Solutions for Intermediate Algebra 7th Us Edition by Martin Gay by 6alsm

Chapter 2: Equations, Inequalities, and Problem Solving

b. ⎛⎞ =+⎜⎟ ⎝⎠ = ≈

42 8 0.05 25,0001 4

25,000(1.0125)

25,000(1.104486101)

27,612.15

Theamountintheaccountis$27,612.15.

c. ⎛⎞ =+⎜⎟ ⎝⎠

25,0001 12

122 24 0.05

25,000(1.00416666)

25,000(1.104941335)

27,623.53

A Theamountintheaccountis$27,623.53.

30. Roundtripdistance=154+154=308miles 1 2 1 3085 2 308 5 56 drt r r r =⋅

= = Theiraveragespeedwas56mph.

32. Usingtheformula 9 32, 5 FC=+ wehave

9932(15)3227325

55 FC=+=−+=−+=

Thetemperaturewas5°F.

34. Thetotalareaoftheceilingis 18(12)=216squarefeet.Eachpackagecan coverupto50squarefeet.Thus,thenumberof packagesneededis 216 4.32. 50 = Therefore, 5packagesmustbepurchased.

36. Usingtheformula1, nt r AP n ⎛⎞ =+⎜⎟ ⎝⎠ wehave ⎛⎞ =+⎜⎟ ⎝⎠ = ≈ ≈ 23 6 0.055 40001 2

4000(1.0275)

4000(1.176768361) 4707.07 A A A A

Yes,theamountisenough.

38. Notethatthewallcovers218=168square feet.Becausewewishtopaintthreecoats,we actuallymustcoveratotalof

1683=504squarefeet.Sinceeachgallon covers300squarefeet,weneed

504 1.68gallons 300 = ofpaint.2gallonsshouldbe purchased.

40. 2 82552 82525 82525 33 () Vrh h h h h =π π=π π=π = = Theheightis33mm.

42. a. 43 3 ; Vr =π 18 9 22 d r ===

729 3 972 () () V V V =π =π =π Thevolumeis972π cubiccm.

b. V =972π ≈ 3053.63cubiccm

44. a. 2 2 415 75398 ()() . Vrh V V =π =π ≈ Thevolumeofthecylinderis753.98cubic millimeters. b. 3 3 4 3

4 3 26808 () Vr V V =π =π ≈ Thevolumeofthesphereis268.08cubic millimeters.

c. V =753.98+268.08=1022.06 Thevolumeofthevitaminis1022.06cubic millimeters.

46. Notethattheradiusofthecircleisequalto 22,248+4000=26,248.

C =2πr

C =2π(26,248)

C =52,496π C ≈ 164,921.0479 The“length”oftheClarkebeltisapproximately 164,921miles.

48. 5280ft 8miles42,240ft 1mile ×= 60min60sec 7.5hours27,000sec 1hour1min ××=

Using d = rt wehave: 42,240(27,000) 42,2401.6 27,000 r r = =≈

Thedrillcanberemovedatarateof1.6ft/sec.

50. Usingtheformula3 4 3,Vr =π wehave 43 (20.6) 3 36,618 V V =π ≈

ThevolumeofEarthaisabout36,618cuft.

52. 13560 2.25 drt t t = = = ItwilltakeMark2.25hoursor2hours 15minutes.

54. 494 494 94 4 Chfp hCfp Cfp h =++ =−− =

56. C =4h +9f +4p C =4(30)+9(9)+4(2) C =209 Thereare209caloriesinthisserving.

58. 44 9 1204(21)4(5) 9 1.8 Chp f f f = = ≈ Thereare1.8gramsoffatperserving.

60. 2,3satisfy x >1.

62. 3,2,1,0,1,2,3,satisfy x 3 ≥ 7or x ≥ 4.

64. answersmayvary

PlanetAUfromSun

66. Earth92.992.91.000 =

68. Jupiter483.392.95.202 ≈

70. Uranus178392.919.193 ≈

72. Pluto367092.939.505 =

74. answersmayvary

76. answersmayvary

78. Twoofthe8sectorsareyellow. 21 (yellow)84 P ==

80. Threeofthe8sectorsareblue. 3 blue) 8 ( P =

82. Threeofthesectorsareblackoryellow. 3 blackoryellow)8 ( P =

84. Sixofthesectorsareyellow,blue,orblack. 63 (yellow,blue,orblack)84 P ==

86. Allofthesectorsarered,yellow,green,blue,or black. 8 red,yellow,green,blue,orblack)1 8 ( P ==

88. P(eventsuretooccur)=1

Section 2.4 Practice Exercises

1. a. {x|x <3.5}(−∞,3.5) 3.5 )

b. {x|x ≥ 3}[3, ∞) –3

c. {x|1 ≤ x <4}[1,4) 4 ) –1

Chapter 2: Equations, Inequalities, and Problem Solving

2. 59 5595 4 x x x +> +−>− > (4, ∞) 4 ( 3. 3123 312232 13 1131 4 +≤− +−≤−− +≤− +−≤−− ≤− xx xxxx x x x (−∞,4] –4

4. a. 24 515 5254 25215 2 3 x x x ≥ ⋅≥⋅ ≥ 2 3, ⎡⎞ ∞ ⎟ ⎢ ⎣⎠ 2 3 b. 2.49.6 2.49.6 2.42.4 4 x x x −< > >− (4, ∞) –4 ( 5. −+≤++ −−≤++ −−≤+

(46)2(59)2 4610182 461218 46412184 61618 618161818 2416 2416 1616 3 2 xxx xxx xx xxxx x x x x x 3 2, ⎡⎞ −∞ ⎟ ⎢ ⎣⎠ 3 2

ISM: Intermediate Algebra

6. 3 5(3)7 3 5(3)5(7) 5 3(3)5(7) 39535 3955355 2935 299359 226 226 22 13 xx xx xx xx xxxx x x x x x −≥− ⎡⎤−≥−⎢⎥ ⎣⎦ −≥− −≥− −−≥−− −−≥− −−+≥−+ −≥− ≤ ≤ (−∞,13] 13

7. 4(2)45 4845 484454 85 xx xx xxxx −<+ −<+ −−<+− −<

Thisisatruestatementforallvaluesof x.Thesolutionsetis{x|x isarealnumber}or(−∞ , ∞). 0

8. Inwords:900+commission (15%ofsales) ≥ 2400

Translate:900+0.15x ≥ 2400

9000.152400 9000.159002400900 0.151500 10,000 x x x x +≥ +−≥− ≥ ≥

Salesmustbegreaterthanorequalto$10,000permonth.

9. 118390175 118215 approximately18.2 t t t −+< −<− >

Theannualconsumptionofcigaretteswillbelessthan175billionmorethan18.2yearsafter2004,orin approximately18+2004=2022andafter.

Vocabulary, Readiness & Video Check 2.4

1. d.(−∞,5)

2. c.[11, ∞)

3. b. 7 2.5, 4 ⎛⎤ ⎜ ⎥ ⎝⎦

4. a. 10 ,0.2 3 ⎡⎞ ⎟ ⎢ ⎣⎠

5. Theset{x|x ≥ 0.4}writteninintervalnotation is[0.4, ∞)

6. Theset{x|x <0.4}writteninintervalnotation is(−∞,0.4).

7. Theset{x|x ≤ 0.4}writteninintervalnotation is(−∞,0.4].

8. Theset{x|x >0.4}writteninintervalnotation is(0.4, ∞)

9. ThegraphofExample1isshadedfrom −∞ to, butnotincluding,3,asindicatedbya parenthesis.Towriteintervalnotation,write downwhatisshadedfortheinequalityfromleft toright.Aparenthesisisalwaysusedwith −∞ , sofromthegraph,theintervalnotationis (−∞,3).

10. Wecanaddthesamenumberto(orsubtractthe samenumberfrom)bothsidesofalinear inequalityinonevariableandhaveanequivalent inequality;additionpropertyofequality.

11. Ifyoumultiplyordividebothsidesofan inequalitybythesamenonzeronegativenumber, youmustreversethedirectionoftheinequality symbol.

12. maximum,orless

Exercise Set 2.4

2. {x|x >5} (5, ∞) 5 ( 4. {x|x <0.2} (−∞,0.2) –0.2 )

6. {x|7 ≥ x} (−∞,7]

8. {x|5 ≤ x ≤ 1} [5,1] –1 –5

10. {x|3> x ≥ 7} [7,3) –3 ) –7 12. 21 3 x x +≤− ≤− (−∞,3] –3 14. 11105 5 xx x <+ < (−∞,5) 5 )

5 5 6 656 5 565

⋅≥⋅

[6, ∞)

x x x

20. 411.2 2.8 x x >− >− (2.8, ∞) –2.8 ( 22. 48 48 44 2 x x x −≥ ≤ ≤− (−∞,2] –2 24. 8523 515 3 x x x −≤ −≤ ≥− [3, ∞)

26. 20615 20515 535 535 55 7 xx x x x x +<− −<− −<− > > (7, ∞)

28. 6(23)12 121812 180 0 x x x x −≥ −≥ −≥ ≤ (−∞,0]

30. 5(4)4(23) 520812 38 8 3 xx xx x x +≤+ +≤+

32. 1237 1 37 1237 2121(1) 37 7(12)3(37)21 71492121 52821 57 7 5 xx xx xx xx x x x −+ +> −+ ⎛⎞

34. 2(42)5[12(1)] 845(122) 845(21) 84105 245 29 9 2 xx xx xx xx x x x −+>−+− −−>−+− −−>−− −−>−+ −> > > 9 2, ⎛⎞ ∞ ⎜⎟ ⎝⎠

36. 912 99129 3 x x x −<− −+<−+ <− (−∞,3)

38. 2 2 11 2 x x x −>− < < (−∞,2)

40. 64.2 64.2 66 0.7 x x x −≤ ≥ ≥− [0.7, ∞)

42. −≥ ⎛⎞⎛⎞ −≥ ⎜⎟⎜⎟ ⎝⎠⎝⎠ −≥ ≥ ≥ 32 436 32 1212 436 982 12 1 2 x x x x x ⎛⎤ −∞ ⎜ ⎥ ⎝⎦ 1 ,2

44. −+<−+ −+<−− <− < < 623(4) 62312 2312 143 14 3 xx xx x x x 14 3, ⎛⎞ ∞ ⎜⎟ ⎝⎠

46. 4 5(1)1 4 5(1)5(1) 5 4(1)5(1) 4455 45 1 1 xx xx xx xx x x x +≤+ ⎡⎤+≤+⎢⎥ ⎣⎦ +≤+ +≤+ −+≤ −≤ ≥− [1, ∞)

48. 0.70.45 0.30.45 1.5 xx x x −> −> <− (−∞,1.5)

Chapter 2: Equations, Inequalities, and Problem Solving

50. 7(23)475(34) 1421471520 18211316 22113 234 17 xxxx xxxx xx x x x ++≤+−+ ++≤+−+ +≤−+ +≤− ≤− ≤− (−∞,17]

52. 13(92)5(6)10 139253010 42520 220 18or18 yyy yyy yy y yy −+≤−+ −−≤−+ −≤− −≤− ≤≥ [18, ∞)

54. 8(3)7(5) 824735 824835 2435 xxx xxx xx +≤++ +≤++ +≤+ ≤ (−∞ , ∞)

56. 77(2) 7714 014False <− <− <− xx xx Nosolution; ∅

58. 0.2(82)1.2(3) 10[0.2(82)]10[1.2(3)] 2(82)12(3) 1641236 4436 432 8 xx xx xx xx x x x −<− −<− −<− −<− −<− <− <− (−∞,8)

60. 7135 12386 7135 2424 12386 2783345 148920 5820 512 12 5 xx xx xx xx x x x −≤− ⎡⎤⎡⎤ −≤− ⎢⎥⎢⎥ ⎣⎦⎣⎦ ⋅−≤⋅−⋅ −≤− −≤− ≤− ≤− 12 ,5 ⎛⎤ −∞− ⎜ ⎥ ⎝⎦

62. 4 28 4 21 (3)(28)236 21 6(3)6(28)2 36 4(3)(28)12 41224 212 x x xx xx xx xx x < <− <− +<−+ ⎡ ⎤⎡⎤ +<−+ ⎢ ⎥⎢⎥ ⎣ ⎦⎣⎦ +<−+ +<+ + (−∞,4)

64. 3412 2 612 3412 1212(2) 612 2(34)(12)24 681224 5624 629 29 6 xx xx xx xx x x x −≤− ⎛⎞−≤−⎜⎟ ⎝⎠ −−−≤− −−+≤− −≤− −≤− ≥

29 6, ⎡ ⎞ ∞ ⎟ ⎢ ⎣ ⎠

66. 425 236 425 66 236 3(4)2(2)5 312245 85 13 xx xx xx xx x x −> ⎛⎞⎛⎞ −> ⎜⎟⎜⎟ ⎝⎠⎝⎠ −−−> −−+> −> > (13, ∞)

68. 32121 1862 32121 1818 1862 323(12)9 32369 319 38 8 3 xx xx xx xx x x x ++ −≤− ++ ⎛⎞⎛⎞ −≤− ⎜⎟⎜⎟ ⎝⎠⎝⎠ +−+≤− +−−≤− −−≤− −≤− ≥ 8 , 3 ⎡ ⎞ ∞ ⎟ ⎢ ⎣ ⎠

70. a. Let x beHolden’stimeonhislasttrial. 6.857.046.92 7 4 6.857.046.92 44(7) 4 6.857.046.9228 20.8120.812820.81 7.19 x x x x x +++ < +++ ⎛⎞ <

+++< +−<− < Thesolutionis{x|x <7.19}.

b. Atimeof7.19minutesorlesswillresultin anaveragetimeunder7.0minutes.

72. a. Let x bethenumberofadditionalouces. 9821300 21202 approximately9.6 x x x +≤ ≤ ≤ Thesolutionis{x|x ≤ 9.6}.

b. Since x representsthenumberofounces afterthefirstounce,youcanmailatmost 1ounceplus9additionalounces,or 10ounces.

74. a. Let x bethenumberofadditionalhalf-hour intervalsparked. 10064 10640 630 5 .. x x x x +≤ +≤ ≤ ≤ Thesolutionis{x|x ≤ 5}.

b. Since x representsthenumberofhalfhours afterthefirsthour,youcanparkforatmost 1hourplus5additionalhalfhours,or 1+2.5=3.5hourstotal.

76. a. Let n =numberofcallsmadeinagiven month.

25130.06 120.06 200 <+ < < n n n {n|n >200}

b. Plan1ismoreeconomicalthanPlan2when 200ormorecallsaremade.

78. Giventhat F ≥ 977,weknowthefollowing: 5 9(32) 5 9(97732) 5 9(945) 525 CF C C C ≥− ≥− ≥ ≥ {C|C ≥ 525}

Sostibnitemeltswhenthetemperatureisatleast 525°C.

80. a. 11839050 118340 approximately28.8 . t t t −+< −<− > 2004+28.8=2032.8 Theconsumptionwillbelessthan50billion duringtheyear2032andafter.

b. answersmayvary

82. Consumptionofskimmilkisdecreasingover time;answersmayvary.

84. 2024is20yearsafter2004,so2024corresponds to t =20.

s =0.22t +27.4

s =0.22(20)+27.4=4.4+27.4=23 Theaverageconsumptionofskimmilkis predictedtobe23poundsperpersonperyearin 2024.

86. answersmayvary

88. answersmayvary

90. answersmayvary

92. x ≥ 0and x ≤ 7 Theintegersare0,1,2,3,4,5,6,7.

94. x <6and x <5 Theintegersare6,7,8,....

96. 3123 31212312 315 315 33 5 x x x x x −= −+=+ = = =

Chapter 2: Equations, Inequalities, and Problem Solving

100. {x|x >4};(4, ∞)

102. 5 (−∞,5]

104. {x|3.7 ≤ x <4} 4 ) –3.7

106. Tosolve3x >14,bothsidesmustbedividedby 3,sotheinequalitysymbolwillnotbereversed.

108. Tosolve x ≤ 9,bothsidesmustbedividedby 1,sotheinequalitysymbolwillbereversed.

110. 235 28 4 x x x −> > > Thesolutionsetis(4, ∞).

112. answersmayvary

114. answersmayvary

116. answersmayvary

⎞ ∞ ⎟ ⎢ ⎣ ⎠ 4. 5324 32 1 xx x x +≥+ +≥ ≥− [1, ∞)

5. 6(4)3(8) 624324 30 0 yy yy y y −=− −=− = = 6. 2 4 5 202 1 10 x x x −≤ −≤ ≥− 1 10, ⎡ ⎞ −∞ ⎟ ⎢ ⎣ ⎠ 7. −≥ ⎛⎞ −≥ ⎜⎟ ⎝⎠ −≥ ≤− 1 3 2 1 2(3)2 2 61 1 6 x x x x 1 ,6 ⎛⎤ −∞− ⎜ ⎥ ⎝⎦ 8. 5(4)4(5) 520420 0 yy yy y +=+ +=+ = 9. <− <− <− 77(2) 7714 014(False) xx xx Nosolution; ∅

ISM: Intermediate Algebra

10. 511 7 2 511 22(7) 2 51114 53 3 5 x x x x x −+ ≤ −+ ⎛⎞ ≤ ⎜⎟ ⎝⎠ −+≤ −≤ ≥− 3 5, ⎡⎞ −∞ ⎟ ⎢ ⎣⎠

11. −+=− −+−=−− −=− = = 51.519.5 51.51.519.51.5 521 521 55 4.2 x x x x x

12. 5426 530 6 x x x −+=− −=− =

13. 52314 511 5211 216 8 xxx xx x x x +−=−+− +=−− +=− =− =−

14. 1214112 142 16 xx x x +<− +<− <− (−∞,16)

15. 2 542 2 2020 542 4510(2) 1020 1120 20 11 xxx xxx xxx xx x x −= ⎛⎞⎛⎞ −= ⎜⎟⎜⎟ ⎝⎠⎝⎠ −=− −=− −=− =

Chapter 2: Equations, Inequalities, and Problem Solving

16. 12128(1) 121288 4128 44 1 xx xx x x x −=− −=− −=− = = 17. 2(3)70 2670 276 38 x x x x −> −> > > (38, ∞)

18. −−= −−+=+ −= = =− 34.711.8 34.74.711.84.7 316.5 316.5 33 5.5 x x x x x

19. −−−−=+ −+−+=+ −+=+ −=− == 2(4)(31)53 283153 5953 106 63 105 bbb bbb bb b b

20. 8(3)7(5) 824735 824835 2435(Trueforall) xxx xxx xx x +<++ +<++ +<+ < Allrealnumbers;(−∞ , ∞) 21. 3152 2 87 3152 565656(2) 87 7(31)8(52)112 2174016112 21716152 5145 29 tt tt tt tt tt t t ++ =+ ++ ⎛⎞⎛⎞ =+⎜⎟⎜⎟ ⎝⎠⎝⎠ +=++ +=++ +=+ = =

22. 4(6)8(3)5 4248245 324324 2424(Trueforall) xxxx xxxx xx x −−=−− −−=−− −=− −=− Thesolutionisallrealnumbers.

Chapter 2: Equations, Inequalities, and Problem Solving

23. 322 623 322 66 623 3(32)4 964 1064 1010 1 xx xx xx xx x x x +< ⎛⎞⎛⎞ +< ⎜⎟⎜⎟ ⎝⎠⎝⎠ +−< +−< −< < < (−∞,1)

24. + += + ⎛⎞⎛⎞⎛⎞ += ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠ +=+ =+ = = 3 3510 3 303030 3510 1063(3) 1639 139 9 13 yyy yyy yyy yy y y

25. 5(6)23(21)4 5302634 73067 23 xxx xxx xx x −+>−− −+>−− −>− > (23, ∞)

26. 14(1)72(36)4 141476124 71468 6 xxx xxx xx x −−≤−+ −−≤−+ −≤− ≤ (−∞,6]

27. 13 (32)(5)248 13 8(32)8(5)2 48 2(32)83(5)16 64831516 2431 35 33 or 55 xxx xxx xxx xxx xx x xx +−≥−+ ⎡⎤⎡⎤ +−≥−+ ⎢⎥⎢⎥ ⎣⎦⎣⎦ +−≥−+ +−≥−+ −+≥+ ≥ ≥≤ 3 ,5 ⎛⎤ −∞ ⎜ ⎥ ⎝⎦

28. 15 (10)4(21)1 36 15 6(10)46(21)1 36 2(10)245(21)6 220241056 2220101 1932 1919 or 3232 xxx xxx xxx xxx xx x xx −−>+− ⎡ ⎤⎡⎤ −−>+− ⎢ ⎥⎢⎥ ⎣ ⎦⎣⎦ −−>+− −−>+−

Section 2.5 Practice Exercises

1. A ={1,3,5,7,9}and B ={1,2,3,4} Thenumbers1and3areinsets A and B. Theintersectionis{1,3}. A ∩ B ={1,3}.

2. 38213 524 52 +<−< << << xandx xandx xandx

{x|x <5},(−∞,5)

5 )

{x|x <2},(−∞,2)

2 )

{x|x <5 and x <2}={x|x <2}

2 )

Thesolutionsetis(−∞,2).

3. 40328 036 02 ≤+> ≤> ≤> xandx xandx xandx

{x|x ≤ 0},(−∞,0]

{x|x >2},(2, ∞)

2 (

{x|4x ≤ 0 and 3x +2>8}={}or ∅

ISM: Intermediate Algebra

4. 359 355595 24 24 111 24 or42 x x x x x x <−< −<−−<− −<−< >> >>− −<< Thesolutionsetis(4,2).

5. 413 2 2(4)212(3) 2 826 822262 68 x x x x x −≤−≤ ⎛⎞ −≤−≤ ⎜⎟ ⎝⎠ −≤−≤ −+≤−+≤+ −≤≤ Thesolutionsetis[6,8].

6. A ={1,3,5,7,9}and B ={2,3,4,5,6}. Thenumbersthatareineithersetorbothsetsare {1,2,3,4,5,6,7,9}.Thissetistheunion, A ∪ B

7. 85812 833 3 3 8 +≤−≥ ≤≥ ≤≥ xorx xorx xorx

33 ,, 88 xx ⎧⎫⎛⎤ ≤−∞⎨⎬ ⎜ ⎥ ⎩⎭⎝⎦

3 8

{x|x ≥ 3},[3, ∞) 3

33 or3,[3,) 88 xxx ⎧⎫⎛⎤≤≥=−∞∪∞⎨⎬ ⎜ ⎥ ⎩⎭⎝⎦ 3 3 8

Thesolutionsetis 3 ,[3,). 8 ⎛⎤ −∞∪∞ ⎜ ⎥ ⎝⎦

8. 32850 360 20 −−>−> −>−> <> xorx xorx xorx

{x|x <2},(−∞,2) 2 )

{x|x >0},(0, ∞)

0 (

Chapter 2: Equations, Inequalities, and Problem Solving

{x|x <2or x >0},(−∞ , ∞)

Thesolutionsetis(−∞ , ∞).

Vocabulary, Readiness & Video Check 2.5

1. Twoinequalitiesjoinedbythewords“and”or “or”arecalledcompoundinequalities.

2. Thewordandmeansintersection.

3. Thewordormeansunion.

4. Thesymbol ∩ meansintersection.

5. Thesymbol ∪ representsunion.

6. Thesymbol ∅ istheemptyset.

7. Foranelementtobeintheintersectionofsets A and B,theelementmustbeinset A andinset B.

8. Graphthetwointervals,eachonitsownnumber line,soyoucanseetheirintersection.Graphthis intersectiononthethirdnumberlinethis intersectionisthesolutionset.

9. Foranelementtobeintheunionofsets A and B,theelementmustbeinset A orinset B

10. Graphthetwointervals,eachonitsownnumber line,soyoucanseetheirunion.Graphthisunion onthethirdnumberlinethisunionisthe solutionset.

Exercise Set 2.5

2. C ∩ D ={4,5}

4. A ∪ D ={x|x isanevenintegeror x =5or x =7}

6. A ∩ B = ∅

8. B ∪ D ={x|x isanoddintegeror x =4or x =6}

10. B ∩ C ={3,5}

12. A ∪ C ={x|x isanevenintegeror x =3or x =5}

14. x ≤ 0 and x ≥ 2 2 ≤ x ≤ 0 [2,0] –20

Chapter 2: Equations, Inequalities, and Problem Solving

16. x <2 and x >4 ∅

18. x ≥ 4 and x >1 x >1 (1, ∞) 1 (

20. 23519 1510 2 +≥−≥ ≥≥ ≥ xandx xandx x x ≥ 2 [2, ∞)

22. 24040 240 2 +>> >−> >− xandx xandx x (0, ∞)

24. 7212015 35 −≤−−≤− ≥≤ xandx xandx

3 ≤ x ≤ 5 [3,5]

26. 230 53 x x −≤+≤ −≤≤− [5,3]

28. 1427 1442474 323 33 22 x x x x <+< −<+−<− −<< << 33 , 22 ⎛⎞ ⎜⎟ ⎝⎠

30. 1 251 2 1 36 2 612 x x x −<−< << << (6,12)

32. 25 41 3 25 3(4)33(1) 3 12253 1722 17 1 2 17 1 2 x x x x x x −+ −≤≤ −+ ⎛⎞ −≤≤ ⎜⎟ ⎝⎠ −≤−+≤ −≤−≤− ≥≥ ≤≤ 17 1, 2 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

34. x ≥ 2 or x ≤ 2 (−∞ , ∞)

36. x <0 or x <1 (−∞,1) 1 )

38. x ≥ 3 or x ≤ 4 (−∞,4] ∪ [3, ∞) –3 –4

40. 510351 236 2 −≤−≥ ≥−≥ ≥ xorx xorx x x ≥ 2 [2, ∞)

42. 90412 93 +<>− <−>− xorx xorx (−∞,9) ∪ (3, ∞)

44. 5(1)5511 116 0 −≥−+≤ −≥−≤ ≥ xorx xorx x (−∞ , ∞)

46. 5 7 x < and x <1 5 7 x < 5 , 7 ⎛⎞ −∞ ⎜⎟ ⎝⎠

Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving

48. 5 7 x < or x <1 x <1 (−∞,1)

50. 35111 2510 2 2 5 x x x <+< << << 2 ,2 5 ⎛⎞ ⎜⎟ ⎝⎠

52. 21 4 32 21 666(4) 32 46324 1621 17 62 x x x x x <+< ⎛⎞⎛⎞ <+<⎜⎟⎜⎟ ⎝⎠⎝⎠ <+< << << 17 , 62 ⎛⎞ ⎜⎟ ⎝⎠

54. 2132 242 22 −≥−> ≥<− ≥<− xandx xandx xandx ∅

56. 3 1024 8 3 12 8 8 2 3 +≤−<− ≤−> ≤−> xorx xorx xorx 8 ,(2,) 3 ⎛⎤ −∞−∪∞ ⎜ ⎥ ⎝⎦

58. 21 22 3 21 3(2)33(2) 3 6216 527 57 22 75 22 x x x x x x −<< ⎛⎞ −<< ⎜⎟ ⎝⎠ −<−−< −<−< >> −<< 75 , 22 ⎛⎞ ⎜⎟ ⎝⎠

60. 52(4)8 5288 1320 13 0 2 x x x x −<+< −<+< −<< −<< 13 ,0 2 ⎛⎞ ⎜⎟ ⎝⎠

62. 5058 03 03 ≤−+< ≤−< ≤>− xandx xandx xandx (3,0]

64. 73120 7321 77 −<+<− >−<− >−<− xorx xorx xorx (−∞,7) ∪ (7, ∞)

66. 2612 33 33 −<−−>− >−>− >< xorx xorx xorx (−∞,3) ∪ (3, ∞)

68. 1311 2102 1311 101010 2102 5315 436 4 2 3 x x x x x −≤< ⎛⎞⎛⎞⎛⎞ −≤< ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠ −≤−< −≤< −≤< 4 ,2 3 ⎡ ⎞ ⎟ ⎢ ⎣ ⎠

Chapter 2: Equations, Inequalities, and Problem Solving

70. 161 4126 161 121212 4126 362 98 98 x x x x x −<<− ⎛⎞⎛⎞⎛⎞ −<<− ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠ −<−<− −<−<− >> (8,9)

72. 0.70.40.80.5 1.50.40.3 3.750.75 x x x −≤+< −≤<− −≤<− [3.75,0.75)

74. |719|=|26|=26

76. |4|(4)+|20|=4+4+20=28

78. 5 5,5 x x = =−

80. |x|=2 ∅

82. Fromthegraph,weseethatthenumberof single-familyhousingstartswerelessthan500 orthenumberofsingle-familyhousing completionsgreaterthan1500arefortheyears 2004,2005,2006,2009,2010,and2011.

84. answersmayvary

86. x +3<2x +1<4x +6 3212146 252 5 2 2 5 2 2 +<++<+ <−< >−< >>− xxandxx xandx xandx xandx (2, ∞)

88. 7x 1 ≤ 7+5x ≤ 3(1+2x) 71757536 284 44 −≤++≤+ ≤≤ ≤≥ xxandxx xandx xandx {4}

90. 1+2x <3(2+ x)<1+4x 12636314 55 55 +<++<+ −<< >−> xxandxx xandx xandx (5, ∞)

92. 1018 5 10(32)18 9 9959 (10)(32)(18) 5595 162 1832 5 1464.4 C F F F F

−≤≤ −≤−≤ ⎛⎞ −≤−≤ ⎜⎟ ⎝⎠ −≤−≤ ≤≤ 14° ≤ F ≤ 64.4°

94. Let x beWendy’sgradeonthefinalexam. 1 80(280908275)89 6 4802327534 1532207

76.5103.5 76.5100 x x x x x ≤++++≤ ≤+≤ ≤≤ ≤≤ ≤≤ IfWendyscoresbetween76.5and100inclusive onherfinalexam,shewillreceiveaBinthe course.

Section 2.6 Practice Exercises

1. |q|=13 q =13or q =13 Thesolutionsetis{13,13}.

2. |2x 3|=5 235or235 28or22 4or1 xx xx xx −=−=− ==− ==− Thesolutionsetis{1,4}.

3. 115 5 x += 115or115 55 14or16 55 70or80 xx xx xx +=+=− ==− ==− Thesolutionsare80and70.

4. 3814 36 x x += = 36or36 2or2 xx xx ==− ==− Thesolutionsare2and2.

5. |z|=0 Thesolutionis0.

Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving

6. 397 32 2 3 z z z += =− =−

Theabsolutevalueofanumberisnever negative,sothereisnosolution.Thesolutionset is{}or ∅ .

7. 53 8 4 x + =−

Theabsolutevalueofanumberisnever negative,sothereisnosolution.Thesolutionset is{}or ∅ .

8. |2x +4|=|3x 1| 2431 41 5 5 xx x x x +=− −+=− −=− = or24(31) 2431 541 53 3 5 xx xx x x x +=−− +=−+ += =− =−

Thesolutionsare 3 5 and5.

9. |x 2|=|8 x| 28 228 210 5 xx x x x −=− −= = = or2(8) 28 28False xx xx −=−− −=−+ −=−

Thesolutionis5.

Vocabulary, Readiness & Video Check 2.6

1. |x 2|=5

C. x 2=5or x 2=5

2. |x 2|=0

A. x 2=0

3. |x 2|=|x +3|

B. x 2= x +3or x 2=(x +3)

4. |x +3|=5

E. x +3=5or x +3=5

5. |x +3|=5

D. ∅

6. If a isnegative,|X|= a hasnosolution.(Also,if a is0,wesolve X =0.)

Exercise Set 2.6

2. |y|=15 y =15or y =15

4. |6n|=12.6 612.6or612.6 2.1or2.1 nn nn ==− ==−

6. |6+2n|=4 +=−+= =−=− =−=− 624or624 210or22 5or1 nn nn nn 8. 24 3 n += 24or24 33 6or2 33 18or6 nn nn nn +=−+= =−= =−= 10. 13 2 2or2 x x xx += = =−= 12. 264 210 x x −= = 210or210 5or5 xx xx =−= =−= 14. 70 70 0 z z z = = = 16. 3281 327 z z −+= −=− whichisimpossible. Thesolutionsetis ∅ . 18. 320 320 32 2 3 y y y y += += =− =−

Chapter 2: Equations, Inequalities, and Problem Solving

20. |9y +1|=|6y +4| 91(64)or9164 9164or33 155or1 1 or1 3 yyyy yyy yy yy +=−++=+ +=−−= =−= =−=

22. |2x 5|=|2x +5| 25(25)or2525 2525or55 40orfalse 0 xxxx xx x x −=−+−=+ −=−−−= = = Theonlysolutionis0.

24. |x|=1 x =1or x =1

26. |y|=8 y =8or y =8

28. Theabsolutevalueofanyexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅ .

30. |4m +5|=5 455or455 40or410 10 0or 4 5

32. 7122 721 z z += = 721or721 3or3 zz zz ==− ==−

34. Theabsolutevalueofanyexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅

36. 441 45 x x +−= += 45or45 1or9 xx xx +=+=− ==−

38. Theabsolutevalueofanyexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅ .

40. Theabsolutevalueofanyexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅

42. 520 520 52 2 5 x x x x −= −= = =

44. 2397 232 m m +−=− += 232or232 30or34 4 0or 3 mm mm mm +=+=− ==− ==−

46. |86c|=1 861or861 67or69 79 or 66 73 or 62 cc cc cc cc −=−=− −=−−=− == ==

48. 354 354 x x +=− += 354or354 31or39 1 or3 3 xx xx xx +=+=− =−=− =−=−

50. |3+6n|=|4n +11| 36411or36(411) 28or36411 4or1014 7 4or 5 nnnn nnn nn nn +=++=−+ =+=−− ==− ==−

52. |45y|=|3| |45y|=3

Theabsolutevalueofanyexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅ .

54. |4n +5|=|4n +3| 45(43)or4543 4543or53 88orfalse 1 nnnn nn n n +=−++=+ +=−−= =− =− Theonlysolutionis1.

56. 13 4 4 n + = 1313 4or4 44 1316or1316 315or317 17 5or 3 nn nn nn nn

58. 8424 416 m m += = 416or416 4or4 mm mm ==− ==−

60. 52 6 2 52 6 2 x x + =− + = 5252 6or6 22 5212or5212 510or514 14 2or 5 xx xx xx xx ++ ==− +=+=− ==− ==−

62. |5z 1|=|7 z| −=−−−=− −=−+= =−= =− 51(7)or517 517or68 4 46or 3 3 2 z zzz zzz zz z 64. 26 2 5 26 2 5 r r =− = 2626 2or2 55 2610or2610 216or24 8or2 rr rr rr rr ==− −=−=− ==− ==−

66. |8 y|=|y +2| 8(2)or82 82or62 82or3 falseor3 yyyy y yy y y −=−+−=+ −=−−= =−=

Theonlysolutionis3.

68. 51 9 6 51 9 6 d d + =−− + =−

Theabsolutevalueofanyexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅ .

70. Fromthecirclegraph,mozzarellacheesehadthe highestU.S.productionin2014.

72. In2014,creamcheeseaccountedfor7.6%ofthe totalcheeseproduction.

7.6%of11,201,000,000is

0.076(11,201,000,000)=851,276,000 Therefore,851,276,000poundsofcreamcheese wasproducedintheU.S.in2014.

74. answersmayvary

76. nosolution

78. Sinceabsolutevalueisnevernegative,the solutionsetis ∅

80. Allnumberswhosedistancefrom0is2unitsis writtenas|x|=2.

82. answersmayvary

84. |x 7|=2

86. answersmayvary

88. |2x 1|=4

Chapter 2: Equations, Inequalities, and Problem Solving

90. |ax + b|= c

a. onesolutionif c =0

b. nosolutionif c isanegativenumber

c. twosolutionsif c isapositivenumber

Section 2.7 Practice Exercises

1. |x|<5

Thesolutionsetofthisinequalitycontainsall numberswhosedistancefrom0islessthan5. Thesolutionsetis(5,5). ( ( –5 5

2. |b +1|<3 313 311131 42 b b b −<+< −−<+−<− −<< (4,2) ( ( –4 2

3. 3259 3295 324 x x x −+≤ −≤− −≤ 4324 4232242 236 2 2 3 x x x x −≤−≤ −+≤−+≤+ −≤≤ −≤≤ 2 ,2 3 ⎡⎤ ⎢⎥ ⎣⎦ 2 2 3

4. 5 34 8 x +<−

Theabsolutevalueofanumberisalways nonnegativeandcanneverbelessthan4.The solutionsetis{}or ∅ 5. ≤ = ⎡⎤ = ⎢⎥ ⎣⎦ −= −= = = 3(2)0 5 3(2)0 5 3(2) 55(0) 5 3(2)0 360 36 2 x x x x x x x Thesolutionsetis{2}.

6. |y +4| ≥ 6 46or46 4464or4464 10or2 yy yy yy +≤−+≥ +−≤−−+−≥− ≤−≥ (−∞,10] ∪ [2, ∞) 2 –10

7. 4353 435535 432 x x x ++> ++−>− +>−

Theabsolutevalueofanynumberisalways nonnegativeandthusisalwaysgreaterthan2. (−∞ , ∞) 8. 352 2 35525 2 33 2 x x x −−>− −−+>−+ −> 33or33 22 232(3)or232(3) 22 66or66 0or12 xx xx xx xx −<−−> ⎛⎞⎛⎞ −<−−> ⎜⎟⎜⎟ ⎝⎠⎝⎠ −<−−> <> (−∞,0) ∪ (12, ∞) 12 ( ( 0

Vocabulary, Readiness & Video Check 2.7

4. B

5. A

6. Theleftsideoftheinequalityisanabsolute value,whichmustbenonnegativeitmustbe0 orpositive.Therefore,thereisnovalueof x that canmakethevalueofthisabsolutevaluebeless thanthenegativevalueontherightsideofthe inequality.

7. Thesolutionsetinvolves“or”and“or”means “union.”

Exercise Set 2.7

2. |x|<6 6< x <6

Thesolutionsetis(6,6).

( ( –6 6

4. |y 7| ≤ 5 575 212 y y −≤−≤ ≤≤

Thesolutionsetis[2,12].

212

6. |x +4|<6

646 102 x x −<+< −<<

Thesolutionsetis(10,2). ( ( –10 2

8. |5x 3| ≤ 18 185318 15521 21 3 5 x x x −≤−≤ −≤≤ −≤≤

Thesolutionsetis 21 3,. 5 ⎡⎤ ⎢⎥ ⎣⎦ 21 5 –3

10. 67 1 x x +≤ ≤ 1 ≤ x ≤ 1

Thesolutionsetis[1,1]. –1 1

12. |8x 3|<2

Theabsolutevalueofanexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅

14. 273 24 z z +−<− +< 424 422242 62 z z z −<+< −−<+−<− −<<

Thesolutionsetis(6,2). ( ( –62

16. |y| ≥ 4 y ≤ 4or y ≥ 4

Thesolutionsetis(−∞,4] ∪ [4, ∞). 4 –4

18. |x 9| ≥ 2 92or92 7or11 xx xx −≤−−≥ ≤≥

Thesolutionsetis(−∞,7] ∪ [11, ∞). 11 7

20. 13 4 x x −> > x <4or x >4

Thesolutionsetis(−∞,4) ∪ (4, ∞). ( ( –44

22. |4x 11|>1

Anabsolutevalueisalwaysgreaterthana negativenumber.Thus,theansweris(−∞ , ∞).

24. 10312 1031 x x ++> +>

1031or1031 311or39 11 or3 3 xx xx xx +<−+> <−>− <−>−

Thesolutionsetis 11 ,(3,). 3 ⎛⎞ −∞−∪−∞⎜⎟ ⎝⎠ ( ( –3 11 3

Chapter 2: Equations, Inequalities, and Problem Solving

26. |x| ≥ 0

Anabsolutevalueisalwaysgreaterthanorequal to0.Thus,theansweris(−∞ , ∞). 0

28. |5x 6|<0

Theabsolutevalueofanexpressionisnever negative,sonosolutionexists.Thesolutionset is ∅ . 0

30. |z|<8 8< z <8 (8,8) ( ( –8 8

32. |x| ≥ 10 x ≤ 10or x ≥ 10 (−∞,10] ∪ [10, ∞) 10 –10

34. |3+ x| ≤ 10 10310 713 x x −≤−+≤ −≤≤ [7,13] –7 13

36. |1+0.3x| ≥ 0.1 10.30.1or10.30.1 0.31.1or0.30.9 0.31.10.30.9 or 0.30.30.30.3 11 or3 3 xx xx xx xx +≤−+≥ ≤−≥− ≤−≥− ≤−≥− 11 ,[3,) 3 ⎛⎤ −∞−∪−∞ ⎜ ⎥ ⎝⎦ –3 11 3

38. 81 7 x x +< <−

Anabsolutevalueisnevernegative,sono solutionexists.Thesolutionsetis ∅ . 0

40. |x| ≤ 7

Anabsolutevalueisnevernegative,sono solutionexists.Thesolutionsetis ∅ . 0

42. |5x +2|<8 8528 1056 6 2 5 x x x −<+< −<< −<<

Thesolutionsetis 6 2,. 5 ⎛⎞ ⎜⎟ ⎝⎠ ( ( –2 6 5

44. 162 16626 18 x x x −+−> −+−+>+ −+> 18or18 7or9 xx xx −+<−−+> <−> (−∞,7) ∪ (9, ∞) ( ( –7 9

46. |x|<0

Anabsolutevalueisnevernegative,sono solutionexists.Thesolutionsetis ∅ 0

48. 54 1 x x +≥ ≥−

Anabsolutevalueisalwaysgreaterthanorequal to0.Thus,theansweris(−∞ , ∞). 0

50. 3524 527 x x −+−≤ −≤ 7527 559 9 1 5 x x x −≤−≤ −≤≤ −≤≤

Thesolutionsetis 9 1,. 5 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ –1 9 5

ISM: Intermediate Algebra

52. 3 12 4 x −≥ 33 12or12 44 33 1or3 44 4 or4 3 xx xx xx −≤−−≥ ≤−≥ ≤−≥ 4 ,[4,) 3 ⎛⎤ −∞−∪∞ ⎜ ⎥ ⎝⎦ 4 4 3

54. |4+9x| ≥ 6

Anabsolutevalueisalwaysgreaterthanorequal to0.Thus,theansweris(−∞ , ∞). 0

56. 56 0 2 56 0 2 560 56 6 5 x x x x x + ≤ + = += =− =− 6 5 ⎧⎫ ⎨⎬ ⎩⎭ 6 5

58. 73110 7311 x x −−≤ −≤ 117311 8714 8 2 7 x x x −≤−≤ −≤≤ −≤≤ 8 ,2 7 ⎡⎤ ⎢⎥ ⎣⎦ 2 8 7

Chapter 2: Equations, Inequalities, and Problem Solving

60. + ≥ 7 4 2 x 77 4or4 22 78or78 15or1 xx xx xx ++ ≤−≥ +≤−+≥ ≤−≥

Thesolutionsetis(−∞,15] ∪ (1, ∞]. –15 1

62. 9344 934949 345 x x x −++<− −+++<−+ +< 5345 842 2 2 4 1 2 2 x x x x −<+< −<< −<< −<< 1 2, 2 ⎛⎞ ⎜⎟ ⎝⎠ ( ( –2 1 2

64. 3 461 5 3 45 5 x x +−<− +< 3 545 5 2532025 282022 282022 202020 711 510 x x x x x −<+< −<+< −<< −<< −<< 711 , 510 ⎛⎞ ⎜⎟ ⎝⎠ ( ( 7 5 11 10

66. |2x 3|>7 237or237 24or210 2or5 xx xx xx −<−−> <−> <−> (−∞,2) ∪ (5, ∞)

Chapter 2: Equations, Inequalities, and Problem Solving

68. |56x|=29 5629or5629 634or624 17 or4 3 xx xx xx −=−−= −=−−= ==−

70. |x +4| ≥ 20 420or420 24or16 xx xx +≤−+≥ ≤−≥

Thesolutionsetis(−∞,24] ∪ [16, ∞).

72. |9+4x| ≥ 0

Anabsolutevalueisalwaysgreaterthanorequal to0.Thus,theansweris(−∞ , ∞).

74. 85311 533 x x +−≥ −≥ 533or533 50or56 6 0or 5 xx xx xx −≤−−≥ ≤≥ ≤≥

Thesolutionsetis 6 (,0],. 5 ⎡⎞ −∞∪∞ ⎟ ⎢ ⎣⎠

76. 5324 532 x x −+= −= 532or532 51or55 1 or1 5 xx xx xx −=−−= == ==

78. |4x 4|=3

Anabsolutevalueisnevernegative,sono solutionexists.Thesolutionsetis ∅

80. 6 5 4 x = 66 5or5 44 620or620 26or14 xx xx xx =−= −=−−= =−=

82. 47 2 5 x < 47 22 5 104710 3417 317 44 x x x x −<< −<−< −<< −<<

Thesolutionsetis 317 ,. 44 ⎛⎞ ⎜⎟ ⎝⎠

84. 1 (rollinga5)6 P =

86. (rollinga0)0 P =

88. P(rollinga1,2,3,4,5,or6)=1

90. 3412 34(1)12 3412 38 8 3 xy x x x x −= −−= += = =

92. 3412 3(4)412 12412 40 0 xy y y y y −= −= −= −= =

94. |x|>4

96. |x|>1

98. answersmayvary

100. 51 0.20.20.19921875 256 0.00078125 0.00078125 −=− = = Theabsoluteerroris0.00078125.

Chapter 2 Vocabulary Check

1. Thestatement“x <5 or x >7”iscalleda compoundinequality

2. Anequationinonevariablethathasnosolution iscalledacontradiction.

Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving

3. Theintersectionoftwosetsisthesetofall elementscommontobothsets.

4. Theunionoftwosetsisthesetofallelements thatbelongtoeitherofthesets.

5. Anequationinonevariablethathasevery number(forwhichtheequationisdefined)asa solutioniscalledanidentity.

6. Theequation d = rt isalsocalledaformula.

7. Anumber’sdistancefrom0iscalleditsabsolute value

8. Whenavariableinanequationisreplacedbya numberandtheresultingequationistrue,then thatnumberiscalledasolutionoftheequation.

9. Theintegers17,18,19areexamplesof consecutiveintegers.

10. Thestatement5x 0.2<7isanexampleofa linearinequalityinonevariable.

11. Thestatement5x 0.2=7isanexampleofa linearequationinonevariable

Chapter 2 Review

1. 4(5)214 420214 26 3 xx xx x x −=− −=− = =

2. 72(8) 7216 323 23 3 xx xx x x +=−+ +=−− =− =−

3. 3(21)8(6) 63488 1445 45 14 y y y y y y −=−+ −=−− =− =−

4. (12)5(21) 12105 117 7 11 zz zz z z −+=− −−=− −= =−

5. (84)2(34) 8468 36 90 0 nnn nnn nn n n −+=− −−=− −= −= =

6. 4(92)6(16)10 36863610 368364 84 vv vv vv +=+− +=+− +=− =−

Nosolution,or ∅

7. 0.3(2)1.2 10[0.3(2)10(1.2) 3(2)12 3612 318 6 x x x x x x −= −= −= −= = =

8. 1.50.2(0.3) 1.50.20.06 100(1.5)100(0.20.06) 150206 15620 7.8 c c c c c c =− =− =− =− = =

9. 4(23)2(34)6 812686 812128 88 xxx xxx xx −−=−+ −+=−+ −+=− −=−

Allrealnumbers

10. 6(1)3(2)0 66630 30 0 mm mm m m −+−= −+−= = =

11. 63(24)45(12) 66124510 610510 65 ggg ggg gg −+−=− −−−=− −−=− −=

Nosolution, ∅

12. −++=−− −−+=−+ −=−+ = 205(1)3(215) 20553215 152215 1515 ppp ppp pp

Allrealnumbers

Chapter 2: Equations, Inequalities, and Problem Solving

13. 42 3 343(2) 3 1236 26 3 x x x x xx x x −=− ⎛⎞ −=−⎜⎟ ⎝⎠ −=− −= =−

14. 92 43 92 1212 43 278 190 0 yy y y yy y y = ⎛⎞⎛⎞ = ⎜⎟⎜⎟ ⎝⎠⎝⎠ = = =

15. 3 13 86 3 241243 86 924724 596 96 5 nn nn nn n n −=+ ⎛⎞⎛⎞ −=+ ⎜⎟⎜⎟ ⎝⎠⎝⎠ −=+ = =

16. 12 62 6162 62 6312 26 3 zz zz zz z z +=+ ⎛⎞⎛⎞ +=+ ⎜⎟⎜⎟ ⎝⎠⎝⎠ +=+ −= =−

17. 8 42 44(8) 42 232 32 32 yy yy yy y y −=− ⎛⎞ −=−⎜⎟ ⎝⎠ −=− −=− =

18. 28 33 283 8 x x xx x −= −= −=

19. 22 35 5(2)3(2) 51036 216 8 bb bb bb b b −+ = −=+ −=+ = = 20. 2132 315 2132 1515 315 5(21)32 10532 77 1 tt tt tt tt t t −+ = −+ ⎛⎞⎛⎞ = ⎜⎟⎜⎟ ⎝⎠⎝⎠ −=+ −=+ = = 21. 2(1)2(1) 33 2(1)2(1)33 33 2(1)2(1) 2222 22 tt tt tt tt +− = +− ⎡ ⎤⎡⎤ = ⎢ ⎥⎢⎥ ⎣ ⎦⎣⎦ +=− +=− =− Nosolution, ∅ 22. 3341 2 615 3341 30302 615

5(33)2(41)30(2) 15158260 1515862 777 11 aa aa aa aa aa a a −+ =+ −+ ⎛⎞⎛⎞ =+ ⎜⎟⎜⎟ ⎝⎠⎝⎠ −=++ −=++ −=+ = =

23. Let x =thenumber. 2(3)31 2631 7 xx xx x −=+ −=+ −= Thenumberis7.

24. Let x =smallernumber,then x +5=largernumber. 5285 2280 140 xx x x ++= = = x +5=145 Thenumbersare140and145.

25. 40%130=0.40130=52

26. 1.5%8=0.0158=0.12

27. Let x =widthoftheplayingfield,then 2x 5=lengthoftheplayingfield. +−= +−= = = 22(25)230 2410230 6240 40 xx xx x x

Then2x 5=2(40)5=75.Thefieldis 75meterslongand40meterswide.

28. Let x bethemedianweeklyearningsforayoung adultwithanassociate’sdegreein2013. 0431108 1431108 775 xx x x += = ≈

Themedianweeklyearningsforayoungadult withanassociate’sdegreein2013was$775.

29. Let n =thefirstinteger,then n +1=thesecondinteger, n +2=thethirdinteger,and n +3=thefourthinteger. (1)(2)(3)216 616 10 nnnn n n +++++−= += =

Therefore,theintegersare10,11,12,and13.

30. Let x =smalleroddinteger,then x +2=largeroddinteger. 53(2)54 53654 260 30 xx xx x x =++ =++ = =

Sincethisisnotodd,nosuchconsecutiveodd integersexist.

31. Let m =numberofmilesofdriven. +−= +−= += = = 2(19.95)0.12(200)46.86 39.900.122446.86 0.1215.9046.86 0.1230.96 258 m m m m m

Hedrove258miles.

32. Solve R = C. 16.504.503000 123000 250 xx x x =+ = = Thus,250calculatorsmustbeproducedandsold inordertobreakeven.

33. = = Vlwh V w lh

34. 2 2 Cr C r =π = π

35. 5412 5124 512 4 xy xy x y −=− += + =

36. 5412 5412 412 5 xy xy y x −=− =− =

37. 11 1 1 () y ymxx yy m xx −=− =

38. 11 11 11 11 () y ymxx y ymxmx yymxmx yymx x m −=− −=− −+= −+ =

39. () E IRr E IRIr IIRIr EIR r I =+ =+ −= =

40. 2 2 2 Svtgt Svtgt Svt g t =+ −= =

41. () Tgrgvt Tgrvt T g rvt =+ =+ = +

42. (1) 1 IPrtP IPrt I P rt =+ =+ = +

Chapter 2: Equations, Inequalities, and Problem Solving

43. 0.037 130001 ntn r AP nn ⎛⎞⎛⎞ =+=+⎜⎟⎜⎟ ⎝⎠⎝⎠

a. 0.0314

30001$3695.27 2 A ⎛⎞ =+≈ ⎜⎟ ⎝⎠

b. 0.03364

30001$3700.81 52 A ⎛⎞ =+≈ ⎜⎟ ⎝⎠

44. 5 9(32) 5 9(9032) 5 9(58) 290 32.2 9 CF C C C =− =− = =≈

90°Fis 290 C32.2C. 9 ⎛⎞ °≈° ⎜⎟ ⎝⎠

45. Let x =originalwidth,then x +2=originallength. 2 22 (4)(24)(2)88 (4)(6)288 1024288 864 8 xxxx xxxx xxxx x x +++=++ ++=++ ++=++ = = x +2=10

Theoriginalwidthis8in.andtheoriginallength is10in.

46. Area18213782 ft =×= 378 Packages15.75 24 == Thereare16packagesneeded.

47. 3(5)(3) 3153 412 3 xx xx x x −>−+ −>−− > > (3, ∞)

48. 2(7)3(2) 21436 520 4 xx xx x x −+≥+ −−≥+ −≥ ≤− (−∞,4]

49. 4(52)31 45231 2531 4 4 xxx xxx xx x x −+<− −−<− −<− −< >− (4, ∞)

50. 3(8)72(5) 3247102 324510 234 17 xxx xxx xx x x −<+− −<+− −<+ −< >− (17, ∞)

51. ≥−−+ ≥−++ ≥+ ≥ ≥ 2462(35)2 2466102 24102 142 7 xxx xxx x x x (−∞,7]

52. 12 323 12 66 323 234 21 1 2 x x x x x +> ⎛⎞⎛⎞ +> ⎜⎟⎜⎟ ⎝⎠⎝⎠ +> > > 1 , 2 ⎛⎞ ∞ ⎜⎟ ⎝⎠

53. +<−+ ⎛⎞⎛⎞ +<−+ ⎜⎟⎜⎟ ⎝⎠⎝⎠ +<−+ < < 39 424 39 44 424 4329 66 1 x x x x xx x x (−∞,1)

54. 53 28(26) 53 88(26) 28 4(5)3(26) 420618 238 19 x x x x xx xx x x ≤+ ⎛⎞⎡⎤ ≤+ ⎜⎟ ⎢ ⎥ ⎝⎠⎣⎦ −≤+ −≤+ −≤ ≥− [19, ∞)

ISM: Intermediate Algebra

55. Let n =numberofpoundsoflaundry.

150.5(10)0.4(10) 1550.44 1510.4 140.4 35 n n n n n <+− <+− <+ < <

Itismoreeconomicaltousethehousekeeperfor morethan35poundsoflaundryperweek.

56. Let x =thescorefromthelastjudge. 9.59.79.99.79.79.69.59.65 8 67.677.2 9.6 x x x +++++++ ≥ +≥ ≥

ThelastjudgemustgiveNanaatleasta9.6for hertowinthesilvermedal.

57. 1473 8410 5 2 2 x x x ≤−≤ ≤≤ ≤≤ 5 2, 2 ⎡⎤ ⎢⎥ ⎣⎦

58. 2851 1059 9 2 5 x x x −≤+<− −≤≤− −≤≤− 9 2, 5 ⎡⎞ ⎟ ⎢ ⎣⎠

59. 34(21)12 38412 1816 1 2 8 x x x x −<−< −<−< << << 1 ,2 8 ⎛⎞ ⎜⎟ ⎝⎠

60. 6(34)3 6343 6533 350 3 0 5 xx xx x x x −<−−<− −<−+<− −<−<− −<< −<< 3 ,0 5 ⎛⎞ ⎜⎟ ⎝⎠

Chapter 2: Equations, Inequalities, and Problem Solving

61. 1434 635 1434 303030 635 510(43)24 5403024 354054 727 820 x x x x x x <≤ ⎛⎞⎛⎞⎛⎞ <≤ ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠ <−≤ <−≤ << <≤ 727 , 820 ⎛⎤ ⎜ ⎥ ⎝⎦

62. x ≤ 2 and x >5 5< x ≤ 2 (5,2]

63. 3565 3115 11 5 3 −>−<− >> >> xorx xorx xorx 11 3 x > 11 , 3 ⎛⎞ ∞ ⎜⎟ ⎝⎠

64. 5001000 9 500321000 5 9 468968 5 260538 F C C C ≤≤ ≤+≤ ≤≤ ≤≤ Roundedtothenearestdegree,firing temperaturesrangefrom260 °Cto538°C.

65. Let x =theamountsavedeachsummer. 400025008000 350027500 17503750 x x x ≤+≤ ≤≤ ≤≤ Shemustsavebetween$1750and$3750each summer.

66. |x 7|=9 79or79 16or2 xx xx −=−=− ==−

67. |8 x|=3 83or83 5or11 5or11 xx xx xx −=−=− −=−−=− ==

Chapter 2: Equations, Inequalities, and Problem Solving

68. |2x +9|=9 299or299 20or218 0or9 xx xx xx +=+=− ==− ==−

69. |3x +4|=7 347or347 33or311 11 1or 3 xx xx xx −+=−+=− −=−=− =−=

70. 32610 324 x x −+= −= 324or324 36or32 2 2or 3 xx xx xx −=−=− ==− ==−

71. 5615 610 610 61 1 6 x x x x x ++= += += =− =−

72. 5=|4x 3|

Thesolutionsetis ∅ .

73. 5683 565 x x −+= −=−

Thesolutionsetis ∅ .

74. 8310 23 x x −=−− =− 32or32 5or1 xx xx −=−=− ==

75. 37 2 4 x = 3737 2or2 44 378or378 315or31 1 5or 3 xx xx xx xx ==− −=−=− ==− ==−

Intermediate Algebra

76. |6x +1|=|15+4x| 61154or61(154) 214or61154 7or1016 8 5 xxxx xxx xx x +=++=−+ =+=−− ==− =−

77. |5x 1|<9 9519 8510 8 2 5 x x x −<−< −<< −<< 8 ,2 5 ⎛⎞ ⎜⎟

( ( 8 5 2

78. |6+4x| ≥ 10 6410or6410 416or44 4or1 xx xx xx +≤−+≥ ≤−≥ ≤−≥ (−∞,4] ∪ [1, ∞) –4 1

79. 381 39 x x −> > 39or39 3or3 xx xx <−> <−> (−∞,3) ∪ (3, ∞) ( ( –3 3

80. 9524 515 x x +< < 15515 33 x x −<< −<< (3,3) ( ( –33

81. |6x 5| ≤ 1

Thesolutionsetis ∅ . 0

ISM: Intermediate Algebra

82. 2 34 5 x +≥ 22 34or34 55 22 535(4)or535(4) 55 15220or15220 1522or1518 226 or 155 xx xx xx xx xx +≤−+≥

22 15 6 5

83. 685 3 63 3 x x +−>− +> 63or63 33 9or3 33 27or9 xx xx xx +<−+> <−>− <−>− (−∞,27) ∪ (9, ∞) ( ( –27 –9

84. 4(1)102 7 4(1)8 7 x x +< <− Thesolutionsetis ∅ 0

85. 224 523 224 3030 523 6(2)15(2)10(4) 61215301040 21181040 1122 2 xxx xxx xxx xxx xx x x −++ += −++ ⎛⎞⎛⎞ += ⎜⎟⎜⎟

Chapter 2: Equations, Inequalities, and Problem Solving

86. 2341 423 2341 1212 423 3(23)6(4)4(1) 6924644 123344 837 37 8 zzz zzz zzz zzz zz z z −−+ −= −−+ ⎛⎞⎛⎞ −= ⎜⎟⎜⎟

89. Let x =numberoftouristsforFrance,then x +9=numberoftouristsforUnitedStates,and x +44=numberoftouristsforChina. (9)(44)332 353332 3279 93 xxx x x x ++++= += = = x +9=102 x +44=137 Chinaispredictedtohave137milliontourists, whereastheUnitedStatesispredictedtohave 102millionandFrance,93million.

90. d = rt or d r t = 11:00a.m.to1:15p.m.is2.25hours. 130 58 2.25 r =≈ Hisaveragespeedwas58mph.

91. 3 box853120in,Vlwh==⋅⋅= while 223 cyl3654170inVrh=π=π⋅⋅=π≈ Therefore,thecylinderholdsmoreicecream.

Chapter 2: Equations, Inequalities, and Problem Solving

92. +≥+− +≥+− +≥+ ≥ ≥ 485(24)2 4810202 48820 287 4 xxx xxx xx x x (−∞,4]

93. 3(2)5(2) 53

3(2)5(2)1515 53

9(2)25(2) 9182550 3468 2 xx xx xx xx x x > ⎡⎤⎡⎤ > ⎢⎥⎢⎥ ⎣⎦⎣⎦ −>−− −>−+ > > (2, ∞)

94. 2(34) 03 5 2(34) 5(0)55(3) 5 02(34)15 06815 867 47 36 x x x x x x + ≤≤ + ⎡⎤ ≤≤ ⎢⎥ ⎣⎦ ≤+≤ ≤+≤ −≤≤ −≤≤ 47 , 36 ⎡⎤ ⎢⎥ ⎣⎦

95. x ≤ 2 or x >5 (−∞ , ∞)

96. 26237 3210 35 −≤−+<− ≥−−<− ≥−> xandx xandx xandx x >5 (5, ∞)

97. 7265 721 x x −=− = 721or721 3or3 xx xx ==− ==−

98. 92 3 5 x =− Thesolutionsetis ∅ .

99. |x 3|=|7+2x| 372or3(72) 10or372 34 4 3 xxxx xxx x x −=+−=−+ −=−=−− =− =−

100. |6x 5| ≥ 1 Since|6x 5|isnonnegativeforallnumbers x, thesolutionsetis(−∞ , ∞).

101. 43 1 5 x < 43 11 5 5435 248 1 2 2 x x x x −<< −<−< −<< −<< 1 ,2 2

Chapter 2 Getting Ready for the Test

1. 969 70 0 xx x x −=−− = =

Thesolutionis0;C.

2. 4824 4828 20 ()xx xx x +=+ +=+ = Thesolutionis0;C.

3. 524102 10201020 ()() xx xx −=− −=− Bothsidesoftheequationareidentical,soall realnumbersaresolutions;A.

4. 32411 36441 4643 63False ()() xxx xxx xx ++=−+ ++=−+ +=− =−

6=3isfalseforallvaluesof x,sotheequation hasnosolution;B.

5. {x|x ≤ 11}is(−∞,11];A.

6. {x|5< x}is(5, ∞);B.

Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving

7. A. 737

Dgivesthecorrectsolutions.

8. |5x 2| ≤ 4isequivalentto4 ≤ 5x 2 ≤ 4;C.

9. |5x 2|=4isequivalentto5x 2=4or 5x 2=4;A.

10. |5x 2| ≥ 4isequivalentto5x 2 ≥ 4or 5x 2 ≤ 4;E.

11. |5x|2=4or|5x|=6isequivalentto5x =6or 5x =6;B.

12. Anabsolutevaluewillneverbenegative,so |x +3|=9hasnosolution,or ∅ ;A.

13. Anabsolutevaluewillneverbenegative,so |x +3|<9hasnosolution,or ∅ ;A.

14. Anabsolutevaluewillalwaysbegreaterthan equalto0,so|x +3|>9hasallrealnumbersas solutions,or(−∞ , ∞);B.

Chapter 2 Test

1. 814544 330 10 xx x x +=+ = =

2. 9(2)5[112(2)3] 9185[11423] 9185[102] 9185010 32 32 xx xx xx xx x x +=−−+ +=−++ +=+ +=+ −= =−

3. 3(4)2(62) 312124 412124 1212 y yy y yy y y −+=+ −+=+ −=+ −= Nosolution, ∅

4. 762(43) 8686 66 nnn nn −+=− −=− −=− Allrealnumbers

5. 73 51 410 73 205201 410 35100620 2980 80 29 ww ww ww w w +=+ ⎛⎞⎛⎞ +=+ ⎜⎟⎜⎟

+=+ =− =−

6. ++ += ++ ⎛⎞⎛⎞ += ⎜⎟⎜⎟ ⎝⎠⎝⎠ ++=+ ++=+ +=+ −=− −=− = 721 1 96 721 18118 96 2(7)183(21) 2141863 23263 26332 429 29 4 zz zz zz zz zz zz z z

7. 6532 651 x x −−=− −= 651or651 66or64 2 1or 3 xx xx xx −=−=− == ==

8. |82t|=6 Nosolution, ∅

9. |2x 3|=|4x +5| 2345or23(45) 2453or2345 28or2453 4or62 1 4or 3 xxxx xxxx xxx xx xx −=+−=−+ −=+−=−− −=+=−+ =−=− =−=−

Chapter 2: Equations, Inequalities, and Problem Solving

10. |x 5|=|x +2| 52or5(2) 52Falseor52 23 3 2 xxxx xx x x −=+−=−+ −=−=−− = = Since5=2isnotpossible,theonlysolutionis 3 . 2

11. 348 384 38 4 xy xy x y −= −= =

12. 2 2 2 () Sgtgvt Sgtvt S g tvt =+ =+ = +

13. 9 32 5 9 32 5 5 (32) 9 FC FC CF =+ −= =−

14. 3(27)4(6) 62146 2216 315 5 xxx xxx xx x x −−>−+ −−>−− −>−− > > (5, ∞)

15. 3251 0 34 3251 1212(0) 34 4(32)3(51)0 1281530 3110 311 11 3 xx xx xx xx x x x −+ −≥ −+ ⎡⎤−≥⎢⎥ ⎣⎦ −−+≥ −−−≥ −−≥ −≥ ≤− 11 , 3 ⎛⎤ −∞− ⎜ ⎥ ⎝⎦

16. 32(3)4 3264 3210 3 5 2 x x x x −<−≤ −<−≤ <≤ <≤ 3 ,5 2 ⎛⎤ ⎜ ⎥ ⎝⎦

17. |3x +1|>5 +<−+> <−> <−> 315or315 36or34 4 2or 3 xx xx xx 4 (,2), 3 ⎛⎞ −∞−∪∞⎜⎟ ⎝⎠

18. 542 52 x x −−<− −< 252 37 x x −<−< << (3,7)

19. x ≥ 5 and x ≥ 4 [5, ∞)

20. x ≥ 5 or x ≥ 4 [4, ∞)

21. 25 12 3 25 3(1)33(2) 3 3256 3525565 2211 2211 222 11 1 2 x x x x x x x −≤< ⎛⎞ −≤< ⎜⎟ ⎝⎠ −≤−< −+≤−+<+ ≤< ≤< ≤< 11 1, 2 ⎡ ⎞ ⎟ ⎢ ⎣ ⎠

22. 615414 35 +>+−>− >> xxorx xorx (−∞ , ∞)

23. 12%80=0.1280=9.6

24. Let x bethenumberofnewvehiclessoldby Fordin2010.Thenumberofnewvehiclessold isincreasedby29.1%,orby0.291x 02912480942 12912480942 1922000 .,, .,, ,, xx x x += = ≈

Fordsoldapproximately1,922,000newvehicles in2010.

25. Recallthat C =2πr.Here C =78.5. 78.52 78.539.25 2 r r =π == ππ

Alsorecallthat2. A r=π 222 39.2539.2539.25 491 3.14 A ⎛⎞ =π=≈≈ ⎜⎟ ππ⎝⎠

Theareaofthepenisabout491squarefeet. Eachdogrequiresatleast60squarefeetof space,and 491 8.18. 60 ≈ Atmost8dogscouldbe keptinthepen.

26. Let x bethenumberofpeopleemployedas registerednursesin2012.Thenumberofpeople employedinthisfieldin2022is x increasedby 19%.

0193240000 1193240000 2723000 .,, .,, ,, xx x x += = ≈

In2012,therewere2,723,000registerednurses employed.

27. Use1 nt r AP n ⎛⎞=+⎜⎟ ⎝⎠ where P =2500, r =3.5%=0.035, t =10,and n =4. 410 40 0.035 25001 4 2500(1.00875) $3542.27 A A A ⎛⎞=+⎜⎟ ⎝⎠ = ≈

28. Let x betheamountofmoneyinternational travelersspendinNewYork.Then x +4isthe amountofmoneyinternationaltravelersspendin Californiaand2x 1istheamountofmoney internationaltravelersspendinFlorida. (4)(21)39 4339 436 9 xxx x x x +++−= += = = x +4=9+4=13

2x 1=2(9)1=181=17 Internationaltravelersspend$9billioninNew York,$13billioninCalifornia,and$17billion inFlorida.

Chapter 2 Cumulative Review

1. a. {101,102,103,...}

b. {2,3,4,5}

2. a. {2,1,0,1,2,3,4}

b. {4}

3. a. |3|=3

b. 11 77 −=

c. |2.7|=2.7

d. |8|=8

e. |0|=0

4. a. Theoppositeof 2 3 is 2 3

b. Theoppositeof9is9.

c. Theoppositeof1.5is1.5.

5. a. 3+(11)=14

b. 3+(7)=4

c. 10+15=5

d. 8.3+(1.9)=10.2

e. 11121 42444 −+=−+= f. 231495 37212121 −+=−+=−

6. a. 2(10)=2+10=8

b. 1.78.9=7.2

c. 11213 24444 −−=−−=−

Chapter 2: Equations, Inequalities, and Problem Solving

7. a. 93 = since239. =

b. 255 = since2525. =

c. 11 42 = since 2 11 24 ⎛⎞ = ⎜⎟ ⎝⎠

d. 366−=− since = 2 636.

e. 36isnotarealnumber.

8. a. 3(2)=6

b. 343 477 ⎛⎞ −−= ⎜⎟ ⎝⎠

c. 0 0 2 =

d. 20 10 2 =

9. Let x =4, y =3.

a. 3x 7y =3(4)7(3)=12+21=33

b. 22 22(3)2(9)18 y −=−−=−=−

c. 43 34 23 34 89 1212 1 12 xy yx −=− =−+ =−+ =

10. a. 411 = since411. =

b. 382 = since328. =

c. 4813 = since4381. =

11. a. x +5=20

b. 2(3+ y)=4

c. x 8=2x

d. 9 9 z z =+

12. a. 3>5since3istotherightof5ona numberline.

b. 12 3 4 =

c. 0>2since0istotherightof2ona numberline.

13. 7x +5=5+7x 14. 5(7x)=(57)x =35x 15. 259 24 2 x x x += = =

16. 11.21.25 105 2 x x x =− =− −=

17. 6426(1) 64266 6464 44,whichisalwaystrue. xx xx xx −=+− −=+− −=− −=− Allrealnumbers

18. 21.50.21.6 0.41.7 4.25 xx x x +=−+ =− =−

19. a. Let x =thefirstinteger.Then x +1=thesecondintegerand x +2=thethirdinteger. x +(x +1)+(x +2)=3x +3

b. x +(5x)+(6x 3)=12x 3

20. a. Let x =thefirstinteger.Then x +2=thesecondevenintegerand x +4=thethirdeveninteger.

x +(x +2)+(x +4)=3x +6

b. 4(3x +1)=12x +4

21. Let x =firstnumber,then 2x +3=secondnumberand 4x =thirdnumber. (23)4164 73164 7161 23 xxx x x x +++= += = = 2x +3=2(23)+3=49

ISM: Intermediate Algebra

4x =4(23)=92 Thethreenumbersare23,49and92.

22. Let x =firstnumber,then 3x +2=secondnumber. (32)24 2224 222 11 xx x x x +−= += = = 3x +2=3(11)+2=35 Thetwonumbersare11and35.

23. 327 327 2727 ,or 333 yx yx xx yy −= =+ + ==+

24. 7410 7410 410410 ,or 777 xy xy yy xx −= =+ + ==+

25. 1 2() 2() 2 2 2 A Bbh A Bbh A Bhbh ABhbh ABh b h =+ =+ =+ −= =

26. 22 22 2 2 Plw Pwl Pw l =+ −= =

27. a. {x|x ≥ 2} 2 [2, ∞)

b. {x|x <1} –1 ) (−∞,1)

c. {x|0.5< x ≤ 3} 3 ( ] 0.5 (0.5,3]

28. a. {x|x ≤ 3} –3 (−∞,3]

Chapter 2: Equations, Inequalities, and Problem Solving

b. {x|2 ≤ x <0.1} 0.1 ) –2 [2,0.1)

29. (3)23(25) 32615 5715 208 5 2 xxx xxx xx x x −−+≤−+ −++≤−+ −+≤− ≤ ≤ 5 2, ⎡ ⎞ ∞ ⎟ ⎢ ⎣ ⎠

30. 2(71)5(7)4 142574 9274 26 3 xxx xxx xx x x −−>−−+ −−>+ −>+ > > (3, ∞)

31. 2(3)21 2621 61;Trueforallrealnumbers. xx xx x +>+ +>+ > (−∞ , ∞)

32. 4(1)341 44341 4141 11Nevertrue xx xx xx +−<+ +−<+ +<+ < ∅

33. A ={2,4,6,8}, B ={3,4,5,6};thenumbers4 and6areinbothsetssotheintersectionof A and B is{4,6}.

34. Theelementsineithersetorbothsetsare2,1, 0,1,2,3,4,and5,sotheunionis {2,1,0,1,2,3,4,5}.

35. 72219 928 4 −<+< << < xandx xandx x x <4 (−∞,4)

36. 31318 239 3 +≤−< ≤−< < xorx xorx x x <3 (−∞,3)

Chapter 2: Equations, Inequalities, and Problem Solving

37. A ={2,4,6,8}and B ={3,4,5,6},sotheunion of A and B is{2,3,4,5,6,8}.

38. ∅ ;therearenoelementsincommon.

39. 25360 220 1 −−<−< −<< >− xorx xorx x Allrealnumbers (−∞ , ∞)

40. 25360 220 1 −−<−< −<< >− xandx xandx x 1< x <0 (1,0)

41. |p|=2 p =2or p =2

42. |x|=5 x =5or x =5

43. 111 2 x −= 111or111 22 12or10 22 24or20 xx xx xx −=−=− ==− ==−

44. 210 3 y += 210or210 33 8or12 33 24or36 yy yy yy +=+=− ==− ==−

Intermediate Algebra

45. |x 3|=|5 x| 35or3(5) 28or35 4or35 xxxx xxx x −=−−=−− =−=−+ =−=−

Since3=5isnotpossible,theonlysolutionis 4.

46. |x +3|=|7 x| 37or3(7) 24or37 2or37 xxxx xxx x +=−+=−− =−=−+ =−=−

Since3=7isnotpossible,theonlysolutionis 2.

47. |x| ≤ 3 3 ≤ x ≤ 3 [3,3]

48. |x|>1 x <1or x >1 (−∞,1) ∪ (1, ∞)

49. 2953 292 x x ++> +>−

Since|2x +9|isnonnegativeforallnumbers x, thesolutionsetis(−∞ , ∞).

50. 3191 318 x x ++< +<−

Thesolutionsetis ∅