis$1.25morethanthepriceofabagofpotatochips.

Thecostofasoftdrinkis$2.25lessthanthepriceoftwohotdogs.Findthecostofeachitem.

A) $1.75forahotdog;$0.50forabagofpotatochips;$1.25 forasoftdrink

B) $0.50forahotdog;$1.75forabagofpotatochips;$1.25 forasoftdrink

C) $1.75forahotdog;$1.25forabagofpotatochips;$0.50 forasoftdrink

D) $2.00forahotdog;$0.75forabagofpotatochips;$1.25 forasoftdrink

8) Astoresellstents,sleepingbags,andcampstools.Acustomerbuysatent,5 sleepingbags,and2 campstools

for$227.Thepriceofthetentis9timesthecostofacampstool.Thecostofasleepingbagis$23morethanthe

costofacampstool.Findthecostofeachitem.

A) $63foratent;$30forasleepingbag;$7 foracampstool

B) $63foratent;$30forasleepingbag;$8 foracampstool

C) $63foratent;$35forasleepingbag;$12 foracampstool

D) $72foratent;$30forasleepingbag;$8 foracampstool

9) Threetrainsoneeastbound,onewestbound,andonenorthboundleaveacityatthesametime.Thespeed

ofthenorthboundtrainis10milesperhourgreaterthanthespeedoftheeastboundtrain.After3hours,the

distancebetweenthewestboundtrainandtheeastboundtrainis240miles.Twicethespeedofthewestbound

trainis60milesperhourmorethanthespeedofthenorthboundtrain.Findthespeedsofthethreetrains.

A) eastbound,30mph;westbound,50 mph;northbound,40 mph

B) eastbound,40mph;westbound,50 mph;northbound,50 mph

C) eastbound,20mph;westbound,60 mph;northbound,40 mph

D) eastbound,40mph;westbound,50 mph;northbound,30 mph

Page13

10) Ronattendsacocktailparty(withhisgraphingcalculatorinhispocket).Hewantstolimithisfoodintaketo

120gprotein,105gfat,and159gcarbohydrate.Accordingtothehealthconscioushostess,themarinated

mushroomcapshave3gprotein,5gfat,and9gcarbohydrate;thespicymeatballshave14gprotein,7gfat,

and15gcarbohydrate;andthedeviledeggshave13gprotein,15gfat,and6gcarbohydrate.Howmanyof

eachsnackcanheeattoobtainhisgoal?

A) 8mushrooms;5meatballs;2eggs B) 5 mushrooms;2 meatballs;8eggs

C) 2mushrooms;8meatballs;5eggs D) 9 mushrooms;6 meatballs;3eggs

11) Ms.Adamsreceivedabonuscheckfor$12,000.Shedecidedtodividethemoneyamongthreedifferent

investments.Withsomeofthemoney,shepurchasedamunicipalbondpaying5.5%simpleinterest.She

investedtwicetheamountshepaidforthemunicipalbondinacertificateofdepositpaying4.1%simple

interest.Ms.Adamsplacedthebalanceofthemoneyinamoneymarketaccountpaying3.1%simpleinterest.If

Ms.Adamsʹtotalinterestforoneyearwas$460,howmuchwasplacedineachaccount?

A) municipalbond:$2000

certificateofdeposit:$4000

moneymarket:$6000

B) municipalbond:$1500

certificateofdeposit:$3000

moneymarket:$7500

C) municipalbond:$2500

certificateofdeposit:$5000

moneymarket:$4500

D) municipalbond:$1750

certificateofdeposit:$3500

moneymarket:$6750

12) TheLittleTownArtsCentercharges$22 foradults,$18 forseniorcitizens,and$7forchildrenunder12for

theirliveperformancesonSundayafternoon.ThispastSunday,thepaidrevenuewas$10,932for716tickets

sold.Therewere41morechildrenthanadults.Howmanychildrenattended?

A) 256children B) 215children C) 245 children D) 246 children

13) Aceramicsworkshopmakesservingbowls,platters,andbreadbasketstosellatitsWinterFestival.Aserving

bowltakes3hourstoprepare,2hourstopaint,and8hourstofire.Aplattertakes16hourstoprepare,3hours

topaint,and4hourstofire.Abreadbaskettakes4hourstoprepare,16hourstopaint,and7hourstofire.If

theworkshophas123hoursforpreptime,97hoursforpainting,and120hoursforfiring,howmanyofeach

canbemade?

A) 9servingbowls,5platters,4breadbaskets B) 5 servingbowls,4platters,9breadbaskets

C) 4servingbowls,9platters,5breadbaskets D) 10 servingbowls,6platters,5breadbaskets

14) Aceramicsworkshopmakeswreaths,trees,andsleighsforsaleatChristmas.Awreathtakes3hoursto

prepare,2hourstopaint,and8hourstofire.Atreetakes14hourstoprepare,3hourstopaint,and4hoursto

fire.Asleightakes4hourstoprepare,15hourstopaint,and7hourstofire.Iftheworkshophas85hoursfor

preptime,56hoursforpainting,and86hoursforfiring,Howmanyofeachcanbemade?

A) 7wreaths,4trees,2sleighs B) 4 wreaths,2 trees,7sleighs

C) 2wreaths,7trees,4sleighs D) 8 wreaths,5 trees,3sleighs

15) ThreeshrimpboatssupplytheshrimpwholesalersonHiltonHeadwithfreshcatch.TheAnnabelletakes50%

ofitscatchtoHudsonʹs,20%toCaptainJʹs,and30%toMainstreet.TheCurlyQtakes40%ofitscatchto

Hudsonʹs,40%toCaptainJʹs,and20%toMainstreet.TheSloJoetakes30%ofitscatchtoHudsonʹs,40%to

CaptainJʹs,and30%toMainstreet.OneweekHudsonʹsreceived248.8poundsofshrimp,CaptainJʹsreceived

216.8pounds,andMainstreetreceived163.4pounds.Howmanypoundsofshrimpdideachboatcatch?

A) Annabelle174lbs,CurlyQ253lbs,SloJoe202 lbs

B) Annabelle253lbs,CurlyQ202lbs,SloJoe174 lbs

C) Annabelle202lbs,CurlyQ253lbs,SloJoe174 lbs

D) Annabelle202lbs,CurlyQ174lbs,SloJoe253 lbs

Page14

16) Adelisellsthreesizesofturkeysandwiches:thesmallturkey sandwichcontains4ouncesofmeatandsellsfor

$3.50;theregularturkeysandwichcontains7ouncesofmeatandsellsfor$4.00;andthelargeturkeysandwich

contains11ouncesofmeatandsellsfor$4.50.Acustomerrequestsaselectionofeachsizeforareception.She

andthemanageragreeonacombinationof42sandwichesmadefrom18pounds6ouncesofturkeyforatotal

costof$166.Howmanyofeachsizesandwichwillbeinthiscombination?(Note:1pound=16ounces)

A) 16smallsandwiches,14mediumsandwiches,12 largesandwiches.

B) 18smallsandwiches,8mediumsandwiches,16 largesandwiches.

C) 12smallsandwiches,10mediumsandwiches,20 largesandwiches.

D) 14smallsandwiches,20mediumsandwiches,8 largesandwiches.

17) Arealestateinvestorisexaminingatriangularplotofland.Shemeasureseachangleofthefield.Thesumof

thefirstandsecondanglesis160°morethanthemeasureofthethirdangle.Ifthemeasureofthethirdangleis

subtractedfromthemeasureofthesecondangle,theresultisthricethemeasureofthefirstangle.Findthe

measureofeachangle.(Note:Thesumoftheanglesofatriangleis180°.)

A) Firstangleis40°

,

secondangleis130°

,

thirdangleis10°

B) Firstangleis46°

,

secondangleis126°

,

thirdangleis8°

C) Firstangleis38°

,

secondangleis130°

,

thirdangleis8°

D) Firstangleis44°

,

secondangleis132°

,

thirdangleis4°

8.3 PartialFractions

1 DecomposeP/Q,WhereQHasOnlyDistinctLinearFactors

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writetheformofthepartialfractiondecompositionoftherationalexpression.Itisnotnecessarytosolveforthe

constants.

1) 2x–1

(x+6)(x+2)

A) A

x+6+B

x+2B) A

x+6+B

x+2+C

(x+6)(x+2)

C) A

x+6+B

x+2+Cx+D

(x+6)(x+2) D) A

x+6+B

x+2+C

(x+6)2(x+2)2

2) 6x–3

x2–13x+42

A) A

x–6+B

x–7B) A

x–6+B

x–7+C

x2–13x+42

C) A

x–6+B

x–7+Cx+D

x2–13x+42 D) A

x–6+B

x–7+C

(x–6)2(x–7)2

Writethepartialfractiondecompositionoftherationalexpression.

3) x–7

(x–2)(x–3)

A) 5

x–2+–4

x–3B) –4

x–2+5

x–3C) 5

x–2+4

x–3D) 4

x–2+–5

x–3

4) x

(x–2)(x–3)

A) –2

x–2+3

x–3B) 2

x–2+–3

x–3C) –3

x–2+2

x–3D) –2

x–2+–3

x–3

Page15

5) x

x2+3x+2

A) 2

x+2+–1

x+1B) –2

x+2+1

x+1C) 1

x+2+–2

x+1D) 2

x+2+1

x+1

6) 2x2–x–15

x(x+1)(x–1)

A) 15

x+–6

x+1+–7

x–1B) 15

x+6

x+1+–7

x–1C) 15

x+–6

x+1+7

x–1D) 15

x+–7

x+1+6

x–1

7) 4x2–x–17

x3–x

A) 17

x+–6

x+1+–7

x–1B) 17

x+6

x+1+–7

x–1C) 17

x+–6

x+1+7

x–1D) 17

x+–7

x+1+6

x–1

8) 13x–18

(x–1)(x–2)

A) 5

x–1+8

x–2B) 12

x–1+1

x–2C) 4

x–1+9

x–2D) 5

x–1+–8

x–2

9) 8x–12

x2+2x–24

A) 6

x+6+2

x–4B) 6

x+6+–2

x–4C) 7

x+6+1

x–4D) 6

x–6+2

x+4

10) 14x2–61x–90

x(x–6)(x–5)

A) –3

x+8

x–6+9

x–5B) –3

x+9

x–6+8

x–5C) –3

x+7

x–6+10

x–5D) x+8

x2–6

+9

x–5

2 DecomposeP/Q,WhereQHasRepeatedLinearFactors

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writetheformofthepartialfractiondecompositionoftherationalexpression.Itisnotnecessarytosolveforthe

constants.

1) 4x–1

(x–2)2

A) A

x–2+B

(x–2)2B) A

x–2+Bx+C

(x–2)2

C) A

x–2+B

x–2+Cx+D

(x–2)2D) A

x–2+B

x–2+C

(x–2)2

Page16

2) 5x+2

(x+7)(x+1)2

A) A

x+7+B

x+1+C

(x+1)2B) A

x+7+B

x+1+C

x+1+Dx+E

(x+1)2

C) A

x+7+B

x+1+Cx+D

(x+1)2D) A

x+7+B

x+1+C

x+1+D

(x+1)2

Writethepartialfractiondecompositionoftherationalexpression.

3) 8x+6

(x–3)2

A) 8

x–3+30

(x–3)2B) 8

x–3+48

(x–3)2C) 8

x–3+x+30

(x–3)2D) 1

x–3+x+30

(x–3)2

4) 2x2–2x+4

(x–1)3

A) 2

x–1+2

(x–1)2+4

(x–1)3B) 2

x–1+2

(x–1)2

C) 2

x–1+x+2

(x–1)2+x2+4

(x–1)3D) 2

x–1–2

(x–1)2+4

(x–1)3

5) x+2

x3–2x2+x

A) 2

x+–2

x–1+3

(x–1)2B) 2

x+3

x–1+–2

(x–1)2

C) 2

x+–2

x–1+5

(x–1)2D) –2

x+2

x–1+3

(x–1)2

6) 36–7x

x3–6x2+9x

A) 4

x+–4

x–3+5

(x–3)2B) 4

x+5

x–3+–4

(x–3)2

C) 4

x+–4

x–3+10

(x–3)2D) –4

x+4

x–3+5

(x–3)2

7) 10x2+24x+13

(x+2)(x+1)2

A) 5

x+2+5

x+1+–1

(x+1)2B) –5

x+2+5

x+1+1

(x+1)2

C) 5

x+2+–5

x+1+–1

(x+1)2D) –1

x+2+5

x+1+–5

(x+1)2

Page17

8) 64–11x

x(x–4)2

A) 4

x+–4

x–4+5

(x–4)2B) 4

x+5

x–4+–4

(x–4)2

C) 4

x+–4

x–4+10

(x–4)2D) –4

x+4

x–4+5

(x–4)2

3 DecomposeP/Q,WhereQHasaNonrepeatedPrimeQuadraticFactor

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writetheformofthepartialfractiondecompositionoftherationalexpression.Itisnotnecessarytosolveforthe

constants.

1) 3x+4

(x+7)(x2+x+5)

A) A

x+7+Bx+C

x2+x+5B) A

x+7+B

x2+x+5

C) A

x+7+B

x2+x+5

+C

(x+7)(x2+x+5) D) A

x+7+Bx+C

x2+x+5

+D

(x+7)(x2+x+5)

Writethepartialfractiondecompositionoftherationalexpression.

2) 8x+1

x3–1

A) 3

x–1+–3x+2

x2+x+1B) 3

x–1+–3

x+1+2

x–1

C) –3

x–1+3x+2

x2+x+1D) 3

x–1+2x–3

x2+x+1

3) 8x2+10x+28

(x+2)(x2+6)

A) 4

x+2+4x+2

x2+6B) 4

x+2+4

x2+6

C) 4

x+2+4x–2

x2+6D) 4

x+2+4

x+6+2

(x+6)2

4) 7x2+21x+10

x3+4x2+3x+12

A) 2

x+4+5x+1

x2+3B) 2

x+4+5

x2+3

C) 2

x+3+5x+1

x2+4D) 2

x+4+5

x+3+1

(x+3)2

Page18

5) –13x+84

(x+4)2(x2+1)

A) 3

x+4+8

(x+4)2+–3x+4

x2+1B) 3

x+4+8

(x+4)2+–3

x2+1

C) 3

x+4+12

(x+4)2+–3x–4

x2+1D) 12x–4

(x+4)2+–3x+4

x2+1

6) 12x+3

(x–1)(x2+x+1)

A) 5

x–1+–5x+2

x2+x+1B) 5

x–1+–5

x+1+2

x–1

C) –5

x–1+5x+2

x2+x+1D) 5

x–1+2x–5

x2+x+1

7) x2–111

x4–x2–72

A) 1

x+3+1

x–3–7

x2+8B) 1

x+3–1

x–3+7

x2+8

C) 1

x+3–1

x–3–7

x2+8D) 1

x+3+1

x–3+7

x2+8

4 DecomposeP/Q,WhereQHasaPrime,RepeatedQuadraticFactor

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writetheformofthepartialfractiondecompositionoftherationalexpression.Itisnotnecessarytosolveforthe

constants.

1) 5x+5

(x2+x+6)2

A) Ax+B

x2+x+6

+Cx+D

(x2+x+6)2B) A

x2+x+6

+Bx+C

(x2+x+6)2

C) A

x2+x+6

+B

(x2+x+6)2D) Ax+B

x2+x+6

+C

(x2+x+6)2

2) 2x+3

(x+6)(x2+x–5)2

A) A

x+6+Bx+C

x2+x–5

+Dx+E

(x2+x–5)2B) A

x+6+B

x2+x–5

+C

(x2+x–5)2

C) A

x+6+B

x2+x–5

+Cx+D

(x2+x–5)2D) A

x+6+Bx+C

x2+x–5

Writethepartialfractiondecompositionoftherationalexpression.

3) x2+2x–1

(x2+5)2

A) 1

x2+5

+2x–6

(x2+5)2B) x+1

x2+5

+2x–6

(x2+5)2C) 1

x2+5

+–x–6

(x2+5)2D) –1

x2+5

+2x–6

(x2+5)2

Page19

4) 5x3+26x–6

(x2+4)2

A) 5x

x2+4

+6x–6

(x2+4)2B) 5x+1

x2+4

+6x–6

(x2+4)2C) 5x

x2+4

+–6x–6

(x2+4)2D) 5x

x2+4

+6x+6

(x2+4)2

5) 3x3+4x2

(x2+5)2

A) 3x+4

x2+5

+–15x–20

(x2+5)2B) 3x–4

x2+5

+–15x+20

(x2+5)2C) 3x+4

x2+5

+15x+20

(x2+5)2D) 3x+4

x2+5

+15x–20

(x2+5)2

6) 4x3–6x2+14x–28

(x2+5)3

A) 4x–6

(x2+5)2+–6x+2

(x2+5)3B) x+1

x2+5

+4x–6

(x2+5)2+–6x+2

(x2+5)3

C) 4x+6

(x2+5)2+–6x–2

(x2+5)3D) x

x2+5

+4x–6

(x2+5)2+–6x+2

(x2+5)3

8.4 SystemsofNonlinearEquationsinTwoVariables

1 RecognizeSystemsofNonlinearEquationsinTwoVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Decideifthesystemofequationsintwovariablesislinearornonlinear.

1) x2=2y+6

3x–y=6

A) Nonlinear B) Linear

2) y=x2–4

x2+y2=9

A) Nonlinear B) Linear

3) x=7y+8

2x–y=4

A) Linear B) Nonlinear

4) xy=–7

x+3y=6

A) Nonlinear B) Linear

5) x3+2x=y

x+4y=9

A) Nonlinear B) Linear

Page20

2 SolveNonlinearSystemsBySubstitution

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystembythesubstitutionmethod.

1) –2x–y=–52

y=x2+4

A) {(6

,

40),(–8

,

68)} B) {(–6

,

40),(8

,

68)} C) {(6

,

32),(–8

,

60)} D) {(–6

,

40),(–8

,

68)}

2) x+y=4

y=x2–8x+16

A) {(3

,

1),(4

,

0)} B) {(–3

,

7),(–4

,

8)} C) {(4

,

0)} D) {(3

,

7),(4

,

0)}

3) y=x–2

y2=–8x

A) {(–2

,

–4)} B) {(–2

,

–4),(2

,

0)}

C) {(–2

,

–4),(–2

,

4)} D) {(–2

,

–4),(–2

,

4),(2

,

0)}

4) x2+y2=13

x+y=5

A) {(3

,

2),(2

,

3)} B) {(–3

,

2),(–2

,

3)} C) {(3

,

–2),(2

,

–3)} D) {(–3

,

–2),(–2

,

–3)}

5) xy=56

x+y=15

A) {(8

,

7),(7

,

8)} B) {(–8

,

7),(–7

,

8)} C) {(8

,

–7),(7

,

–8)} D) {(–8

,

–7),(–7

,

–8)}

6) xy=1

–16x–y=–10

A) 1

8,8 ,1

2,2 B) 8,1

8,2,1

2C) {(–8

,

8),(–2

,

2)} D) 1

2,2

7) x2=y2+39

x–y=3

A) {(8,5)} B) {(8,–5)} C) {(–8,5)} D) {(–8,–5)}

8) xy=54

x2+y2=117

A) {(6

,

9),(–6

,

–9),(9

,

6),(–9

,

–6)} B) {(6

,

9),(–6

,

–9),(6

,

–9),(–6

,

9)}

C) {(6

,

9),(9

,

6),(6

,

–9),(9

,

–6)} D) {(–6

,

–9),(–9

,

–6),(–6

,

9),(–9

,

6)}

9) x–2y=3

x2–xy=20

A) {(5,1),(–8,–11

2)} B) {(–5,–1),(8,11

2)} C) {(5,1),(–11

2,–8)} D) {(–5,–1),(11

2,8)}

10) x–y=9

(x–2)2+(y+5)2=20

A) {(6

,

–3),(0

,

–9)} B) {(–3

,

6),(–9

,

0)} C) {(6

,

–3),(4

,

–9)} D) {(–3

,

6),(–9

,

4)}

Page21

11) 6x2+12y2=300

y=x+5

A) 0,5,–20

3,–5

3B) 0,–5 ,–20

3,–5

3

C) 0,5 ,20

3,35

3D) 0,–5 ,20

3,35

3

12) y=(x+5)2+1

2x–y+10=0

A) {(–4,2)} B) {(–4,2),(4,18)} C) {(0,10),(0,26)} D) {(–5,0)}

13) 2y–x=10

x2+y2–100=0

A) {(–10

,

0),(6

,

8)} B) {(10

,

0),(–10

,

0),(6

,

8)}

C) 0,5

,

8

,

9 D) 0,5

,

0,–5

,

8

,

9

14) x2+y2=25

x–y=1

A) {(–3

,

–4),(4

,

3)} B) {(3

,

–4),(4

,

–3)} C) {(–3

,

4),(–4

,

3)} D) {(3

,

4),(4

,

3)}

15) y=–x2+13

x2+y2=25

A) {(4

,

–3),(3

,

4),(–3

,

4),(–4

,

–3)} B) {(3

,

4),(–3

,

4)}

C) {(4

,

–3),(–4

,

–3)} D) {(9

,

94),(16

,

269)}

16) x+y=2

x2+y2=4y–2

A) {(1

,

1),(–1

,

3)} B) {(1

,

1),(3

,

–1)} C) {(3

,

–1

,

(5

,

–3)} D) {(–1

,

3),(–3

,

5)}

3 SolveNonlinearSystemsByAddition

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystembytheadditionmethod.

1) x2+y2=16

x2–y2=16

A) {(4

,

0),(–4

,

0)} B) {(0,4),(0,–4)} C) {(4

,

0)} D) {(0,4)}

2) 8x2+y2=64

8x2–y2=64

A) {(2 2,0),(–22,0)} B) {(8

,

0),(–8

,

0)}

C) {(0,22

),(0,–22)} D) {(0,8),(0,–8)}

3) x2+y2=4

4x2+25y2=100

A) {(0,2),(0,–2)} B) {(2

,

0),(–2

,

0)} C) {(0,5),(0,–5)} D) {(5

,

0),(–5

,

0)}

4) x2–y2=9

16x2+9y2=144

A) {(3

,

0),(–3

,

0)} B) {(4

,

0),(–4

,

0)} C) {(0,3),(0,–3)} D) {(0,4),(0,–4)}

Page22

5) x2+y2=125

x2–y2=–75

A) {(5

,

10),(–5

,

10),(5

,

–10),(–5

,

–10)} B) {(5

,

10),(10

,

5),(–5

,

–10),(–10

,

–5)}

C) {(5

,

–10),(5

,

10)} D) {(–5

,

–10),(–10

,

–5)}

6) 2x2–2y2=24

2x2+2y2=40

A) {(4

,

2),(–4

,

2),(4

,

–2),(–4

,

–2)} B) {(4

,

2),(2

,

4),(–4

,

–2),(–2

,

–4)}

C) {(4

,

–2),(4

,

2)} D) {(–4

,

–2),(–2

,

–4)}

7) x2–3y2=1

2x2+3y2=74

A) {(5

,

8),(–5

,

8),(5

,

–8),(–5

,

–8)} B) {(5

,

8),(–5

,

–8)}

C) {(8

,

5),(–8

,

5),(8

,

–5),(–8

,

–5)} D) {(8

,

5),(–8

,

–5)}

8) 2x2+y2=17

3x2–2y2=–6

A) {(2,3),(2,–3),(–2,3),(–2,–3)} B) {(1,3),(1,–3),(–1,3),(–1,–3)}

C) {(2,–3),(–2,3)} D) {(1,3),(–1,–3)}

9) 2x2+y2=66

x2+y2=41

A) {(5,4),(5,–4),(–5,4),(–5,–4)} B) {(4,5),(–4,5),(4,–5),(–4,–5)}

C) {(5,4),(–5,–4)} D) {(4,5),(–4,–5)}

10) x2–2y2=2

3x2+2y2=14

A) {(2,1),(2,–1),(–2,1),(–2,–1)} B) {(1,2),(–1,2),(1,–2),(–1,–2)}

C) {(2,1),(–2,–1)} D) {(–1,2),(1,–2)}

11) x2–3y2–1=0

4x2+3y2–19=0

A) {(2,1),(2,–1),(–2,1),(–2,–1)} B) {(1,2),(–1,2),(1,–2),(–1,–2)}

C) {(2,1),(–2,–1)} D) {(–1,2),(1,–2)}

12) x2–y2=5

x2+y2=15

A) {( 10,5),(–10,5),(10,–5),(–10,–5)}

B) {(0,0)}

C) {( 10,5),(–10,–5)}

D) {( 10,5),(–10,5)}

13) y2+4x2=21

y2–x2=1

A) (2,5),(2,–5),(–2,5),(–2,–5)B)(2,5),(2,–5)

C) (3,5),(3,–5),(–3,5),(–3,–5)D)(3,5),(3,–5)

Page23

14) y=x2+12

y=–x2+14

A) {(1

,

13),(–1

,

13)} B) {(1

,

13),(1

,

–13),(–1

,

13),(–1

,

–13)}

C) {(13

,

1),(13

,

–1)} D) {(1

,

13),(1

,

–13)}

15) x2+y2+6x–2y+9=0

x2–y2+6x+2y+7=0

A) {(–2

,

1),(–4

,

1)} B) {(–2

,

1),(4

,

1)} C) {(–3

,

2),(–3

,

0)} D) {(–3

,

–2),(–3

,

0)}

Solvebythemethodofyourchoice.

16) 2x2+xy–y2=3

x2+2xy+y2=3

A) 23

3,3

3,–23

3,–3

3B) 23

3,–3

3,–23

3,3

3

C) –2

3,1

3,23

3,–1

3D) –2

3,–1

3,23

3, 1

3

17) x2+y2=25

(x–6)2+y2=25

A) {(3

,

4),(3

,

–4)} B) {(3

,

4),(3

,

–4),(–3

,

4),(–3

,

–4)}

C) {(4

,

3),(4

,

–3)} D) {(4

,

3),(4

,

–3),(–4

,

3),(–4

,

–3)}

18) x2+y2=25

10x+y2=46

A) {(3

,

4),(3

,

–4)} B) {(3

,

4)}

C) {(3

,

4),(3

,

–4),(7

,

4),(7

,

–4)} D) {(3

,

4),(3

,

–4),(–3

,

4),(–3

,

–4)}

19) x3+y=0

12x2–y=0

A) {(0,0),(–12

,

1728)} B) {(0,0),(12

,

–1728)}

C) {(0,0),(–12

,

144)} D) {(–1,1),(–12

,

1728)}

4 SolveProblemsUsingSystemsofNonlinearEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Letxrepresentonenumberandletyrepresenttheothernumber.Usethegivenconditionstowriteasystemofnonlinear

equations.Solvethesystemandfindthenumbers.

1) Thesumoftwonumbersis21andtheirproductis20.Findthenumbers.

A) 20and1B)

–20 and–1

C) 20and1;–20and–1D)20and–1;–20 and1

2) Thesumofthesquaresoftwonumbersis65.Thesumofthetwonumbersis–7.Findthetwonumbers.

A) –8and1B)

–8and1;–1 and8C)

–1 and8D)

–8and–1;1 and8

3) Thesumofthesquaresoftwonumbersis90.Thedifferenceofthetwonumbersis6.Findthetwonumbers.

A) –9and–3;3and9B)

–9and–3C)3and9D)

–3and9;–9 and3

4) Thesumoftwosquaresoftwonumbersis90

,

andthedifferenceoftheirsquaresis72.Findthenumbers.

A) 9and3;–9and3;9and–3;–9and–3B)9and3

C) 9and3;–9and–3D)9and3;–9 and3;9and–3

Page24

5) Thedifferencebetweenthesquaresoftwonumbersis5.Twicethesquareofthesecondnumbersubtracted

fromthesquareofthefirstnumberis1.Findthenumbers.

A) 3and2;–3and2;3and–2;–3and–2B)3and2

C) 3and2;–3and–2D)3and2;–3 and2;3and–2

Solvetheproblem.

6) Asystemfortrackingshipsindicatedthatashipliesonahyperbolicpathdescribedby6x2–y2=8.The

processisrepeatedandtheshipisfoundtolieonahyperbolicpathdescribedbyy2–2x2=8.Ifitisknown

thattheshipislocatedinthefirstquadrantofthecoordinatesystem,determineitsexactlocation.

A) (2

,

4) B) (–2

,

–4) C) (4

,

2) D) (–4

,

–2)

7) Findthedimensionsofarectanglewhoseperimeteris26 feetandwhoseareais40squarefeet.

A) 5feetby8feet B) 4feetby9 feet C) 6 feetby7 feet D) 4feetby7 feet

8) Theareaofagardenis2940squarefeet,andthelengthofitsdiagonalis91 feet.Findthedimensionsofthe

garden.

A) 35feetby84feet B) 5feetby588 feet C) 245 feetby12 feet D) 60feetby49 feet

9) Theareaofarectangularpieceofcardboardshownis893 squareinches.Thecardboardisusedtomakean

openboxbycuttinga3–inchsquarefromeachcornerandturningupthesides.Iftheboxistohaveavolume

of1599cubicinches,findthedimensionsofthecardboardthatmustbeused.

A) 19inchesby47inches B) 22 inchesby50 inches

C) 16inchesby44inches D) 13 inchesby38 inches

10) Arighttrianglehasanareaof10squareinches.Thesquareofthehypotenuseis104.Findthelengthsofthe

legsofthetriangle.Roundyouranswertothenearestinch.

A) 2inchesand10inches B) 4 inchesand100 inches

C) 1inchesand20inches D) 4 inchesand5 inches

11) Apersonatthetopofa600foottallbuildingdropsayellowball.Theheightoftheyellowballisgivenbythe

equationh=–16t2+600wherehismeasuredinfeetandtisthenumberofsecondssincetheyellowballwas

dropped.Asecondperson,inthesamebuildingbutonalowerfloorthatis180feetfromtheground,dropsa

whiteball3.5secondsaftertheyellowballwasdropped.Theheightofthewhiteballisgivenbytheequation

h=–16(t–3.5)2+180wherehismeasuredinfeetandtisthenumberofsecondssincetheyellowballwas

dropped.Findthetimethattheballsarethesamedistanceabovethegroundandfindthisdistance.

A) 5.5seconds;116feet B) 4.5 seconds;276 feet

C) 5seconds;200feet D) 6 seconds;24 feet

12) Thediagonalofthefloorofarectangularofficecubicleis2ftlongerthanthelengthofthecubicleand5ft

longerthantwicethewidth.Findthedimensionsofthecubicle.Roundtothenearesttenth,ifnecessary.

A) width=9.7ft,length=22.4ft B) width=4ft,length=11ft

C) width=2ft,length=9ft D) width=3.9ft,length=9.7ft

Page25

13) Ina1–milerace,thewinnercrossesthefinishline11 feetaheadofthesecond–placerunnerand31 feetahead

ofthethird–placerunner.Assumingthateachrunnermaintainsaconstantspeedthroughouttherace,byhow

manyfeetdoesthesecond–placerunnerbeatthethird–placerunner?(5280feetin1mile.)

A) 20.04ft B) –11.06 ft C) –20.12 ft D) 9.02 ft

8.5 SystemsofInequalities

1 GraphaLinearInequalityinTwoVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphtheinequality.

1) x–y>–4

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page26

2) x+y<–4

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page27

3) x–y<–4

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page28

4) 5x+y≤2

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page29

5) x+5y≥3

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page30

6) –3x–4y≤12

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page31

7) 3x+4y≤12

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page32

8) –2x–4y≤–8

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page33

9) y≥x–6

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page34

10) y+5<x

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page35

11) x

3+y

2≤1

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page36

12) y<–5

4x

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page37

13) y≤5x–2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page38

14) x>7



x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A) 

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page39

15) y≤7



x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page40

2 GraphaNonlinearInequalityinTwoVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphtheinequality.

1) x2+y2≤49

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page41

2) x2+y2>9

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page42

3) (x+1)2+(y–3)2>16

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page43

4) y>x2–2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page44

5) y≤x2+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page45

6) y≤3x

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page46

7) y>2x

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

8) y≥log(x+8)

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

5

4

3

2

1

-1

-2

-3

-4

-5

Page47

A)

x

–20–16–12–8–4 4 8 121620

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

–20–16–12–8–4 4 8 121620

y

5

4

3

2

1

-1

-2

-3

-4

-5

B)

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

5

4

3

2

1

-1

-2

-3

-4

-5

C)

x

–20–16–12–8–4 4 8 121620

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

–20–16–12–8–4 4 8 121620

y

5

4

3

2

1

-1

-2

-3

-4

-5

D)

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

5

4

3

2

1

-1

-2

-3

-4

-5

3 UseMathematicalModelsInvolvingLinearInequalities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Apersonwithnomorethan$5000toinvestplanstoplacethemoneyintwoinvestments,telecommunications

andpharmaceuticals.Thetelecommunicationsinvestmentistobenomorethan5timesthepharmaceuticals

investment.Writeasystemofinequalitiestodescribethesituation.Letx=amounttobeinvestedin

telecommunicationsandy=amounttobeinvestedinpharmaceuticals.

A) x+y≤5000

x≤5y

x≥0

y≥0

B) x+y=5000

x≤5y

x≥0

y≥0

C) x+y=5000

y≥5x

x≥0

y≥0

D) x+y≤5000

5x≤y

x≥0

y≥0

2) Amanisplantingasectionofgardenwithtomatoesandcucumbers.Theavailableareaofthesectionis120

squarefeet.Hewantstheareaplantedwithtomatoestobemorethan20%oftheareaplantedwithcucumbers.

Writeasystemofinequalitiestodescribethesituation.Letx=amounttobeplantedintomatoesand

y=amounttobeplantedincucumbers.

A) x+y≤120

x>0.20y

x≥0

y≥0

B) x+y=120

x≥0.20y

x≥0

y≥0

C) x+y≤120

0.20x>y

x≥0

y≥0

D) x+y≤120

x<0.20y

x≥0

y≥0

Page48

3) Benjaminneverhasmorethan24hoursfreeduringtheweek.Heistryingtomakeaweeklyplanfordividing

hisfreetimebetweenreadingandworkingout.Hewantstospendatleast7hoursperweekreading.Writea

systemofinequalitiestodescribethesituation.Letxrepresentthenumberofhoursforreadingandyrepresent

thenumberofhoursforworkingout.

A) x+y≤24

x≥7

y≥0

B) x+y≤24

x≥7y

x≥0

y≥0

C) x+y≤24

y≥7

x≥0

D) x+y≤24

x≤7y

x≥0

y≥0

4) Avolunteerwantstocrochetbeachhatsandbabyafghansforachurchfund–raisingbazaar.Sheneeds5 hours

tomakeahatand4hourstomakeanafghanandshehas20hoursavailable.Writeaninequalitythatdescribes

thesituationandusetheinequalitytodecidewhethershecanmake0hatsand9afghansinthetimeallowed.

Letxrepresentthenumberofhatsandythenumberofafghansthatshemakes.

A) 5x+4y≤20;no B) 5x+4y≤20;yes C) 5x+4y≥20;no D) 5x+4y≥20;yes

5) Awomanworksoutbyrunningandswimming.Whensheruns,sheburns 7 caloriesperminute.Whenshe

swims,sheburns10caloriesperminute.Shewantstoburnatleast490caloriesinherworkout.Graphan

inequalitythatdescribesthesituation.Letxrepresentthenumberofminutesrunningandythenumberof

minutesswimming.Becausexandymustbepositive,limitthegraphtoquadrantIonly.

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

A)

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

B)

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

Page49

C)

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

D)

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

x

10 20 30 40 50 60 70

y

70

60

50

40

30

20

10

6) Theequationthatrepresentsthepropertrafficcontrolandemergencyvehicleresponseavailabilityinasmall

cityis2P+3F≤23,wherePisthenumberofpolicecarsonactivedutyandFisthenumberoffiretrucksthat

haveleftthefirehouseinresponsetoacall.Inordertocomplywithstaffinglimitations,theequation

2P+F≤17isappropriate.Thenumberofpolicecarsonactivedutyandthenumberoffiretrucksthathaveleft

thefirehouseinresponsetoacallcannotbenegative,soP≥0andF≥0.Graphtheregionssatisfyingallthe

availabilityandstaffingrequirements,usingthehorizontalaxisforPandtheverticalaxisforF.If8policecars

areonactivedutyand2firetruckshaveleftthefirehouseinresponsetoacall,arealloftherequirements

satisfied?

P

12345678910

F

10

9

8

7

6

5

4

3

2

1

P

12345678910

F

10

9

8

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A) No

P

12345678910

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P

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F

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B) Yes

P

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P

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Page50

C) No

P

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6

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P

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F

10

9

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D) Yes

P

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P

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F

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7) Abakeryplanstomarketamixedassortmentofitstwomostpopularcookies,ChocolateChipandToffee

Chunk.Themarketinganalystproposesthatthenewassortmentbeconstrainedbytheinequality3C+4T≤31,

whereCisthenumberofChocolateChipcookiesandTisthenumberofToffeeChunkcookies.Thesales

analystsuggeststhattheassortmentshouldbeconstrainedbytheinequality5C+2T≤33.Thenumberofeach

typeofcookiecannotbenegative,soC≥0andT≥0.Graphtheregionsatisfyingalltherequirementsforthe

assortmentusingCasthehorizontalaxisandTastheverticalaxis.Doesthecombinationof6ChocolateChip

cookiesand3ToffeeChunkcookiessatisfyalloftherequirements?

C

12345678910

T

10

9

8

7

6

5

4

3

2

1

C

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T

10

9

8

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1

A) No

C

12345678910

T

10

9

8

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6

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4

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2

1

C

12345678910

T

10

9

8

7

6

5

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3

2

1

B) Yes

C

12345678910

T

10

9

8

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4

3

2

1

C

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T

10

9

8

7

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Page51

C) No

C

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T

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9

8

7

6

5

4

3

2

1

C

12345678910

T

10

9

8

7

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1

D) Yes

C

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T

10

9

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C

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T

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1

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

8) TheJillsonʹshaveupto$75,000toinvest.Theydecidethattheywanttohaveatleast$40,000investedinstable

bondsyielding6%andthatnomorethan$20,000shouldbeinvestedinmorevolatilebondsyielding12%.

(a)Usingxtodenotetheamountofmoneyinvestedinthestablebondsandytheamountinvestedinthemore

volatilebonds,writeasystemoflinearinequalitiesthatdescribethepossibleamountsofeachinvestment.

(b)Graphthesystemofinequalities.

x

y

x

y

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

9) Theliquidportionofadietistoprovideatleast300calories,36unitsofvitaminA,and90unitsofvitaminC

daily.AcupofdietarydrinkXprovides60calories,12unitsofvitaminA,and10unitsofvitaminC.Acupof

dietarydrinkYprovides60calories,6unitsofvitaminA,and30unitsofvitaminC.Setupasystemoflinear

inequalitiesthatdescribestheminimumdailyrequirementsforcaloriesandvitamins.Letx=numberofcups

ofdietarydrinkX,andy=numberofcupsofdietarydrinkY.Writealltheconstraintsasasystemoflinear

inequalities.

A) 60x+60y≥300

12x+6y≥36

10x+30y≥90

x≥0

y≥0

B) 60x+60y>300

12x+6y>36

10x+30y>90

x>0

y>0

C) 60x+60y≥300

12x+6y>36

10x+30y≥90

D) 60x+60y≤300

12x+6y≤36

10x+30y≤90

x≥0

y≥0

Page52

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

10) Acoffeestorehasavailable75poundsofAgradecoffeeand120poundsofBgradecoffee.Thesewillbe

blendedinto1poundpackagesasfollows:aneconomyblendthatcontains4ouncesofAgradecoffeeand12

ouncesofBgradecoffeeandasuperiorblendthatcontains8ouncesofAgradecoffeeand8ouncesofBgrade

coffee.Usingxtodenotethenumberofpackagesoftheeconomyblendandytodenotethenumberof

packagesofthesuperiorblend,writeasystemoflinearinequalitiesthatdescribesthepossiblenumberof

packagesofeachblend.Graphthesystemofinequalities.

x

y

x

y

Page53

4 GraphaSystemofInequalities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphthesolutionsetofthesystemofinequalitiesorindicatethatthesystemhasnosolution.

1) 3x–y≤3

x+2y≥–6

x

-10 -8 -6 -4 -2 2 4 6 8 10

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

-10 -8 -6 -4 -2 2 4 6 8 10

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page54

2) y<–x+5

y>2x–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

–20 -10 10 20

y

20

10

-10

-20

x

–20 -10 10 20

y

20

10

-10

-20

D)

x

–20 -10 10 20

y

20

10

-10

-20

x

–20 -10 10 20

y

20

10

-10

-20

Page55

3) x+2y≥2

x–y≤0

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page56

4) 1≤y<8

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

-12

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

-12

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page57

5) –x+7y<7

x≥4

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page58

6) y<x+1

2x+7y>28

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page59

7) y>–5

x≥–3

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page60

8) –3x+y>3

–3x+y<–1

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

A) nosolution B)

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

C)

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

D)

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

Page61

9) 2x+y<4

2x+y>1

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D) ∅

Page62

10) y>x2

10x+2y≤20

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page63

11) x2+y2≤100

–6x+5y≤–30

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page64

12) y≥3x

y≤8

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page65

13) x2+y2≤25

x2+y2≥4

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

A)

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

B) nosolution

C)

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

D)

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

x

–8–6–4–2 2468

y

6

4

2

-2

-4

-6

14) x2+y2≤64

y–x2>0

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page66

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

15) x2+y2≤36

y>3x

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page67

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

16) x2+y2≤16

x+y>1

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page68

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

17) x2+y2≤100

5x+5y≤25

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page69

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page70

18) (x+5)2+(y+2)2>16

(x+5)2+(y+2)2<36

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page71

19) x+2y≥6

x>–2

y≤6

x

–10–8–6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8–6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page72

20) x+y≤9

y≥8x–3

x≥0

y≥0

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

A)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

B)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

C)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

D)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

Page73

21) 2x+3y≥6

x–y≤3

y≤2

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page74

22) y≥2x–4

x+2y≤7

x≥–2

y≤1

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

A)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

B)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

C)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

D)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

Page75

23) y≥2x–4

x+2y≤7

y≥–2

x≤1

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

A)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

B)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

C)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

D)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

Page76

24) x≥0

y≥0

2x+2y≤4

2x+y≤5

x

-4 -2 2 4

y

4

2

-2

-4

x

-4 -2 2 4

y

4

2

-2

-4

A)

x

-4 -2 2 4

y

4

2

-2

-4

x

-4 -2 2 4

y

4

2

-2

-4

B)

x

-4 -2 2 4

y

4

2

-2

-4

x

-4 -2 2 4

y

4

2

-2

-4

C)

x

-4 -2 2 4

y

4

2

-2

-4

x

-4 -2 2 4

y

4

2

-2

-4

D)

x

-4 -2 2 4

y

4

2

-2

-4

x

-4 -2 2 4

y

4

2

-2

-4

Page77

25) y≥1

2≤x≤6

x–2y≥–2

x+y≤6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page78

8.6 LinearProgramming

1 WriteanObjectiveFunctionDescribingaQuantityThatMustbeMaximizedorMinimized

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Asteelcompanyproducestwotypesofmachinedies,partAandpartB.Thecompanymakesa$3.00 profiton

eachpartAthatitproducesanda$5.00profitoneachpartBthatitproduces.Letx=thenumberofpartA

producedinaweekandy=thenumberofpartBproducedinaweek.Writetheobjectivefunctionthat

describesthetotalweeklyprofit.

A) z=3x+5y B) z=5x+3y

C) z=8(x+y) D) z=3(x–5)+5(y–3)

2) Adietitianneedstopurchasefoodforpatients.Shecanpurchaseanounceofchickenfor$0.35andanounceof

potatoesfor$0.03.Letx=thenumberofouncesofchickenandy=thenumberofouncesofpotatoes

purchasedperpatient.Writetheobjectivefunctionthatdescribesthetotalcostperpatientpermeal.

A) z=0.35x+0.03y B) z=0.03x+0.35y C) z=35x+3y D) z=3y+35y

2 UseInequalitiestoDescribeLimitationsinaSituation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Asteelcompanyproducestwotypesofmachinedies,partAandpartBandisboundbythefollowing

constraints:

·PartArequires1hourofcastingtimeand10hoursoffiringtime.

·PartBrequires4hoursofcastingtimeand3hoursoffiringtime.

·Themaximumnumberofhoursperweekavailableforcastingandfiringare100and70,respectively.

·Thecosttothecompanyis$0.75perpartAand$3.00perpartB.Totalweeklycostscannotexceed$45.00.

Letx=thenumberofpartAproducedinaweekandy=thenumberofpartBproducedinaweek.Writea

systemofthreeinequalitiesthatdescribestheseconstraints.

A)

x+4y ≤100

10x +3y ≤70

0.75x +3y ≤45

B)

x+10y ≤100

4x +3y ≤70

0.75x +3y ≤45

C)

x+10y ≥100

4x +3y ≥70

0.75x +3y ≤45

D)

x+4y ≤100

10x +3y ≤70

3x +0.75y ≤45

2) Adietitianneedstopurchasefoodforpatients.Shecanpurchaseanounceofchickenfor$0.25andanounceof

potatoesfor$0.02.Thedieticianisboundbythefollowingconstraints.

·Eachounceofchickencontains13gramsofproteinand24gramsofcarbohydrates.

·Eachounceofpotatoescontains5gramsofproteinand35gramsofcarbohydrates.

·Theminimumdailyrequirementsforthepatientsunderthedietitianʹscareare45gramsofproteinand58

gramsofcarbohydrates.

Letx=thenumberofouncesofchickenandy=thenumberofouncesofpotatoespurchasedperpatient.Write

asystemofinequalitiesthatdescribestheseconstraints.

A)

13x +5y ≥45

24x +35y ≥58

B)

13x +24y ≥45

5x +35y ≥58

C)

13x +5y ≥58

24x +35y ≥45

D)

13x +24x ≥45

5y +35y ≥58

Page79

3) Mrs.Whitewantstocrochethatsandafghansforachurchfundraisingbazaar.Sheneeds6hourstomakeahat

and2hourstomakeanafghan,andshehasnomorethan48hoursavailable.Shehasmaterialfornomorethan

14items,andshewantstomakeatleasttwoafghans.Letx=thenumberofhatsshemakesandy=thenumber

ofafghansshemakes.Writeasystemofinequalitiesthatdescribestheseconstraints.

A) 6x+2y≤48

x+y≤14

y≥2

B) 6x+2y≤48

x+y≤14

x≥2

C) 2x+6y≤48

x+y≤14

x≤2

D) 6x+2y≥48

x+y≤14

y≥2

4) Anofficemanagerisbuyingusedfilingcabinets.Smallfilecabinetscost$5 eachandlargefilecabinetscost$11

each,andthemanagercannotspendmorethan$101onfilecabinets.Asmallcabinettakesup5squarefeetof

floorspaceandalargecabinettakesup10squarefeet,andtheofficehasnomorethan95squarefeetoffloor

spaceavailableforfilecabinets.Themanagermustbuyatleast5filecabinetsinordertogetfreedelivery.Letx

=thenumberofsmallfilecabinetsboughtandy=thenumberoflargefilecabinetsbought.Writeasystemof

inequalitiesthatdescribestheseconstraints.

A) 5x+11y≤101

5x+10y≤95

x+y≥5

B) 5x+11y≤101

10x+5y≤95

x≥5

C) 5x+11y≤101

5x+10y≤95

x+y≤5

D) 5x+11y≤101

5x+10y≤95

y≥5

3 UseLinearProgrammingtoSolveProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthemaximumorminimumvalueofthegivenobjectivefunctionofalinearprogrammingproblem.Thefigure

illustratesthegraphofthefeasiblepoints.

1) ObjectiveFunction:z=3x+4y

Findmaximumandminimum.

x

y

(0, 8) (8, 8)

(8, 2)

(

2

,

0

)

(0, 2)

x

y

(0, 8) (8, 8)

(8, 2)

(

2

,

0

)

(0, 2)

A) maximumvalue:56;minimumvalue:6 B) maximumvalue:56;minimumvalue:8

C) maximumvalue:32;minimumvalue:6 D) maximumvalue:32;minimumvalue:8

Page80

Findthemaximumorminimumvalueofthegivenobjectivefunctionofalinearprogrammingproblem.Thefigure

illustratesthegraphoffeasiblepoints.

2) ObjectiveFunction:z=x+7y

Findmaximum.

A) maximum:37 B) maximum:24 C) maximum:18 D) maximum:30

3) ObjectiveFunction:z=–x–5y

Findmaximum.

A) maximum:–14 B) maximum:–18 C) maximum:–27 D) maximum:–22

4) ObjectiveFunction:z=9x+10y

Findminimum.

A) minimum:56 B) minimum:57 C) minimum:68 D) nominimum

5) ObjectiveFunction:z=x+8y+8

Findminimum.

A) minimum:28 B) minimum:50 C) minimum:36 D) nominimum

6) ObjectiveFunction:z=–6x–y

Findmaximum.

A) maximum:–17 B) maximum:–21 C) maximum:–26 D) nomaximum

Anobjectivefunctionandasystemoflinearinequalitiesrepresentingconstraintsaregiven.Graphthesystemof

inequalitiesrepresentingtheconstraints.Findthevalueoftheobjectivefunctionateachcornerofthegraphedregion.

Usethesevaluestodeterminethemaximumvalueoftheobjectivefunctionandthevaluesofxandyforwhichthe

maximumoccurs.

7) ObjectiveFunction z=18x+6y

Constraints 0≤x≤10

0≤y≤5

3x+2y≥6

A) Maximum:210;at(10,5) B) Maximum:180;at(10,0)

C) Maximum:30;at(0,5) D) Maximum:18;at(0,3)

8) ObjectiveFunction z=6x+7y

Constraints x≥0

y≥0

2x+3y≤12

2x+y≤8

A) Maximum32;at(3,2) B) Maximum32;at(2,3)

C) Maximum52;at(4,4) D) Maximum24;at(4,0)

Page81

9) ObjectiveFunction z=3x+5y

Constraints x≥0

y≥0

2x+y≤15

x–3y≥–3

A) Maximum33;at(6,3) B) Maximum75;at(0,15)

C) Maximum22.5;at(7.5,0) D) Maximum38;at(6,4)

10) ObjectiveFunction z=17x–19y

Constraints 0≤x≤5

0≤y≤8

4x+5y≤30

4x+3y≤20

A) Maximum:85;at(5,0) B) Maximum:0;at(0,0)

C) Maximum:–73.75;at(1.25,5) D) Maximum:–114;at(0,6)

11) ObjectiveFunction z=7x+9y

Constraints x≥0

0≤y≤5

2x+3y≥12

2x+3y≤20

A) Maximum:70;at(10,0) B) Maximum:59;at(2,5)

C) Maximum:42;at(6,0) D) Maximum:–28;at(4,0)

12) ObjectiveFunction z=7x–3y

Constraints x≥0

0≤y≤3

x–y≤2

x+2y≤8

A) Maximum:22;at(4

,

2) B) Maximum:14;at(2

,

0)

C) Maximum:5;at(2,3) D) Maximum:–9;at(0,3)

13) ObjectiveFunction z=7x+6y

Constraints x≥0

y≥0

3x+y≤21

x+y≤10

x+2y≥12

A) Maximum65.5;at(5.5,4.5) B) Maximum60;at(6,3)

C) Maximum68;at(8,2) D) Maximum66;at(6,4)

Solvetheproblem.

14) Twokindsofcratedcargo,AandB,aretobeshippedbytruck.Theweightandvolumeofeachtypearegiven

inthefollowingtable:

AB

Volume 50cubicfeet 10cubicfeet

Weight 200pounds 360pounds

Theshippingcompanycharges$75percrateforcargoAand$100percrateforcargoB.Thetruckhasa

maximumloadlimitof7,200poundsand1,000cubicfeet.Howmanyofeachtypeofcargoshouldbeshipped

tomaximizeprofitfortheshippingcompany?

A) 18cratesofcargoAand10cratesofcargoBB)20cratesofcargoAand0cratesofcargoB

C) 0cratesofcargoAand20cratesofcargoBD)10cratesofcargoAand18cratesofcargoB

Page82

15) Avineyardproducestwospecialwines,awhiteandared.Abottleofthewhitewinerequires14poundsof

grapesand1hourofprocessingtime.Abottleofredwinerequires25poundsofgrapesand2hoursof

processingtime.Thevineyardhasonhand2,198poundsofgrapesandcanallot160hoursofprocessingtime

totheproductionofthesewines.Abottleofthewhitewinesellsfor$11.00,whileabottleoftheredwinesells

for$20.00.Howmanybottlesofeachtypeshouldthevineyardproduceinordertomaximizegrosssales?

A) 132bottlesofwhiteand14bottlesofred B) 14bottlesofwhiteand132bottlesofred

C) 76bottlesofwhiteand42bottlesofred D) 42bottlesofwhiteand59bottlesofred

16) Acandycompanyhas120poundsofcashewsand160 poundsofpeanutswhichtheycombineintotwo

differentmixes.Thedeluxemixhashalfcashewsandhalfpeanutsandsellsfor$7perpound.Theeconomy

mixhasonethirdcashewsandtwothirdspeanutsandsellsfor$4.80perpound.Howmanypoundsofeach

mixshouldbepreparedformaximumrevenue?

A) 160poundsofdeluxeand120poundsofeconomy

B) 240poundsofdeluxeand80poundsofeconomy

C) 80poundsofdeluxeand40poundsofeconomy

D) 120poundsofdeluxeand0poundsofeconomy

17) Adoctorhastoldasickpatienttotakevitaminpills.Thepatientneedsatleast32unitsofvitaminA,atleast7

unitsofvitaminB,andatleast27unitsofvitaminC.Theredvitaminpillscost10¢eachandcontain8unitsof

A,1unitofB,and3unitsofC.Thebluevitaminpillscost20¢eachandcontain4unitsofA,1unitofB,and5

unitsofC.Howmanypillsshouldthepatienttakeeachdaytominimizecosts?

A) 4redand3blue B) 3redand4 blue C) 7 redand0blue D) 5redand2 blue

18) Mrs.Whitewantstocrochethatsandafghansforachurchfundraisingbazaar.Sheneeds7hourstomakeahat

and2hourstomakeanafghan,andshehasnomorethan53hoursavailable.Shehasmaterialfornomorethan

14itemsandnomorethan10afghans.Thebazaarwillsellthehatsfor$18eachandtheafghansfor$5each.

Howmanyofeachshouldshemaketomaximizetheincomeforthebazaar?Whatisthemaximumincome?

A) 5hatsand9afghans;$135 B) 9 hatsand5 afghans;$187

C) 10hatsand4afghans;$200 D) 7 hatsand7 afghans;$161

19) Anofficemanagerisbuyingusedfilingcabinets.Eachsmallfilecabinetcosts$5

,

takesup7squarefeetoffloor

space,andholds10cubicfeetoffiles.Eachlargefilecabinetcosts$12,takesup10squarefeetoffloorspace,

andholds14cubicfeetoffiles.Thetotalcostcannotexceed$112,andtheofficehasnomorethan116square

feetoffloorspaceavailableforthecabinets.Howmanyofeachfilecabinetshouldthemanagerbuyto

maximizestoragecapacity?Whatisthemaximumstoragecapacity?

A) 8smallfilecabinetsand6largefilecabinets;164 cubicfeet

B) 6smallfilecabinetsand8largefilecabinets;192 cubicfeet

C) 0smallfilecabinetsand14largefilecabinets;196 cubicfeet

D) 14smallfilecabinetsand0largefilecabinets;140 cubicfeet

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

20) Yourcomputersupplystoresellstwotypesoflaserprinters.Thefirsttype,A,hasacostof$86andyoumakea

$45profitoneachone.Thesecondtype,B,hasacostof$130andyoumakea$35profitoneachone.You

expecttosellatleast100laserprintersthismonthandyouneedtomakeatleast$3850profitonthem.How

manyofwhattypeofprintershouldyouorderifyouwanttominimizeyourcost?

21) TheJillsonʹshaveupto$75,000toinvest.Theydecidethattheywanttohaveatleast$25,000investedinstable

bondsyielding6%andthatnomorethan$45,000shouldbeinvestedinmorevolatilebondsyielding12%.How

muchshouldtheyinvestineachtypeofbondtomaximizeincomeiftheamountinthemorevolatilebond

shouldnotexceedtheamountinthemorestablebond?Whatisthemaximumincome?

Page83

22) TheFiedlerfamilyhasupto$130,000toinvest.Theydecidethattheywanttohaveatleast$40,000investedin

stablebondsyielding5.5%andthatnomorethan$60,000shouldbeinvestedinmorevolatilebondsyielding

11%.Howmuchshouldtheyinvestineachtypeofbondtomaximizeincomeiftheamountinthestablebond

shouldnotexceedtheamountinthemorevolatilebond?Whatisthemaximumincome?

23) EricʹsCarpentrymanufacturestwotypesofbookshelvesthatare4feettalland3feetwide,abasicmodelanda

deluxemodel.Eachbasicbookshelfrequires1.5hoursforassemblyand1hourforfinishing;eachdeluxemodel

requires2.5hoursforassemblyand1hourforfinishing.Twoassemblersandonefinisherareemployedbythe

company,andeachworks40hoursperweek.Ericcansellmorebasicmodelsthandeluxemodels,sohewants

thenumberofbasicmodelsproducedtobe50%morethanthenumberofdeluxemodelsproduced.Ifhe

makes$50profitonthebasicmodelsand$65profitonthedeluxemodels,howmanyshouldhemaketo

maximizetheprofit?Whatisthemaximumprofit?

24)

J

oelyʹsTeaShop,astorethatspecializesinteablends,hasavailable45poundsofAgradeteaand70poundsof

Bgradetea.Thesewillbeblendedinto1poundpackagesasfollows:Abreakfastblendthatcontainsonethird

ofapoundofAgradeteaandtwothirdsofapoundofBgradeteaandanafternoonteathatcontainsonehalf

poundofAgradeteaandonehalfpoundofBgradetea.IfJoelymakesaprofitof$1.50oneachpoundofthe

breakfastblendand$2.00profitoneachpoundoftheafternoonblend,howmanypoundsofeachblendshould

shemaketomaximizeprofits?Whatisthemaximumprofit?

25) Anartistiscreatingamosaicthatcannotbelargerthanthespaceallottedwhichis4feettalland6feetwide.

Themosaicmustbeatleast3feettalland5feetwide.Thetilesinthemosaichavewordswrittenonthemand

theartistwantsthewordstoallbehorizontalinthefinalmosaic.Thewordtilescomeintwosizes:Thesmaller

tilesare4inchestalland4incheswide,whilethelargetilesare6inchestalland12incheswide.Ifthesmall

tilescost$3.50eachandthelargertilescost$4.50each,howmanyofeachshouldbeusedtominimizethecost?

Whatistheminimumcost?

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Ch.8 SystemsofEquationsandInequalities

AnswerKey

8.1 SystemsofLinearEquationsinTwoVariables

1 DecideWhetheranOrderedPairisaSolutionofaLinearSystem

2 SolveLinearSystemsbySubstitution

3 SolveLinearSystemsbyAddition

4 IdentifySystemsThatDoNotHaveExactlyOneOrdered –PairSolution

15) A

5 SolveProblemsUsingSystemsofLinearEquations

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8.2 SystemsofLinearEquationsinThreeVariables

1 VerifytheSolutionofaSystemofLinearEquationsinThreeVariables

2 SolveSystemsofLinearEquationsinThreeVariables

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3 SolveProblemsUsingSystemsinThreeVariables

8.3 PartialFractions

1 DecomposeP/Q,WhereQHasOnlyDistinctLinearFactors

2 DecomposeP/Q,WhereQHasRepeatedLinearFactors

3 DecomposeP/Q,WhereQHasaNonrepeatedPrimeQuadraticFactor

4 DecomposeP/Q,WhereQHasaPrime,RepeatedQuadraticFactor

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8.4 SystemsofNonlinearEquationsinTwoVariables

1 RecognizeSystemsofNonlinearEquationsinTwoVariables

2 SolveNonlinearSystemsBySubstitution

3 SolveNonlinearSystemsByAddition

4 SolveProblemsUsingSystemsofNonlinearEquations

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8.5 SystemsofInequalities

1 GraphaLinearInequalityinTwoVariables

2 GraphaNonlinearInequalityinTwoVariables

8) A

3 UseMathematicalModelsInvolvingLinearInequalities

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4 GraphaSystemofInequalities

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8.6 LinearProgramming

1 WriteanObjectiveFunctionDescribingaQuantityThatMustbeMaximizedorMinimized

2 UseInequalitiestoDescribeLimitationsinaSituation

3 UseLinearProgrammingtoSolveProblems

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