Chapter 1
Equations and Inequalities
Section 1.1
1. Distributive
2. Zero-Product
3. 4 xx
4. False.Multiplyingbothsidesofanequationby zerowillnotresultinanequivalentequation.
5. identity
6. linear;first-degree
7. False.Thesolutionis8 3
8. True 9. b
10. d 11. 721 721
Thesolutionsetis{3}. 12. 624 624 66 4 x x x
Thesolutionsetis{4}. 13. 3150 31515015 315 315 33 5 x x x x x
Thesolutionsetis{5}. 14. 6180 61818018 618 618 66 3 x x x x x
Thesolutionsetis{3}. 15. 230 23303 23 23 22 3 2 x x x x x
Thesolutionsetis3 2
16. 340 34404 34 34 33 4 3 x x x x x
Thesolutionsetis4 3
17. 17 420 17 44 420 287 205
x x x
Thesolutionsetis7 5
18. 29 32 29 66 32 427 427 44 27 4 x x x x x
Thesolutionsetis27. 4
19. 34 3444
20.
Thesolutionsetis{2}.
Thesolutionsetis{3}.
21. 263 26636 29 29 39 39 33 3 tt tt tt tttt t t t
Thesolutionsetis{3}.
Section 1.1: Linear Equations
22. 5618 566186 524 524 624 624 66 4 yy yy yy yyyy y y y
Thesolutionsetis{4}.
23. 629 66296 23 2232 33 33 33 1 xx xx xx xxxx x x x
Thesolutionsetis{1}.
24. 322 32323 21 21 1 1 11 1 xx xx xx xxxx x x x
Thesolutionsetis{1}.
25. 3247 323473 244 24444 24 24 22 2 nn nn nn nnnn n n n Thesolutionsetis{2}.
Chapter 1: Equations and Inequalities
26. 6231 626316 235 23353 55 55 55 1 mm mm mm mmmm m m m
Thesolutionsetis{1}.
30. 7(21)10 72110 8210 828108 22 22 22 1 x x x x x x x
Thesolutionsetis{1}.
27. 3(53)8(1) 15988 9815888 1515815
xx xx xxxx x x Thesolutionsetis{23}.
28. 3(2)21 xx 6321 636216 327 32272 57 57 55 7 5 xx xx xx xxxx x x x
Thesolutionsetis7. 5
29. 8(32)310 xxx 832310 52310 5223102 538 53383 28 28 22 4 xxx xx xx xx xxxx x x x
Thesolutionsetis{4}.
31. 311 2 222 xx 311 222 222 341 34414 33 33 43 43 44 3 4 xx xx xx xx xxxx x x x
Thesolutionsetis3 4
32. 12 2 33 12 332 33 62 2622 36 36 33 2 xx xx xx xxxx x x x
Thesolutionsetis{2}.
33. 13 5 24 13 454 24 2203 220232 20 20 xx xx xx xxxx x x
Thesolutionsetis{20}.
34.
1 16 2 1 2126 2 212 22122 10 10 11 10 x x x x x x x
Thesolutionsetis{10}.
35. 211 323 211 66 323 432 43323 2 pp pp pp
Thesolutionsetis{2}.
36. 114 233 p 114 66 233 328 32383 25 25 22 5 2 p p p p p p
Thesolutionsetis5. 2
37. 0.20.90.5 0.20.50.90.50.5 0.30.9 0.30.9 0.30.3 3
Thesolutionsetis{-3}.
38. 0.91 0.91 0.11 0.11 0.10.1 10 tt tttt t t t
Thesolutionsetis{10}.
39. 12 2 37 xx
12 21212 37 713242 773642 101342 1013134213 1029 1029 1010 29 10 xx xx xx x x x x x
Thesolutionsetis29 10
40. 21 163 3 x x 21 31633 3 21489 2499 249292 497 497 77 7 x x xx xx xxxx x x x
Thesolutionsetis{7}.
Chapter 1: Equations and Inequalities
41. 5111 (3)2(23) 8416 10(3)324(23)11 10303281211
42. 122 3515 (1)3(4) 5(1)456(4)2 55456242 540626
wwww w w w w
Thesolutionsetis
Since y =2doesnotcauseadenominatorto equalzero,thesolutionsetis{2}. 44. 45 5 2 yy 45 252 2 8105 810858 103
Since3 10 y doesnotcauseadenominatorto
equalzero,thesolutionsetis310
Since x =8doesnotcauseanydenominatorto equalzero,thesolutionsetis{8}.
Since6 x doesnotcauseadenominatorto equalzero,thesolutionsetis{6}.
47. (7)(1)(1)2 xxx 22 2222 6721 6721 6721 677217 628 62282 48 48 44 2 xxxx xxxxxx xx xx xx xxxx x x x
Thesolutionsetis{2}.
48. (2)(3)(3)2 xxx 22 2222 669 669 669 66696 615 66156 715 715 77 15 7 xxxx xxxxxx xx xx xx xxxx x x x
Thesolutionsetis15 7
49. 22 2222 (23)(21)(4) 23274 2322742 374 37747 44 44 44 1 xxxx xxxx xxxxxx xx xxxx x x x
Thesolutionsetis{1}.
Section 1.1: Linear Equations
50. (12)(21)(2) xxxx 22 2222 2252 222522 52 5525 62 621 663
xxxx xxxxxx xx xxxx x x x Thesolutionsetis1 3
51. 23 33 3333 312 312 312 312 312 4 33
ppp ppp ppppp p p p
Thesolutionsetis{4}.
52. 23 33 3333 (4)8 48 48 48 48 44 2 www www wwwww w w w
Thesolutionsetis{2}.
53. 2 3 22 x xx 2 322 22 322 362 462 46626 48 48 44 2 x xx xx xx xx x x x x x
Since x =2causesadenominatortoequalzero, wemustdiscardit.Thereforetheoriginal equationhasnosolution.
Since x =–3causesadenominatortoequalzero, wemustdiscardit.Thereforetheoriginal equationhasnosolution. 55.
Since x =2causesadenominatortoequalzero,wemustdiscardit.Thereforetheoriginalequationhasnosolution.
56.
Since x =3causesadenominatortoequalzero,wemustdiscardit.Thereforetheoriginalequationhasnosolution.
Since x =–6doesnotcauseanydenominatorto equalzero,thesolutionsetis{6}.
Since x =–2doesnotcauseanydenominatorto equalzero,thesolutionsetis{2}.
Since x =41doesnotcauseanydenominatorto equalzero,thesolutionis{41}. 60.
Since1 x doesnotcauseanydenominatortoequalzero,thesolutionsetis{1}.
Since3 x doesnotcauseanydenominatortoequalzero,thesolutionsetis{3}.
Since6 x doesnotcauseanydenominatortoequalzero,thesolutionsetis{14}.
Chapter 1: Equations and Inequalities
69.
Thesolutionsetisapproximately{0.41}.
Thesolutionsetisapproximately{0.94}.
73. ,0axbca axbbcb axbc axbc aa bc x a
74. 1,0 111 1 1 11 axba axb axb axb aa bb x aa
75. ,0,0, () () xxcabab ab xx ababc ab bxaxabc abxabc abxabc abab abc x ab
76. ,0 ab cc xx ab xxc xx abcx abcx cc ab x c
Chapter 1: Equations and Inequalities
77. 2166,if4 4216(4)6 421646 42162 412 412 44 3 xaaxax aaa aaa aa a a a
78. 242,for2 22242(2) 22244 2224 42 4 2 2
79. 12 111 RRR 1212 12 1221 1221 1221 2121 12 21 111 () () RRRRRR RRR RRRRRR RRRRR RRRRR RRRR RR R RR
80. (1)APrt
APPrt
APPrt APPrt PtPt AP r Pt
82. PVnRT PVnRT nRnR PV T nR
1 Sa r (1)(1) 1 a Srr r SSra SSrSaS SraS SraS SS Sa r S
0 0 0 00 vgtv vvgt vvgt gg
85. AmountinbondsAmountinCDsTotal 300020,000 xx
300020,000 2300020,000 223,000 11,500 xx x x x
$11,500willbeinvestedinbondsand$8500 willbeinvestedinCD's.
86. Sean'sAmountGeorge'sAmountTotal 300010,000 xx
Seanwillreceive$6500andGeorgewill receive$3500.
87. DollarsHoursMoney perhourworkedearned Regular4040 wage
Sandra’sregularhourlywageis$17.50.
88. DollarsHoursMoney perhourworkedearned Regular4040 wage Overtime1.566(1.5) wage Sunday244(2) wage xx xx xx
Leigh’sregularhourlywageis$19.00.
89. Let x representthescoreonthefinalexam. 8083716195 80 7 3902 80 7 3902560 2170 85 xx x x x x
Brookeneedsascoreof85onthefinalexam.
90. Let x representthescoreonthefinalexam. Note:sincethefinalexamcountsfortwo-thirds oftheoverallgrade,theaverageofthefourtest scorescountforone-thirdoftheoverallgrade. ForaB,theaveragescoremustbe80.
1868084902 80 343 13402 80 343 852 80 33 852 3380 33 852240 2155 77.5 x x x x x x x
Mikeneedsascoreof78toearnaB. ForanA,theaveragescoremustbe90.
1868084902 90 343 13402 90 343 852 90 33 852 3390 33 852270 2185
Mikeneedsascoreof93toearnanA.
91. Let x representtheoriginalpriceofthephone. Then0.12x representsthereductionintheprice ofthephone. Thenewpriceofthephoneis$572. originalpricereductionnewprice
xx x x
650
Theoriginalpriceofthephonewas$650. Theamountofthereduction(i.e.,thesavings)is 0.12($650)=$78.
92. Let x representtheoriginalpriceofthecar. Then0.15x representsthereductionintheprice ofthecar. Thenewpriceofthecaris$8000. listpricereductionnewprice
Thelistpriceofthecarwas$21,176.47. Theamountofthereduction(i.e.,thesavings)is 0.15($21176.47)$3176.47
93. Let x representthepricethetheaterpaysforthe candy.
Then2.75 x representsthemarkuponthecandy. Thesellingpriceofthecandyis$4.50. suppierpricemarkupsellingprice
xx x x Thetheaterpaid$1.20forthecandy.
94. Let x representsellingpriceforthenewcar. Thedealer’scostis0.85($24,000)$20,400.
Themarkupis$300. sellingprice=dealer’scost+markup 20,400300$20,700 x At$300overthedealer’scost,thepriceofthe careis$20,700.
95. TicketsPriceperMoney soldticketearned Adults7.507.50 Children52004.504.50(5200) xx xx
7.504.50520029,961 7.5023,4004.5029,961 3.0023,40029,961 3.006561 2187 xx xx x x x
Therewere2187adultpatrons.
96. Let p representtheoriginalpricefortheboots. Then,0.30p representsthediscountedamount. originalpricediscountclearanceprice 0.30399 0.70399 570 pp p p
Thebootsoriginallycost$570.
97. Let w representthewidthoftherectangle. Then8 w isthelength. Perimeterisgivenbytheformula22. Plw 2(8)260 216260 41660 444 11 ww ww w w w
Now,11+8=19. Thewidthoftherectangleis11feetandthe lengthis19feet.
98. Let w representthewidthoftherectangle. Then2 w isthelength. Perimeterisgivenbytheformula22. Plw 2(2)242 4242 642 7 ww ww w w
Now,2(7)=14. Thewidthoftherectangleis7metersandthe lengthis14meters.
99. WewillletBbethecaloriesfrombreakfast,L thecaloriesfromlunchandDthecaloriesfrom dinner.Sowehavethefollowingequations: 125 2300 2025 BL DL BLD
Nowwesubstitutethefirsttwointothelastone andsolveforL.
2025(125)(2300) 20254175 22004 550 LLL L L L
NowwesubstituteLintothefirsttwoequations togetBandD. 550125675 2(550)300800 B D
SoHerscheltookin675caloriesfrombreakfast, 550caloriesfromlunchand800caloriesfrom dinner.
100. WewillletBbethecaloriesfrombreakfast,L thecaloriesfromlunch,Dthecaloriesfrom dinnerandSthecaloriesfromsnacks.Sowe havethefollowingequations:
Nowwesubstitutethefirstfourintothelastone andsolveforB. 14800.5(200)(200)700 14803.5620 21003.5 600 BBBB B B B
NowwesubstituteBtogetS. 120600120480 SB
101. Judy'sAmountTom'sAmountTotal
xx x x xx x xx x x
Since22isthelargestofthenumbersthenthe largestperimeteris:
422102224032218266
Judypays$10.80andTompays$7.20.
102. Anisoscelestrianglehasthreeequalsides. Therefore:410240318 xxx .Solve eachsetseparately:
MultiplybothsidesbytheLCD80toclear fractions. 608488012064 1087258 10716 16 107
104. Ifahexagonisinscribedinacirclethenthesides ofthehexagonareequaltotheradiusofthe circle.LettheP=6rbetheperimeterofthe hexagon.Letrbetheradiusofthecircle. 610 510 2
Thusr=2inchesistheradiusofthecirclewhere theperimeterofthehexagonis10inchesmore thantheradius.
105. Tomovefromstep(6)tostep(7),wedivided bothsidesoftheequationbytheexpression 2 x .Fromstep(1),however,weknow x =2, sothismeanswedividedbothsidesofthe equationbyzero.
106– 107. Answerswillvary.
Chapter 1: Equations and Inequalities
Section 1.2 1. 25661 xxxx 2.
3. 5,3 3 4. True
5.
6. discriminant;negative
7. False;aquadraticequationmayhavenoreal solutions.
8. False;If2 xp then x couldalsobenegative.
9. b
10. d
11.
290 90 xx xx
0or90 0or9 xx xx
Thesolutionsetis{0,9}.
12. 240 (4)0 xx xx 0or40 0or4 xx xx Thesolutionsetis{–4,0}.
13. 2250 (5)(5)0 x xx
50or50 5or5 xx xx Thesolutionsetis{–5,5}.
14. 290 (3)(3)0 x xx 30or30 3or3 xx xx Thesolutionsetis{–3,3}.
15. 260 (3)(2)0 zz zz 30or20 3or2 zz zz Thesolutionsetis{–3,2}.
16. 2760 (6)(1)0 vv vv 60or10 6or1 vv vv Thesolutionsetis{–6,–1}
17. 2 2530 (21)(3)0 xx xx 210or30 1or3 2 xx xx
Thesolutionsetis 1,3 2
18. 2 3520 (32)(1)0 xx xx
320or10 2 or1 3 xx xx
Thesolutionsetis 1,2 3
19. 2 2 51800 5(36)0 5(6)(6)0
w w ww 60or60 6or6
ww ww Thesolutionsetis{–6,6}.
20. 2 2 2500 2(25)0 2(5)(5)0 y y yy
50or50 5or5 yy yy
Thesolutionsetis{–5,5}.
21.
2 3100 3100 (2)50
Section 1.2: Quadratic Equations
25. 2 2 2 6(1)5 665 6560 (32)(23)0 pp pp pp pp
320or230 23 or 32 pp pp
Thesolutionsetis23 , 32
xx xx
xx xx xx 20or50 2or5
Thesolutionsetis 5,2.
22. 2 (4)12 4120 (6)(2)0 xx xx xx
60or20 6or2 xx xx
Thesolutionsetis 6,2.
23. 2 2 2 4912 41290 (23)0 230 3 2 xx xx x x x
Thesolutionsetis 3 2
24. 2 2 2 251640 2540160 (54)0 540 4 5 xx xx x x x
Thesolutionsetis 4 5 .
26. 2 2 2(24)30 4830 (21)(23)0 uu uu uu
210or230 13 or 22 uu uu
Thesolutionsetis13 , 22
27. 2 2 6 65 6 65 656 6560 (32)(23)0 x x xxx x xx xx xx
320or230 23 or 32 xx xx
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis23 , 32
28. 2 2 12 7 12 7 127 7120 (3)(4)0 x x xxx x xx xx xx
30or40 3or4 xx xx
Chapter 1: Equations and Inequalities
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis{3,4}.
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis
31. 225 25 5 x x x
Thesolutionsetis 5,5
32. 236 36 6 x x x
Thesolutionsetis 6,6
33. 2 14 14 12 12or12 3or1 x x x xx xx
Thesolutionsetis 1,3
34. 2 21 21 21 21or21 1or3 x x x xx xx
Thesolutionsetis 3,1
35. 12416 3 1416 3 144 3 1144or44 33 110or8 33 0or24
h h h hh hh hh
Thesolutionsetis 24,0
36. 2 324 324 322 322or322 34or30 4or0 3
z z z zz zz zz
Thesolutionsetis 0,4 3
37. 2 2 2 421 44214 225 225 25 25 3or7 xx xx x x x x xx
Thesolutionsetis7,3.
38.
Thesolutionsetis322,322.
Section 1.2: Quadratic Equations
The13 solutionsetis,. 44 40. 2210 33 xx 2
The1 solutionsetis1,. 3
41. 21 30 2 xx 2 2 2 2 110 36 11 36 1111 336636 17 636 17 636 17 66 17 6 xx xx xx x x x x
The1717 solutionsetis,. 66
42. 2 2310 xx 2 2 2 310 22 31 22 3919 216216 xx xx xx
Chapter 1: Equations and Inequalities
43. 2420xx 2 1,2,13 (2)(2)4(1)(13)2452 2(1)2 2562214114 22
Thesolutionsetis114,114.
44. 2420xx 2 1,4,2 444(1)(2)4168 2(1)2 48422 22 22 abc x
Thesolutionsetis22,22.
45. 2410xx 2 1,4,1 (4)(4)4(1)(1)4164 2(1)2 42042525 22 abc x
Thesolutionsetis25,25.
46. 2610xx 2 1,6,1 664(1)(1)6364 2(1)2 632642 322 22 abc x
Thesolutionsetis322,322.
47. 2 2530 xx 2 2,5,3 (5)(5)4(2)(3) 2(2) 525245151 444 5151 or 44 64 or 44 3or1 2 abc x xx xx xx
The3 solutionsetis1,. 2
48. 2 2530 xx 2 2,5,3 554(2)(3) 2(2) 525245151 444 5151 or 44 46 or 44 13 or 2 abc x xx xx xx
The3solutionsetis,1. 2
49. 2 420 yy 2 4,1,2 (1)(1)4(4)(2) 2(4) 1132131 88 abc y
Norealsolution.
50. 2 410 tt 2 4,1,1 114(4)(1) 2(4) 1116115 88 abc t
Norealsolution.
51. 2 2 985 9850 xx xx 2 9,8,5 884(9)(5) 2(9) 8641808244 1818 8261461 189
abc x
Thesolutionsetis461461,.99
52. 2 2 212 2210 xx xx 2 2,2,1 224(2)(1)248 2(2)4 21222313 442 abc x
Thesolutionsetis1313,.22
Section 1.2: Quadratic Equations
53. 2 2 49 490 (49)0 xx xx xx 0or490 09 or 4 xx xx
Thesolutionsetis 9 0,. 4
54. 2 2 54 045 0(45) xx xx xx 0or450 05 or 4 xx xx
Thesolutionsetis 5 0,. 4
55. 2 9610 tt 2 9,6,1 (6)(6)4(9)(1) 2(9) 63636601 18183 abc t
Thesolutionsetis 1 3.
56. 2 4690 uu 2 4,6,9 (6)(6)4(4)(9) 2(4) 6361446108 88 abc u Norealsolution.
57. 2 2 2 3110 442 311 440 442 320 xx xx xx
3,1,2abc
Chapter 1: Equations and Inequalities
2 11432 23 112412515 666 1515 or 66 64 or 66 12 or 3 x xx xx xx
The2solutionsetis,1. 3
58. 2230 3 xx
2 2 2 3330 3 2390 2,3,9 xx xx abc
2 33429 22 397238139 444 3939 or 44 126 or 44 33 or 2 x xx xx xx
The3solutionsetis,3. 2
59. 2 2 2 2 51 33 5133 33 531 5310 xx xx xx xx
2 5,3,1 33451 25 3920329 1010 abc x
Thesolutionsetis329329 , 1010
60. 2 2 2 2 31 55 3155 55 351 3510 xx xx xx xx
2 3,5,1 55431 23 52512537 66 abc x
Thesolutionsetis537537 , 66
.
61. 2 2(2)3 2430 xx xx 2 2,4,3 444(2)(3)41624 2(2)4 4404210210 442 abc x
Thesolutionsetis210210,.22
62. 2 3(2)1 3610 xx xx
2 3,6,1 664(3)(1)63612 2(3)6 648643323 663
63. 2 22 2 2 11 40 11 40 410 xx xx xx
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis 117117 , 88
64. 2 83 20 xx
Section 1.2: Quadratic Equations
65. 2 22 2 314 2 31(2)4(2) 2 3()(2)48 3248 092 x xx x xxxx xx xxxxx xxxx xx
1,9,2abc 2 (9)(9)4(1)(2) 2(1) 9818973 22 x
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis 973973 , 22
66. 2 22 2 214 3 21(3)4(3) 3 2()(3)412 23412 02133 x xx x xxxx xx xxxxx xxxx xx
2,13,3abc 2 (13)(13)4(2)(3) 2(2) 131692413145 44 x
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis 1314513145 , 44
67. 24.12.20xx 2 1,4.1,2.2 4.14.1412.2 21 4.116.818.84.18.01 22 3.47or0.63 abc x xx
Thesolutionsetis 0.63,3.47
Chapter 1: Equations and Inequalities
68. 23.91.80xx
2 1,3.9,1.8 3.93.9411.8 21 3.915.217.23.98.01 22 0.53or3.37 abc x xx
Thesolutionsetis 3.37,0.53.
69. 2330xx
Thesolutionsetis 2.80,1.07
70. 2220xx
2 1,2,2 22412 21 228210 22 0.87or2.29 abc x xx
Thesolutionsetis 2.29,0.87.
71. 20xx
2 2 ,1, 114 2 114 2 1.17or0.85 abc x xx
Thesolutionsetis 0.85,1.17.
72. 220xx
2 2 ,,2 42 2 8 2 0.44or1.44 abc x xx
Thesolutionsetis 1.44,0.44
73. 2 2670 xx
22 2,6,7 4(6)4(2)7365620 abc bac
Sincethe240, bac theequationhasnoreal solution.
74. 2470xx
22 1,4,7 4(4)4(1)7162812 abc bac
Sincethe240, bac theequationhasnoreal solution.
75. 2 930250 xx
22 9,30,25 4(30)4(9)259009000 abc bac
Since240, bac theequationhasone repeatedrealsolution.
76. 2 252040 xx
22 25,20,4 4(20)4(25)44004000 abc bac
Since240, bac theequationhasone repeatedrealsolution.
77. 2 3580 xx
22 3,5,8 4(5)4(3)82596121 abc bac
Since240, bac theequationhastwo unequalrealsolutions.
78. 2 2370 xx
22 2,3,7 4(3)4(2)795665 abc bac
Since240, bac theequationhastwo unequalrealsolutions.
79. 2 2 50 5 5 x x x Thesolutionsetis5,5.
80. 2 2 60 6 6 x x x Thesolutionsetis6,6.
81. 2 16810 41410 410 1 4 xx xx x x
Thesolutionsetis 1 . 4
82. 2 91240 32320 320 2 3 xx xx x x Thesolutionsetis 2 3.
83. 2 1019150 53250 xx xx 530or250 35 or 52 xx xx
Thesolutionsetis 35 ,.52
84. 2 67200 34250 xx xx 340or250 45 or 32 xx xx
Thesolutionsetis 54 ,.23
Section 1.2: Quadratic Equations
85. 2 2 26 062 03221 zz zz zz
320or210 21 or 32 zz zz
Thesolutionsetis 12 , 23
86. 2 2 26 062 03221 yy yy yy 320or210 21 or 32 yy yy
Thesolutionsetis 21 , 32
87.
2 2 2 2 21 2 1 20 2 1 2220 2 22210 xx xx xx xx
2,22,1abc
2 (22)(22)4(2)1 2(2) 22882216 44 22422 42 x
The2222 solutionsetis,22
88.
2 2 2 2 121 2 1210 2 1 22120 2 2220 xx xx xx xx
1,22,2abc
Chapter 1: Equations and Inequalities
The1515 solutionsetis,. 22
91. 2 271 212 xx
xx xxxx xx xxxx xxxx xxxx xxxx xxx xx xx
Thevalue1 x causesadenominatortoequalzero,sowedisregardit.Thus,thesolutionsetis{5}.
92.
Thevalue2 x causesadenominatortoequalzero,sowedisregardit.Thus,thesolutionsetis1
93. Sincethisisarighttrianglethenwecanusethe PythagoreanTheorem.So 222 222 2 2 (23)(25)(7) 4129420251449 129674 01865 0(5)(13) xxx xxxxxx xxx xx xx
50or130 5or13 xx xx
Thismeansthereare2possiblethatmeetthese requirements.Substituting x intothegivensides gives:
When x =5:5m,12m,13m
When x =13:20m,21m,29m Thusthereare2solutions.
94. Sincethisisarighttrianglethenwecanusethe PythagoreanTheorem.So 222 222 2 2 (45)(313) 164025978169 6381440 2(31972)0 2(38)(9)0 xxx xxxxx xx xx xx
Thismeansthereare2possiblesolutionsthat meettheserequirements.Substituting x intothe givensidesgives:
When x =9:41m,40m,9m
When x = 8 3 atleastonesideofthetriangle hasanegativemeasurementwhichisimpossible. Thusthereisonly1trianglepossible
95. Let w representthewidthofwindow. Then2 lw representsthelengthofthe window.
Sincetheareais143squarefeet,wehave: 2 (2)143 21430 (13)(11)0 ww ww ww
or11 w
Discardthenegativesolutionsincewidthcannot benegative.Thewidthoftherectangular windowis11feetandthelengthis13feet.
96. Let w representthewidthofwindow. Then1 lw representsthelengthofthe window.
Sincetheareais306squarecentimeters,we have:(1)306 ww 23060 (18)(17)0 ww ww
or17 w
Discardthenegativesolutionsincewidthcannot
Chapter 1: Equations and Inequalities
benegative.Thewidthoftherectangular windowis17centimetersandthelengthis18 centimeters.
97. Let l representthelengthoftherectangle. Let w representthewidthoftherectangle. Theperimeteris26metersandtheareais40 squaremeters. 2226 13so13 lw
Thedimensionsare5metersby8meters.
98. Let r representtheradiusofthecircle. Sincethefieldisasquarewitharea1250square feet,thelengthofasideofthesquareis 1250252 feet.Thelengthofthediagonal is2r
UsethePythagoreanTheoremtosolvefor r :
Theshortestradiussettingforthesprinkleris25 feet.
99. Let x =lengthofsideoforiginalsheetinfeet.
Lengthofbox:2 x feet
Widthofbox:2 x feet
Heightofbox:1foot
Discard0 x sincethatisnotafeasiblelength fortheoriginalsheet.Therefore,theoriginal sheetshouldmeasure4feetoneachside.
100. Let x =widthoforiginalsheetinfeet.
Lengthofsheet:2 x
Lengthofbox:22 x feet
Widthofbox:2 x feet
Heightofbox:1foot
2 2 2 42221 4264 026 03 03 Vlwh xx xx xx xx xx
0or3xx
Discard0 x sincethatisnotafeasiblelength fortheoriginalsheet.Therefore,theoriginal sheetis3feetwideand6feetlong.
101. a. Whentheballstrikestheground,the distancefromthegroundwillbe0. Therefore,wesolve
2 2 2 9680160 1680960 560 610 tt tt tt tt
6or1tt Discardthenegativesolutionsincethetime offlightmustbepositive.Theballwill strikethegroundafter6seconds.
b. Whentheballpassesthetopofthebuilding, itwillbe96feetfromtheground.Therefore, wesolve
2 2 2 96801696 16800 50 50 tt tt tt tt
0or5tt Theballisatthetopofthebuildingattime 0 t whenitisthrown.Itwillpassthetop ofthebuildingonthewaydownafter5 seconds.
102. a. Tofindwhentheobjectwillbe15meters abovetheground,wesolve 2 2 4.92015 4.920150
Theobjectwillbe15metersabovethe groundafterabout0.99seconds(ontheway up)andabout3.09seconds(ontheway down).
b. Theobjectwillstrikethegroundwhenthe distancefromthegroundis0.Therefore,we solve
Theobjectwillstrikethegroundafterabout 4.08seconds.
c. 2 2 4.920100 4.9201000 tt tt
Thereisnorealsolution.Theobjectnever reachesaheightof100meters.
103. Let x representthenumberofcentimetersthe lengthandwidthshouldbereduced. 12 x =thenewlength,7 x =thenewwidth. Thenewvolumeis90%oftheoldvolume.
Section 1.2: Quadratic Equations
2 (19)(19)4(1)(8.4)19327.4 2(1)2 0.45or18.55 x xx
Since18.55exceedsthedimensions,itis discarded.Thedimensionsofthenewchocolate barare:11.55cmby6.55cmby3cm.
104. Let x representthenumberofcentimetersthe lengthandwidthshouldbereduced. 12 x =thenewlength,7 x =thenewwidth. Thenewvolumeis80%oftheoldvolume.
2 2 (12)(7)(3)0.8(12)(7)(3) 357252201.6 35750.40 1916.80 xx xx xx xx
2 (19)(19)4(1)(16.8)19293.8 2(1)2
0.93or18.07 x xx
Since18.07exceedsthedimensions,itis discarded.Thedimensionsofthenewchocolate barare:11.07cmby6.07cmby3cm.
105. Let x representthewidthoftheborder measuredinfeet.Theradiusofthepoolis5 feet.Then5 x representstheradiusofthe circle,includingboththepoolandtheborder. Thetotalareaofthepoolandborderis
2 (5) T Ax Theareaofthepoolis2(5)25 PA Theareaoftheborderis
2 (5)25 BTP AAAx Sincetheconcreteis3inchesor0.25feetthick, thevolumeoftheconcreteintheborderis
2 0.250.25(5)25 B Ax Solvingthevolumeequation:
2 2 2 0.25(5)2527 102525108 101080 x xx xx
35725.20 198.40 xx xx xx xx
2 2 2 (12)(7)(3)0.9(12)(7)(3) 357252226.8
2 2 10(10)4()(108) 2()
2.71or12.71 x xx
31.42100432
6.28
Discardthenegativesolution.Thewidthofthe borderisroughly2.71feet.
Chapter 1: Equations and Inequalities
106. Let x representthewidthoftheborder measuredinfeet.Theradiusofthepoolis5 feet.Then5 x representstheradiusofthe circle,includingboththepoolandtheborder. Thetotalareaofthepoolandborderis
2 (5) T Ax
Theareaofthepoolis2(5)25 PA .
Theareaoftheborderis
108. Let x =thewidthand2x =thelengthofthe patio.Theheightis13footandtheconcrete availableis 827216 cubicfeet.. 2 2 1 (2)216 3 2216 3 32418
2 (5)25
BTP AAAx .
Sincetheconcreteis4inches=13footthick,the volumeoftheconcreteintheborderis 112 33(5)25 B Ax
Solvingthevolumeequation:
2 2 2 1(5)2527 3 10252581 10810 x xx xx
2 2 10(10)4()(81) 2()
Vlwhxx x xx Thedimensionsofthepatioare18feetby36 feet.
109. Let x =thelengthofa12.9-inchiPadProina 16:94:3format. Then9 16 x =thewidthoftheiPad.Thediagonal ofthe12.9-inchiPadis9.7inches,sobythe Pythagoreantheoremwehave:
2.13or12.13 x xx
31.42100324 6.28
Discardthenegativesolution.Thewidthofthe borderisapproximately2.13feet.
107. Let x representthewidthoftheborder measuredinfeet.
Thetotalareais(62)(102) T Axx
Theareaofthegardenis61060 GA .
Theareaoftheborderis (62)(102)60
BTG AAAxx Sincetheconcreteis3inchesor0.25feetthick, thevolumeoftheconcreteintheborderis
0.250.25(62)(102)60 B Axx
Solvingthevolumeequation: 2 2 2 0.25(62)(102)6027 6032460108 4321080 8270 xx xx xx xx
2 884(1)(27)8172 2(1)2
2.56or10.56 x xx
Discardthenegativesolution.Thewidthofthe borderisapproximately2.56feet.
Sincethelengthcannotbenegative,thelengthof theiPadis42600.96337inchesandthewidthis
9 6.32 16 inches.Thus,theareaofthe iPadis42600.96942600.963371633771.11
square inches.
Let y =thelengthofa12-inch3:2format
MicrosoftSurfacePro.Then 2 3 y =thewidthof theSurfacePro.Thediagonalofa12-inch SurfaceProis12inches,sobythePythagorean theoremwehave:
Sincethelengthcannotbenegative,thelengthof theSurfaceProis10.23inchesandthewidthis 21361.61 6.82 313 inches.Thus,theareaofthe 12.3-inch3:2formatSurfaceProis
1361.611361.61 2 13313 69.83squareinches.
TheiPadProformathasthelargerscreensince itsareaislarger.
110. Let x =thelengthofa7.9-inchiPadMiniina 4:3format.
Then3 4 x =thewidthoftheiPad.Thediagonal ofthe7.9-inchiPadis7.9inches,sobythe Pythagoreantheoremwehave:
Sincethelengthcannotbenegative,thelengthof theiPadis6.32inchesandthewidthis
3
4 inches.Thus,theareaofthe iPadis(6.32)(4.74)29.9568 squareinches. Let y =thelengthofa8-inch16:10format AmazonFireHD8™.Then10 16 y =thewidthof theFire.Thediagonalofa8-inchFireis8 inches,sobythePythagoreantheoremwehave:
Section 1.2: Quadratic Equations
Sincethelengthcannotbenegative,thelengthof theFireis163846.78399 356 inchesandthe widthis10163844.240 16356 inches.Thus,thearea oftheAmazonFireis
6.783994.24028.76squareinches.
TheiPadMini™4:3formathasthelargerscreen sinceitsareaislarger.
111. Let h be1.1.Then 2 2 2 0.04(0.04)4(0.00025)(1.1) 2(0.00025) 1.10.000250.04 00.000250.041.1 35.3ftor124.7ft
xx xx x 124.7ftdoesnotmakesenseinthecontextof theproblem,sotheansweris35.3ft.
112. Sincedisexpressedin1000’swewillsetd=15 andsolveforxusingtheQuadraticFormula. 2 2 2 2 0.828(0.828)4(0.012)(9.25) 2(0.012) 0.828.241584 0.024 0.0120.82815.750 250.0120.82815.750 00.0120.8289.25
dxx xx xx x xx
54.98or14.02
Sothenearestyearwhenthedifferencewas $25,000occurredabout14yearsafter1980or 1994.Thevalue55hasnomeaningsinceitisin thefuture.
Chapter 1: Equations and Inequalities
113. Wewillsetg=2.97andsolveforhusingthe QuadraticFormula. 2
0.00060.0153.04
2.970.00060.0153.04
00.00060.0150.07 29or4.02 gxx xx xx
0.015(0.015)4(0.0006)(0.07) 2(0.0006)
0.0150.000393 0.0012
Sotheestimatednumbersofhoursworkedbya studentwithaGPAof2.97is29hours.The value-4.02hasnomeaningsinceitisnegative.
114. Letxbethenumbersofmembersinthe fraternityandsbethesharepaidbyeach member.Then1470 s x . Ifthereare7 memberswhocannotcontributethentheshare goesupby$5.Sowehavethefollowing equation: 51470 7 s x
or 571470sx
Solvingthesetwoequationstogether:
2 5714701470 and 1470571470 10290 14705351470 10290 5350 535102900 sxs x x x x x x x xx
2 2 535102900 720580 (42)(49)0 42or49 xx xx xx xx
Sincexisthenumberofmembers,itmustbe positivesothenumberofmembersis49.
115. Let a betheagetheindividualisabletostart savingmoney.Thenweneedtofindwherethe modelsareequal.Solvingthesetwoequations together: 2 2 2 2240(2240)4(25)(38540) 2(25)
22401163600 50 252400307001607840 252240385400 22401078.7 50
22401078.722401078.7 or 5050 66.4or23.2
Since x istheagetostartsaving,itmakessense thattheanswerisapproximateatage23.
116.Wewillsettheequationequalto10andsolve: 2 2 2 0.034(0.034)4(0.003)(1.914) 2(0.003)
0.034.024124 0.006 0.0030.0348.08610 0.0030.0341.9140
0.034.15532 0.006 0.0340.155320.0340.15532 or 0.0060.006 20.22
x or31.55 x Thepercentagewillreach10%approximately32 yearsafter1960whichis1992. 117.
Sincethenumberofconsecutiveintegerscannot benegative,wediscardthenegativevalue.We mustadd37consecutiveintegers,beginningat1, inordertogetasumof703.
Sincethenumberofsidescannotbenegative,we discardthenegativevalue.Apolygonwith65 diagonalswillhave13sides.
Section 1.2: Quadratic Equations
121. Inordertohaveonerepeatedsolution,weneed thediscriminanttobe0.
Neithersolutionisaninteger,sothereisno polygonthathas80diagonals.
119. Therootsofaquadraticequationare
122. Inordertohaveonerepeatedsolution,weneed thediscriminanttobe0.
2 2 2 40 4140 160 440 bac k k kk
123. For20 axbxc : 24 2 bbac x a
For20 axbxc :
2 2 4 2 4 2
120. Therootsofaquadraticequationare
Chapter 1: Equations and Inequalities
125. If x =originalwidthand y =originallength,then 11 or xyx y .Theratioofsidelengthsis 2 1 x yy .Foldingalongthelongestsideresults insidesoflength1and2 y x y whoseratiois 2 2 12
y y y Equatingtheratiosgives
126. a. 29 x and3 x arenotequivalent becausetheydonothavethesamesolution set.Inthefirstequationwecanalsohave 3 x b. 9 x and3 x areequivalentbecause 93 c.
1212xxx and21 xx are notequivalentbecausetheydonothavethe samesolutionset. Thefirstequationhasthesolutionset 1 whilethesecondequationhasnosolutions.
127. Answerswillvary.Methodsmayincludethe quadraticformula,completingthesquare, graphing,etc.
128. Answerswillvary.Knowingthediscriminant allowsustoknowhowmanyrealsolutionsthe equationwillhave.
129. Answerswillvary.Onepossibility: Twodistinct:23180 xx Onerepeated:214490 xx Noreal:240 xx
130. Answerswillvary.
Section 1.3: Complex Numbers; Quadratic Equations in the Complex Number System
Section 1.3
1. Integers: 3,0 Rationals: 3,0,6 5
2. True;thesetofrealnumbersconsistsofall rationalandirrationalnumbers.
3.
4. real;imaginary;imaginaryunit
5. False;theconjugateof25i is25i
6. True;thesetofrealnumbersisasubsetoftheset ofcomplexnumbers.
7. False;if23i isasolutionofaquadratic equationwithrealcoefficients,thenits conjugate,23i ,isalsoasolution.
8. b
9. a
10. c
11. (23)(68)(26)(38)85 iiii
12. (45)(82)(4(8))(52) 47 iii i
13. (32)(44)(34)(2(4)) 76 iii i
14. (34)(34)(3(3))(4(4)) 606 iii i
15. (25)(86)(28)(56) 611 iii i
16. (84)(22)(82)(4(2)) 106 iii i
17. 3(26)618 ii
18. 4(28)832 ii
19. 2 3(76)21182118(1) 1821 iiiii i
20. 2 3(34)912912(1)129 iiiiii
21. (34)(2)63842 654(1) 105 iiiii i i
22. (53)(2)105632 103(1) 13
iiiii i i
iiiii
23. (5)(5)25552 25(1) 26
24. (3)(3)9332 9(1) 10 iiiii
25. 2 1010343040 3434349121216 30403040 916(1)25 3040 2525 68 55 ii iiiiii ii i i
26. 2 1313512 512512512 65156 256060144 6515665156 25144(1)169 65156 169169 512 1313 i iii i iii ii i i
31. 2 1311332 2 224224 13313 (1) 42422 iii ii
22 (1)1212(1)2
23 2322122211 112 1111 () 111 (1)(1) i iiiiii iii i iii ii
39. 3 62355(1)5156ii
40. 32 444(1)4 iiiii
41. 3532 2 64(64) (64(1))1(10)10 iiii iiii
42. 3222 421421 4(1)2(1)1 421 34 iiiii i i i
43. 32 2 (1)(1)(1)(1)(12)(1) (121)(1)2(1) 2222(1) 22 iiiiiii iiii iii i
44. 44 (3)181181(1)182 ii
45. 7277 (1)(1(1))(0)0 iiii
46. 42 2(1)2(1)(1(1))2(0)0 ii
47. 432 86422222 432 (1)(1)(1)1 1111 0
iiiiiiii
48. 32 753222 32 (1)(1)(1) 0 iiiiiiiiiii iiii iiii
49. 42i 50. 93i
51. 255 i
52. 648 i
53. 124323 ii
54. 189232 ii
55. 2001002102 ii
56. 459535 ii
57. 2 (34)(43)1291612 916(1)
58. 2 (43)(34)1216912 169(1) 25 5 iiiii i
59. 2 2 40 4 4 2 x x x xi
Thesolutionsetis 2,2. ii
63. 26130xx 22 1,6,13, 4(6)4(1)(13)365216 (6)166432 2(1)2
Thesolutionsetis32,32. abc bac i xi ii
64. 2480xx 22 1,4,8 444(1)(8)163216 41644 22 2(1)2 abc bac i xi
Thesolutionsetis 22,22 ii .
65. 26100xx 22 1,6,10 4(6)4(1)(10)36404 (6)4623 2(1)2 abc bac i xi
Thesolutionsetis 3,3ii
66. 2250xx 22 1,2,5 4(2)4(1)(5)42016 (2)162412 2(1)2 abc bac i xi
Thesolutionsetis 12,12 ii .
Thesolutionsetis2,2. xx xx
60. 240 x (2)(2)0 2or2
61. 2160 x
440 4or4
Thesolutionsetis4,4. xx xx
67. 2 251020 xx 22 25,10,2 4(10)4(25)(2)100200100 (10)100101011 505055
abc bac i xi
Thesolutionsetis 1111 , 5555 ii
62. 2250 x 225 255
Thesolutionsetis5,5. x xi ii
Chapter 1: Equations and Inequalities
68. 2 10610 xx 22 10,6,1 464(10)(1)36404 646231 2(10)201010
abc
bac i xi
Thesolutionsetis 3131 , 10101010 ii
69. 2 2 512 5210 xx xx
abc
bac i xi
22 5,2,1 424(5)(1)42016 (2)162412 2(5)1055
Thesolutionsetis 1212 , 5555 ii
70. 2 2 1316 13610 xx xx
abc
bac i xi
22 13,6,1 4(6)4(13)(1)365216 (6)166432 2(13)261313
Thesolutionsetis 3232 , 13131313 ii
71. 210xx 1,1,1,abc 22414(1)(1)143 131313 2(1)222
bac i xi
Thesolutionsetis1313 , 2222 ii
72. 210xx 1,1,1abc 224(1)4(1)(1)143 (1)31313 2(1)222 bac i xi
Thesolutionsetis1313 , 2222 ii
73. 3640 x 2 2 (4)4160 404 or4160
abc bac i xi
xxx xx xx 22 1,4,16 444(1)(16)166448 448443 223 2(1)2
Thesolutionsetis 4,223,223. ii
74. 3270 x 2 2 (3)390 303 or390 xxx xx xx
22 1,3,9 4(3)4(1)(9)93627 (3)27333333 2(1)222 abc bac i xi
Thesolutionsetis3333333,,. 2222 ii
75. 4 4 16 160 x x
22 2 440 (2)(2)40 xx xxx
2 2 20or20or40 2or2or4 2or2or42 xxx xxx xxxi
Thesolutionsetis 2,2,2,2. ii
76. 4 4 1 10 x x
22 2 110 (1)(1)10 xx xxx
2 2 10or10or10 1or1or1 1or1or1 xxx xxx xxxi
Thesolutionsetis 1,1,,. ii
77. 4213360xx
22 22 22 940 90or40 9or4 9or4 3or2 xx xx xx xx xixi
78. 42340xx
Thesolutionsetis 1,1,2,2. ii
79. 2 3340 xx 22 3,3,4 4(3)4(3)(4)94839 abc bac
Theequationhastwocomplexsolutionsthatare conjugatesofeachother.
80. 2 2410 xx 22 2,4,1 4(4)4(2)(1)1688 abc bac
Theequationhastwounequalrealnumber solutions.
81. 2 2 234 2340 xx xx 22 2,3,4 434(2)(4)93241 abc bac
Theequationhastwounequalrealsolutions.
82. 2 2 62 260 xx xx 22 1,2,6 4(2)4(1)(6)42420 abc bac
Theequationhastwocomplexsolutionsthatare conjugatesofeachother.
83. 2 91240 xx 22 9,12,4 4(12)4(9)(4)1441440 abc bac
Theequationhasarepeatedrealsolution.
84. 2 41290 xx 22 4,12,9 4124(4)(9)1441440 abc bac
Theequationhasarepeatedrealsolution.
85. Theothersolutionis2323. ii
86. Theothersolutionis44. ii
87. 343434346zziiii
88. 8383 83(83) 8383 066 wwii ii ii ii
89. 2 (34)(34) (34)(34) 9121216 916(1)25 zzii ii iii
90. 34(83) 3483 57 57 zwii ii i i
91. 2 2 181834 343434 54723454754 9121216916 5075 23 25 Viii Z Iiii iiii iii i i
Theimpedanceis23i ohms.
92. 12 2 11111(43)(2) 243(2)(43) 626262 8643823112 ii ZZZiiii iii iiiii
Chapter 1: Equations and Inequalities
So, 2 2 11211262 626262 662212466104 3612124364
701071 4044 iii Z iii iiii iii i i
Thetotalimpedanceis71 44 i ohms.
2222 22 22 () 2()0
abab ab abba Anycomplexnumberoftheform aai or aai willwork.
98. Let32 u in320 x sothat330 xu
93. ()() 2 zzabiabi abiabi a
() () 2 zzabiabi abiabi abiabi bi
94. zabiabiabiz
95. ()() ()() ()() ()() zwabicdi acbdi acbdi abicdi abicdi zw
96. 2 ()() ()() ()() zwabicdi acadibcibdi acbdadbci acbdadbci
2 ()() ()() zwabicdi abicdi acadibcibdi acbdadbci
uuu x uuuuii
Then,22()()0 xuxuxu .Fromthefirst factorwefind32 xu .Fromthesecond factor,usethequadraticformulatoget 22 233 ()()41 21 33223 22222
Thesolutionsetis: 33 3223 2,22
99. 2 (5)(5)(); xyxy let5 ux (so 5 xu and5 vy so5 yv
Substitutinggives2 () uvuv or 220 uuvv whichisquadraticinu.Using thequadraticformulagives
223 41 212 vvvvv x .Since x isarealnumber, u mustalsobearealnumber. Thisisonlypossibleifv=0whichthenmakes u =0.Therefore,055 x and 055 y ,so5510 xy
100 – 102. Answerswillvary.
103. Answerswillvary.Acomplexnumberisthe sumordifferenceoftwonumbers(realand imaginarypartsofthecomplexnumber)justasa binomialisthesumordifferenceoftwo monomialterms.Wemultiplytwobinomialsby usingtheFOILmethod,anapproachwecanalso usetomultiplytwocomplexnumbers.
97. 22 2222 222222 2222 2()2() 22 22
abiabi aabibiaabibi aabibiaabibi aabibaabib
104. Althoughthesetofrealnumbersisasubsetof thesetofcomplexnumbers,notallrulesthat workintherealnumbersystemcanbeusedin thelargercomplexnumbersystem.Therulethat allowsustowritetheproductoftwosquare rootsasthesquarerootoftheproductonly worksintherealnumbersystem.Thatis, abab onlywhen a and b arereal numbers.Inthecomplexnumbersystemwe mustfirstconverttheradicalstocomplexform. Inthiscasethismeansweneedtowrite9as 19913
Section 1.4
1. True 2. 33 xx 3. 322 62231 xxxx
4. False;youcanalsousetheQuadraticFormulaor completingthesquare.
5. quadraticinform
6. True
7. a 8. c 9. 211 t
Check:3(0)442
Thesolutionsetis{0}.
11. 346 t Sincetheprincipalsquarerootisnevernegative, theequationhasnorealsolution.
12. 532 t
Sincetheprincipalsquarerootisnevernegative, theequationhasnorealsolution.
13. 31230 x 3 333 123 123 1227 226 13 x x x x x
Check:3312(13)32730
Thesolutionsetis{13}.
14. 31210 x 3 333 121 121 121 20 0 x x x x x
Check:3312(0)1110
Thesolutionsetis{0}.
Check:2(1)111 Thesolutionsetis{1}.
Check:
Thesolutionsetis{1}. 16.
Check3:316916255
Check3:316916255
Thesolutionsetis
Check0:080
Check64:64864 6464
Thesolutionsetis 0,64 18.
Check0:030 00 Check9:939 99
Thesolutionsetis 0,9 19. 152 xx
22 2 2 152 152 2150 (5)(3)0 xx xx xx xx
Check–5:152(5)2555
Check3:152(3)933
Disregard5asextraneous. Thesolutionsetis{3}. x
20. 12 xx
22 2 2 12 12 120 (4)(3)0 xx xx xx xx
4or3xx
Check–4:12(4)1644
Check3:123933
Disregard4asextraneous. Thesolutionsetis{3}. x
21. 21xx
22 2 2 2 2 21 4(1) 44 440 (2)0 2 xx xx xx xx x x
Check:2221 22
Thesolutionsetis{2}.
Theequationhasnorealsolution.
Sincetheprincipalsquarerootisalwaysanonnegativenumber;1 3 x doesnotcheck. Thereforethisequationhasnorealsolution.
25. 331xx 22 2 2 313 31(3) 3169 098 0(1)(8) 1or8 xx xx xxx xx xx xx
Check1:33(1)13451
Check8:33(8)132588
Discard1 x asextraneous. Thesolutionsetis{8}.
26. 2122 xx
22 2 2 1222 122(2) 12244 028 (2)(4)0 2or4 xx xx xxx xx xx xx
Check2:2+122(2)21662
Check4:2122(4)2444
Discard2 x asextraneous.
Thesolutionsetis{4}.
27. 22 2 2 3(10)4 3(10)4 3(10)(4) 330816 0514 0(7)(2) 7or2 xx xx xx xxx xx xx xx
Check7:3(710)49417
Check2:3(210)436422
Discard7 x asextraneous. Thesolutionsetis{2}.
Chapter 1: Equations and Inequalities
28.
0(3)(8) 3or8 xx xx xx xxx xx xx xx
22 2 2 132 15 1(5) 11025 01124
30. 3721 xx
Check3:1(3)33211
Check8:1(8)38206
Discard8 x asextraneous. Thesolutionsetis{-3}.
29. 3572 xx
46425616(7) 46425616112 4801440 20360 (2)(18)0 xx xx xxx xx xx xxx xxx xx xx xx
(216)47
2or18xx
Check2:3(2)527 191322
Check18:3(18)5187 49257522
Discard2 x asextraneous. Thesolutionsetis{18}.
22 22 2 2 3712 3712 371222 2422 22 (2)2 442 320 (1)(2)0 1or2 xx xx xxx xx xx xx xxx xx xx xx
Check–1:3(1)712 412131
Check2:3(2)722 101011
Discard1 x asextraneous. Thesolutionsetis{2}.
31. 3112 xx
22 22 2 2 2 3121 3121 314411 2241 (22)41 48416(1) 2144 650 (1)(5)0 xx xx xxx xx xx xxx xxx xx xx
1or5xx
Check1:3(1)111 402022
Check5:3(5)151 1644222
Thesolutionsetis 1,5
Check3:2(3)331
Check4:Check25: 10344 10322 162 42
Discard4 x asextraneous. Thesolutionsetis{25}.
35. 1/2 314 x 2 1/22314 3116 315 5 x x x x
Check: 1/21/2 351164 Thesolutionsetis{5}.
36. 1/2 352 x 2 1/22352 354 39 3 x x x x
Check: 1/21/2 33542 Thesolutionsetis{3}.
Chapter 1: Equations and Inequalities
37. 1/3 522 x
3 1/33522 528 510 2 x x x x
Check: 1/31/3 52282
Thesolutionsetis{2}.
38. 1/3 211 x
3 1/33211 211 22 1 x x x x
Check:
1/31/3 21111
Thesolutionsetis{1}.
39. 21/295 x
21/222 2 2 95 925 16 164 x x x x
Check4:
21/21/2 49255
Check4: 21/21/2 49255
Thesolutionsetis 4,4
40. 21/2169 x
21/222 2 2 169 1681 97 97 x x x x
Check97: 21/21/2 9716819
Check97: 21/21/2 9716819
Thesolutionsetis
41.
42.
3/21/2 1/2 30 30 xx xx 1/20or30 0or3 xx xx
Check0:3/21/2030000
Check3:3/21/233333330
Thesolutionsetis 0,3
3/41/4 1/41/2 90 90 xx xx 1/40 0 x x or 1/29 81 x x
Check0:3/41/4090000
Check81:3/41/48198127270
Thesolutionsetis 0,81
43.
42 22 22 540 410 40or10 2or1 xx xx xx xx
Thesolutionsetis 2,1,1,2.
44.
42 22 2 10250 550 50 5 xx xx x x
Thesolutionsetis 5,5.
45.
42 22 6510 6110
xx xx 22 22 610or10 61or1 Notrealor1
xx xx x Thesolutionsetis 1,1.
46.
42 22 25120 2340 xx xx
22 22 230or40 23or4 Notrealor2 xx xx x
Thesolutionsetis 2,2.
47. 63 33 780 810 xx xx
Section 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations
33 33 80or10 8or1 2or1 xx xx xx
Thesolutionsetis 2,1.
48.
63 33 780 810 xx xx
33 33 80or10 8or1 2or1 xx xx xx
Thesolutionsetis 1,2.
49. 2 272120xx
Let22 2,sothat2.uxux
27120 340 uu uu 30or40 3or4 23or24 5or6 uu uu xx xx
Thesolutionsetis 6,5.
50. 2 252560 xx Let22 25sothat25.uxux
260 320 uu uu
30or20 3or2 253or252 7 1or2 uu uu xx xx
Thesolutionsetis 7,1. 2
51. 2 491049250 xx
Let22 49sothat49. uxux
uu u u u x x x
2 2 10250 50 50 5 495 414 7 2
Thesolutionsetis 7 2.
52. 2 22200 xx Let22 2sothat2.uxux
2200 540 uu uu 50or40 5or4 25or24 7or2 uu uu xx xx
Thesolutionsetis 2,7.
53. 2 21513 ss Let22 1sothat1.usus
2 2 253 2530 2130 uu uu uu
210or30 1or3 2 11or13 2 3or2 2 uu uu ss ss
Thesolutionsetis 3,2. 2
54. 2 315120 yy Let22 1sothat1.uyuy
2 3520 3210 uu uu
Chapter 1: Equations and Inequalities
320or10 2or1 3 12or11 3 5or2 3 uu uu yy yy
Thesolutionsetis 5,2. 3 55.
40 140 xxx xx
1 4 122 4 1 16 0or140 14 xx x x x x
Check:
1111 16161616 111 16164 11 1616 0:04(0)00 00 :40 40 0 00 x x
Thesolutionsetis 1 0,. 16
56. 80xx
0or64xx
Check:0:0800 00 64:648640 64640 x x
Thesolutionsetis 0
57. 20 xx Let2sothat. uxux
2 2 20 200 540 uu uu uu
5040 or 54 or or 54 notorpossible16 uu uu xx x
Check:161620 16420
Thesolutionsetis 16
58. 6 xx Let2sothat. uxux
2 2 6 60 320 uu uu uu
3020 or 32 or or 32 notorpossible4 uu uu xx x
Check:446 426
Thesolutionsetis 4.
59. 1/21/4210tt Let1/421/2 sothat.utut
2 2 1/4 210 10 10 1 1 1 uu u u u t t
Check: 1/21/4 12110 1210 00
Thesolutionsetis 1
Section 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations
60. 1/21/4440zt Let1/421/2 sothat.uzuz
Check:
1/21/4 1641640 4840 00
Thesolutionsetis 16.
61. 1/21/4320xx Let1/421/2 sothat.uxux
1/21/4 1/21/4 16:1631620 4620 00 1:13120 1320 00 x x
Thesolutionsetis 1,16.
62. 1/21/4 4940 xx Let1/421/2 sothat.uxux 2 4940 uu
2 (9)(9)4(4)(4)917 2(4)8 u
1/21/444 2 2 2 917917 494
1/21/444 2 917917 4940 88 917917 4940 88 481181717729172560
Since
isafourthroot,541 2 x isalso notreal.Therefore,wehaveonlyonepossible solutiontocheck:541: 2
Section 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations
Thesolutionsetis 4,1.
66. 22 222 332 Let3sothat3. xxxx uxxuxx
Chapter 1: Equations and Inequalities
Thesolutionsetis
Thesolutionsetis
Thesolutionsetis
77.
78.
80.
81.
or50 5 x x or40 4 x x
Thesolutionsetis 5,0,4
32 2 670 670 710 xxx xxx xxx
0 x or70 7 x x or10 1 x x
Thesolutionsetis 7,0,1
32 2 2 10 1110 110 1110 xxx xxx xx xxx
10 1 x x or10 1 x x
Thesolutionsetis 1,1
82.
32 2 2 440 4140 410 4110 xxx xxx xx xxx
40 4 x x or10 1 x x or10 1 x x
Thesolutionsetis 4,1,1.
83.
xxx xxx xx xxx
32 2 2 316480 31630 3160 3440
30 3 x x or40 4 x x or40 4 x x Thesolutionsetis 4,3,4.
84.
85.
32 2 2 330 3130 310 3110 xxx xxx xx xxx
30 3 x x or10 1 x x or10 1 x x Thesolutionsetis 1,1,3
32 32 2 2 248 2840 214210 2140 21220 xxx xxx xxx xx xxx
210 21 1 2 x x x or20 2 x x or20 2 x x
Thesolutionsetis 1 2,,2 2
Chapter 1: Equations and Inequalities
Thesolutionsetis
370 37 7 3
x x x or 2 2 40 4 norealsolutions
Thesolutionsetis
350 35 5 3 x x x
or 2 2 40 4 norealsolutions x
Thesolutionsetis
5 3.
1/34/322 1/3 22 1/3 22 1/3 22 3230 3230 3260 3250 xxxxx xxxxx xxxxx xxxx
1/3 22 22 30or250 30or250 30or250 05 or3or0or 2 xxxx xxxx xxxx xxxx
Thesolutionsetis
1/23/222 1/2 22 1/2 22 1/2 22 32220 23220 23240 220 xxxxx xxxxx xxxxx xxxx
1/2 22 22 20or20 20or20 20or210 01 or2or0or 2 xxxx xxxx xxxx xxxx
Check0 x :
1/23/222 1/23/2 3002020200 300200 00
Check2 x
1/23/222 1/23/2 1/23/2 3(2)(2)2(2)2(2)2(2)0 3(2)442440 3(2)0200 3(2)0200 00
Check1 2 x
1/23/222 11111 22222 1/23/2111 244 1/23/2133 244 32220 31210 320 Notreal
Thesolutionsetis
91. 1/2 1/222 420 Letsothat. xx
2 2242220 442284220 00
Thesolutionsetis
22 22,220.34,11.66
92. 2/31/3 1/322/3 420 Letsothat. xx uxux
2420uu 2 444(1)(2) 2(1) 4842222 22 u
1/31/3 33 22or22 22or22 22or22 uu xx xx
3 Check22: x
Check1142
Check1142
Thesolutionsetis5
Thesolutionsetis7,3
Thesolutionsetis52,,2 3
101. 433 4123 315 5 ww ww w w
Thesolutionsetis
102.
63212 618212 46 3 2 kk kk k
Thesolutionsetis3 2
103. 2 2 8 11 vv vv
Let 1 v u v .Rewritetheequation:
2 2 28 280 240 uu uu uu
Thesolutionsetis4 2, 5
104. 2 6 7 11 yy yy
Thesolutionsetis17 , 26
109.
2 88443 24 811284727 882 y
42 24 42 22 56 56 560 230
424 4 5(3)65(3)6 933
Thesolutionsetis 2,3
Neitherofthesevaluescausesadenominatorto equalzero,sothesolutionsetis
113. Solvetheequation4 41100 ss
40 11004 1100401100 11004 27544000 ss
Let2,sothat. usus 227544000uu
2 275275414400 2 27593,225 2 15.1638or290.1638 u uu
Since us ,itmustbepositive,so 2215.1638229.94 su
Thedistancetothewater'ssurfaceis approximately229.94feet.
114. 2 4 25 TLH
Let4 T and10 H ,andsolvefor L.
2 4 4 44 4 4(10) 25 44 44 2564 64 L L L L L
Thecrushingloadis64tons.
115. 2 32 Tl
Let16.5 T andsolvefor l
equalzero,sothesolutionsetis 1331 ,.56
118.
x x x x x x x x
Thesolutionsetis 11
79 2 5510 1449 101010 41 102 4 10 12313 123130 121330 00
xxx xxx xxx xx
Tosolve1 2 121330 xx ,let1 2 ux 11 22 Then2121330 (43)(31)0 31 or 43 91 169 uu uu uxx xx
Thesolutionsetis 910,,169
119. 63 33 22 28270 (27)(1)0 (3)(39)(1)(1)0
zz zz zzzzzz 30or10 3or1 zz zz
Chapter 1: Equations and Inequalities
abc x i Also abc x i Thesolutionsetis133333,1,,.
120. Answerswillvary.Oneexample:11. x
121. Answerswillvary.Oneexample:20. xx
122. Answerswillvary.
123. Janedidnotcheckhersolutionsandincludedthe extraneoussolution,1 x
Section 1.5
1. 2 x
2. False.
3. closedinterval
4. multiplicationproperties(forinequalities)
5. True.Thisfollowsfromtheadditionproperty forinequalities.
6. True.Thisfollowsfromtheadditionproperty forinequalities.
7. True;.Thisfollowsfromthemultiplication propertyforinequalities.
8. False.Sincebothsidesoftheinequalityare beingdividedbyanegativenumber,thesense, ordirection,oftheinequalitymustbereversed. Thatis, ab cc .
9. True
10. False;eitherorbothendpointscouldbeanyreal number.
11. d
12. c
13. Interval: 0,2 Inequality:02 x
14. Interval: 1,2 Inequality:12 x
15. Interval: 2, Inequality:2 x
Thesolutionsetis 3
16. Interval: ,0 Inequality:0 x
17. Interval: 0,3 Inequality:03 x
18. Interval: 1,1 Inequality:11 x
19. a. 35 3353
c.
20. a. 21 2313
c.
21. a. 43 4333
b. 43 4535 18
c.
d.
43 3433
43 2423 86
22. a. 35 3353 02
b. 35 3555 810
c.
35 3335 915
d. 35 2325 610
23. a. 212 21323 245 x x x
b. 212 21525 243 x x x
c.
d.
212 32132 636 x x x
212 22122 424 x x x
24. a. 125 12353 428 x x x b. 125 12555 420 x x x
c.
d.
25. [0,4]
125 31235 3615 x x x
125 21225 2410 x x x
Chapter 1: Equations and Inequalities
26. (–1,5)
27. [4,6)
(–2,0)
25 x
35. 43 x 36. 01 x
40. 8 x
41. If5,then50. xx
42. If4,then40. xx
43. If4,then40. xx
44. If6,then60. xx
45. If4,then312. xx
46. If3,then26. xx
47. If6,then212. xx
48. If2,then48. xx
49. If5,then420. xx
50. If4,then312. xx
51. If840,then5. xx
52. If312,then4. xx
53. If13,then6. 2 xx
54. If11,then4. 4 xx
55. If11 05,then05 x x
56. 04,11then0 4 x x
57. 50,11then0 5 x x
58. 010,11 then010 x x
59. 15 1151 4 x x x
4or(,4) xx
60. 61 6616 7 x x x
Thesolutionsetis 7or(,7) xx .
65. 313 24 2 xx x x
Section 1.5: Solving Inequalities
Thesolutionsetis 2or[2,) xx .
66. 223 5 xx x
Thesolutionsetis 5or[5,) xx
61. 357 510 2
x x x
Thesolutionsetis 2or[2,) xx .
62. 235 33 1 x x x
Thesolutionsetis 1or[1,) xx
63. 372 39 3 x x x
Thesolutionsetis 3or(3,) xx .
64. 251 24 2 x x x
Thesolutionsetis 2or(2,) xx
67. 2(3)8 268 214 7 x x x x
Thesolutionsetis 7or(7,) xx .
68. 3(1)12 3312 315 5 x x x x
Thesolutionsetis 5or(,5) xx
69. 43(1)3 4333 313 32 2 3 x x x x x
Thesolutionsetis22 or, 33 xx
Chapter 1: Equations and Inequalities
70. 84(2)2 8842 42 60 0 xx xx xx x x
Thesolutionsetis
71. 1(4)8 2 128 2 110 2 20 xx xx x x
Thesolutionsetis 20or(,20) xx
72. 1 34(2) 3 3412 33 9122 814 7 4 xx xx xx x x
Thesolutionsetis77or(,)44 xx
73. 1 24 24 34 4 3 xx xx x x
Thesolutionsetis44or,33 xx
74. 2 36 212 12 xx xx x
Thesolutionsetis 12or12, xx
x x x
75. 0375 7312 74 3
Thesolutionsetis774or,433
76. 42210 228 14 x x x
Thesolutionsetis
77. 5432 932 32 3 x x x
14or1,4xx .
Thesolutionsetis223or,333 xx
78. 3329 626 33 x x x
Thesolutionsetis
33or3,3xx
79. 21 30 4 12210 1121 111 22 x x x x
Thesolutionsetis111 22 xx
or 111 , 22
80. 32 04 2 0328 236 22 3 x x x x
Thesolutionsetis222or,233 xx
81. 1 114 2 1 03 2 06or60 x x xx
Thesolutionsetis
60or6,0xx
82. 1 011 3 1 10 3 30or03 x x xx
Thesolutionsetis 03or0,3xx
Section 1.5: Solving Inequalities
83. 22 (2)(3)(1)(1) 61 61 5 5 xxxx xxx x x x
Thesolutionsetis 5or,5 xx
84. 22 (1)(1)(3)(4) 112 112 11 11 xxxx xxx x x x
Thesolutionsetis 11or,11 xx
85. 2 22 (43)(21) 43441 341 1 1 xxx xxxx xx x x
Thesolutionsetis 1or1, xx .
86. 2 22 (95)(31) 95961 561 1 xxx xxxx xx x
Thesolutionsetis 1or,1 xx
Chapter 1: Equations and Inequalities
Thesolutionsetis1515
Thesolutionsetis1111
Thesolutionsetis11or,22 xx
91. 1 147 170 14 17(14)0 14 6280 14
x x x x x x
Thezerosandvalueswheretheexpressionis undefinedare31 144 and xx 3311 141444 22 100 422 33
Interval(,)(,)(,) Number01 Chosen Valueof6
Conclusion f NegativePositiveNegative Wewanttoknowwhere()0 fx ,sothe solutionsetis 31 144xxorx or,using intervalnotation,31 144 [,).Notethat14isnotin thesolutionsetbecause14isnotinthedomain of f
Thesolutionsetis 31 144 ,
92. 1 2353 230 (35) 23(35)0 2(35) 1790 (35) x x x x x x
Thezerosandvalueswheretheexpressionis undefinedare175 93 and xx
171755 9933 17 5
Interval Number Chosen Valueof Conclusion (,)(,)(,) 21.70 117 f PositiveNegativePositive
Wewanttoknowwhere()0 fx ,sothe solutionsetis 175 93xxorx or, usingintervalnotation,175 93 [,).Notethat 5 3isnotinthesolutionsetbecause27isnotin thedomainof f
Thesolutionsetis 175 93 ,
93. 023 5 0223 and 5 x xx
Since20 x ,thismeansthat0 x .Therefore, 23 5 2355 5 103 10 3 x xx x x x
Thesolutionsetis1010or,33 xx
94. 042 3 0442 and 3 x xx
Since40 x ,thismeansthat0 x .Therefore, 42 3 4233 3 122 6 x xx x x x
Section 1.5: Solving Inequalities
Thesolutionsetis 6or6, xx
95. 02411 2 011 242 0111 and 24242 x x xx
Since10 24 x ,thismeansthat240 x Therefore, 11 242 11 2(2)2 112(2)2(2)2(2)2 12 3 x x xx x x x
Thesolutionsetis
3or3, xx
96. 03611 3 011 363 0111 and 36363 x x xx
Since10 36 x ,thismeansthat360 x Therefore, 11 363 11 3(2)3 113(2)3(2)3(2)3 12 1 x x xx x x x
Thesolutionsetis
1or1, xx .
Chapter 1: Equations and Inequalities
97. If11, x then 14414 345 x x So,3and5. ab
98. If32, x then 36626 964 x x So,9and4. ab
99. If23, x then 4(2)4()4(3) 1248 x x So,12and8. ab
100. If40, x then
11140 222 1 20 2 x x
So,2and0. ab
101. If04, x then 2(0)2()2(4) 028 032383 32311 x x x x
So,3and11. ab
102. If33, x then 2(3)2()2(3) 626 612161 7125 5127 x x x x x
So,5and7. ab
103. If30, x then 34404 144 111 44 111 44 x x x x
So,1and1. 4 ab
104. If24, x then 26646 462 111 462 111 264 x x x x
So,11 and. 24 ab
105. If6312, x then 222 2 6312 333 24 24 416 x x x x
So,4and16. ab
106. If026, x then 222 2 026 222 03 03 09 x x x x
So,0and9. ab
107. 36 x
Weneed360 36 2 x x x
Tothedomainis 2 xx or 2,.
108. 82 x
Weneed820 28 4 x x x
Tothedomainis 4 xx or 4,.
109. 21<youngadult'sage<30
110. 40≤middle-aged<60
111. a. Let x =ageatdeath. 3052.2 82.2 x x Therefore,theaveragelifeexpectancyfora 30-year-oldmalein2018willbegreater thanorequalto82.2years.
b. Let x =ageatdeath.
3055.8
85.8 x x
Therefore,theaveragelifeexpectancyfora 30-year-oldfemalein2018willbegreater thanorequalto85.8years.
c. Bythegiveninformation,afemalecan expecttolive85.882.23.6 yearslonger.
112. 20 VT
80º120º
80º120º 20 16002400 T V V
Thevolumerangesfrom1600to2400cubic centimeters,inclusive.
113. Let P representthesellingpriceand C representthecommission.
Calculatingthecommission:
45,0000.25(900,000)
45,0000.25225,000
0.25180,000
Calculatethecommissionrange,giventheprice range:
9001100 9008158151100815 85815285 0.22(85)0.228150.22(285) 18.700.2281562.7
Theamountwithheldvariesfrom$104.32to $148.32,inclusive.
116. Let x representthelengthoftimeSueshould exerciseontheseventhday. 20040450502535300 200195300 5105 x x x
SuewillstaywithintheACSMguidelinesby exercisingfrom5to105minutes.
117. Let K representthemonthlyusageinkilowatthoursandlet C representthemonthlycustomer bill.
Calculatingthebill:0.100625 CK
900,0001,100,000
0.25(900,000)0.250.25(1,100,000)
225,0000.25275,000
225,000180,0000.25180,000275,000180,000
45,00095,000 P P P P C
Theagent'scommissionrangesfrom$45,000to $95,000,inclusive.
45,000 900,000 0.055% to95,0001,100,0000.0868.6%, inclusive.
Asapercentofsellingprice,thecommission rangesfrom5%to8.6%,inclusive.
114. Let C representthecommission. Calculatethecommissionrange: 250.4(200)250.4(3000) 1051225 C C
Thecommissionsareatleast$105andatmost $1225.
115. Let W =weeklywagesand T =taxwithheld. Calculatingthewithholdingtaxrange,giventhe rangeofweeklywages:
Calculatingtherangeofkilowatt-hours,given therangeofbills: 140.69231.23
140.69250.1006231.23
115.690.1006206.23 11502050
C W K K
Theusagevariesfrom1150.00kilowatt-hoursto 2050.00kilowatt-hours,inclusive.
118. Let W representtheamountofsewer/waterused (inthousandsofgallons).Let C representthe customercharge(indollars).
Calculatingthecharge: 23.550.40 CW
Calculatingtherangeofwaterusage,giventhe rangeofcharges: 30.3536.75
30.3523.550.4036.75
6.80.4013.2 1733
C W K K
Therangeofwaterusagerangedfrom17,000to 33,000gallons.
Chapter 1: Equations and Inequalities
119. Youhavealreadyconsumed22gramsoffat. LetCrepresentthenumberofcookies.Thenwe havethefollowingequation: 22547 525 5
Youmayeatupto5cookiesandkeepthetotal fatcontentofyourmealnotmorethan47g.
120. Youhavealreadyconsumed145gramsof sodium.Let H representthenumberof hamburgers.Thenwehavethefollowing equation: 1453801285
Youmayeatupto3hamburgersandkeepthe totalsodiumcontentofyourmealnotmorethan 1285g.
121. a. Let T representthescoreonthelasttestand G representthecoursegrade. Calculatingthecoursegradeandsolvingfor thelasttest: 68828789 5 326 5 5326 5326
Calculatingtherangeofscoresonthelast test,giventhegraderange: 8090 4005450 745326124 74124 G G
TogetagradeofB,youneedatleasta74 onthefifthtest.
b. Let T representthescoreonthelasttestand G representthecoursegrade. Calculatingthecoursegradeandsolvingfor thelasttest: 688287892 6 3262 6 163 3 3163 GT GT GT TG
Calculatingtherangeofscoresonthelast
test,giventhegraderange: 8090 2403270 773163107 77107 G G G T TogetagradeofB,youneedatleasta77 onthefifthtest.
122. Let T representthetestscoresofthepeoplein thetop2.5%. 1.96(12)100123.52 T Peopleinthetop2.5%willhavetestscores greaterthan123.52.Thatis,123.52 T or (123.52,).
123. Since ab , and 2222 and 22222222 and 22 abab aaababbb abab ab
So,2ab ab .
124. Fromproblem123,2ab ab ,so 2 ,2222 ababababadaa and 2 ,2222 ababbabbadbb
Therefore,2ab isequidistantfrom a and b.
125. If0,then ab
22 22 22 0and0 and and ababab ababab ababab
Therefore, aabb
126. Showthat 2 abab 2 12 22 10,since. 2 ababaabb abab
Therefore,2abab
Section 1.6: Equations and Inequalities Involving Absolute Value
127. For1111 0,2 ab hab
11 2 11 2 2 ba hh hab bah ab hab ab
22 22() 2 ()0 ababaabhaa abab abaababa abab aba ab
. 22 2()2 2 ()0 bhbabbabab abab abbabbab abab bba ab
Therefore, hb
,andwehave ahb
128. Showthat
2 (geometric2 mean) arithmeticmean1() 2 ab h ab
2 1111 2 211 2 2 1() 2 hab ba habab hab ab habab abab
129. 4235 3 312695 31269and695 33514 114 4
.
Thisisequivalentto14 1 5 x .Thesolution set,inintervalnotation,is14 1,5
130. Thelargestvalueof223 x occursatthelargest valuefor x 259 34 43 34or43
x x x xx
Thelargestvaluefor223 x is 2 2(4)332329 .
131. Answerswillvary
132. Answerswillvary.Onepossibility: Nosolution: 46252 xxx Onesolution: 35231321 xxx
133. Since20 x ,wehave 2 2 101 11 x x Therefore,theexpression21 x canneverbe lessthan5
134. Answerswillvary.
Section 1.6
22
{|55} xx
True
True
d
a
Chapter 1: Equations and Inequalities
9. 315 x 315or315 5or5
xx xx Thesolutionsetis{–5,5}.
10. 312 x 312or312 4or4 xx xx
Thesolutionsetis{–4,4}.
11. 235 x 235or235 22or28 1or4 xx xx xx
Thesolutionsetis{–4,1}.
12. 312 x 312or312 33or31 11 or 3 xx xx xx
Thesolutionsetis 1,1 3
13. 14813 145 t t
145or145 44or46 13 or 2 tt tt tt
Thesolutionsetis 1,3 2
14. 1269 123 z z
123or123 22or24 1or2 zz zz zz
Thesolutionsetis 1,2
15. 28 28 x x
28or28 4or4 xx xx
Thesolutionsetis{–4,4}.
16. 1 1 x x 1or1xx Thesolutionsetis{–1,1}.
17. 24 24 2 x x x Thesolutionsetis{2}.
18. 39 39 3 x x x Thesolutionsetis{3}.
19. 83 7 x 21 8 2121 or 88 x xx
Thesolutionsetis 2121 , 88
20. 39 4 x 12 12or12 x xx Thesolutionsetis{–12,12}.
21. 22 35 x 222or2 3535 5630or5630 524or536 2436 or 55 xx xx xx xx
Thesolutionsetis 3624 , 55
22. 11 23 x 111or1 2323 326or326 38or34 84 or 33 xx xx xx xx
Thesolutionsetis 48 , 33 .
23. 21 2 u
Nosolution,sinceabsolutevaluealwaysyieldsa non-negativenumber.
24. 21 v Nosolution,sinceabsolutevaluealwaysyieldsa non-negativenumber.
25. 544 41 41
x x x 41or41 11 or 44
Thesolutionsetis 11 , 44 .
26.
Thesolutionsetis 4,4
27. 290 x 2 2 90 9 3 x x x
Thesolutionsetis 3,3
28. 2160 x 2 2 160 16 4 x x x Thesolutionsetis 4,4
29. 223xx 22 22 23or23 230or230 3102412 or 2 328 or1ornorealsol. 2 xxxx xxxx xxx xxx
Thesolutionsetis 1,3
30. 212xx 22 22 12or12 120or120 3401148 or 2 3147 or4ornorealsol. 2 xxxx xxxx xxx xxx
Thesolutionsetis 4,3
31. 211xx 22 22 11or11 20or0 120or10 1,2or0,1 xxxx xxxx xxxx xxxx
Thesolutionsetis 2,1,0,1.
32. 2322xx 22 22 2 322or322 34or30 340or30 410or0,3 4,1 xxxx xxxx xxxx xxxx xx
Thesolutionsetis 4,3,0,1
Chapter 1: Equations and Inequalities
33. 53 2 35 x x
5353 2or2 3535 53235or53235 53610or53610 7or1113 713 or 11
xx xx xxxx xxxx xx xx Neitherofthesevaluescausethedenominatorto equalzero,sothesolutionsetis 13,7. 11
34. 21 1 34 x x
2121 1or1 3434 21134or21134 2134or2134 3or55 3or1 xx xx xxxx xxxx xx xx
35. 2232 xxxx
2222 22 2 32or32 32or32 50or20 0or(21)0 01 or0or 2 xxxxxxxx xxxxxx xxx xxx xxx
Thesolutionsetis 1,0. 2
36. 2226 xxxx
2222 22 2 26or26 26or26 80or240 0or2(2)0 0or0or2 xxxxxxxx xxxxxx xxx xxx xxx
Thesolutionsetis 2,0.
37. 28 x 828 44 x x
44or4,4xx
38. 315 x 15315 55 x x
55or5,5xx
39. 742 x 742or742 6or6 xx xx
6or6or,66, xxx
40. 26 x 26or26 3or3 xx xx
3or3or,33,xxx
41. 223 21 x x 121 13 x x
13or1,3xx
42. 435 42 x x 242 62 x x
62or6,2xx
43. 324 t 4324 236 22 3 t t t
Section 1.6: Equations and Inequalities Involving Absolute Value
47. 1472 145 x x 5145 644 64 44 33 1or1 22 x x x xx
33 122 or1, xx
44. 257 u 7257 1222 61 u u
44. 232 x 232or232 21or25 15 or 22 xx xx
46. 342 x 342or342 36or32 22 or 3 xx xx xx
222 oror,2, 33 xxx
48. 1241 123 x x 3123 422 42 22 21or12 x x x xx
12or1,2xx
3 2
49. 527 x 527or527 212or22 6or1
xx xx xx
1or6or,16, xxx
50. 231 x 231or231 33or31 11 or 3 xx xx xx
11or1or,1,33 xxx
3
Chapter 1: Equations and Inequalities
51. 451 451 44 x x x
Thisisimpossiblesinceabsolutevaluealways yieldsanon-negativenumber.Theinequality hasnosolution.
52.
54. 21 x 21or21 1or3 1or3 xx xx xx
3or1
55. 32521
56.
57. 95 x
Thisisimpossiblesinceabsolutevaluealways yieldsanon-negativenumber.Nosolution.
58. 30 x
Absolutevalueyieldsanon-negativenumber,so thisinequalityistrueforallrealnumbers,(,).
59. 51 x
Absolutevalueyieldsanon-negativenumber,so thisinequalityistrueforallrealnumbers,(,).
60. 62 x
Thisisimpossiblesinceabsolutevaluealways yieldsanon-negativenumber.Nosolution.
61. 2311 32 x
62. 311 2 15 2 15 2 x x x
63. 8413
Thisisimpossiblesinceabsolutevaluealways yieldsanon-negativenumber.Nosolution.
64. 749
Thisisimpossiblesinceabsolutevaluealways yieldsanon-negativenumber.Nosolution.
65. 720 3 x
Sincetheabsolutevaluecannotbenegative,the onlypossiblesolutionwouldbe:
x x
Sincetheabsolutevaluecannotbenegative,the onlypossiblesolutionwouldbe: 4150 6 4150 415 15 4
x x x x 67. 374 5 x 37374or4 55 3720or3720 313or327 13or9 3
1313or9or,9,33
68. 528 9 x 52528or8 99 5272or5272 277or267 7767 or 22
xxx 69. 511 2 19 2 x x
xx xx xx xx 67776777 oror,, 2222
Absolutevalueyieldsanon-negativenumber,so thisinequalityistrueforallrealnumbers,(,).
Chapter 1: Equations and Inequalities
70. 2311 23 x
71. Atemperature x thatdiffersfrom98.6 Fbyat least1.5F . 98.61.5 98.61.5or98.61.5 97.1or100.1
Thetemperaturesthatareconsideredunhealthy arethosethatarelessthan97.1˚Forgreaterthan 100.1˚F,inclusive.
72. ThelengthLmustbewithin0.0025of5.375 inches..
5.3750.0025
0.00255.3750.0025
5.37255.3775
Thelengthsmustbebetween5.3725and5.3775 inches,inclusive.
73. Thepercentagemustbewithin3.9percentage pointsof64percent.Theinequalitythat representsthiswouldbe:
x x x
60.167.9
643.9 3.9643.9
Theactualpercentageislikelytofallbetween 60.1%and67.9%inclusive.
74. Thespeed x variesfrom707mphbyupto55 mph.
a. 70755 x
b. 5570755 5570755 652762 x x x
Thespeedofsoundisbetween652and762 milesperhour,dependingonconditions.
75. differs1 from3bylessthan2 x . 31 2 11 3 22 57 22 x x x 57 22 xx
76. differsfrom4bylessthan1 x (4)1 41 141 53 x x x x
53xx
77. x differsfrom3bymorethan2. (3)2 32 x x 32or32 5or1 xx xx |5or1xxx
78. x differsfrom2bymorethan3. 23 23or23 1or5 x xx xx
|1or5xxx
79. 13 x 313 35(1)535 248 x x x 2,8ab
80. 25 x 525 54(2)454 921 x x x
9,1ab
Section 1.6: Equations and Inequalities Involving Absolute Value
81. 42 x 242 62 1224 15237 x x x x
82. 31 x 131 24 6312 73113 x x x x
83. 27 x 727 59 15101 111 1510 111 1015 x x x x x
1,115
84. 13 x 313 42 157 111 57 111 75 x x x x x
1,1 7 ab
85. Giventhat0,0, ab and ab ,show that ab Notethat
bababa .
Sincemeans0 abba ,wehave
0 bababa Therefore,0whichmeans. baab
86. Showthat aa Weknow0 a .Soif a <0,thenwehave
0whichmeans. aaaa .Now,if 0,then aaa .So aa .
87. Prove abab . Notethat2ababab .
Case1:If0,then, ababab so
22 22 2 2 2 byproblem86 abababab aabb aabb ab
Thus,22 . abab abab
Case2:If
0,then, ababab so
abababab abab aabb aabb ab Thus,22 abab abab
22 22 2 2 2 byproblem86
88. Prove abab
aabbabb bytheTriangle Inequality,so aabb whichmeans abab Therefore, abab
89. Giventhat a >0, 2 20 0 xa xa xaxa
If xa ,then0 xa and 20xaa .Therefore, 0 xaxa ,whichisacontradiction.
If axa ,then02xaa and 20 axa Therefore, 0 xaxa .
Chapter 1: Equations and Inequalities
If xa ,then20 xaa and 0 xa .Therefore, 0 xaxa , whichisacontradiction.Sothesolutionsetfor 2is xaxaxa .
90. Giventhat a >0, 2 20 0 xa xa xaxa
If xa ,then0 xa and 20xaa
Therefore, 0 xaxa .
If axa ,then02xaa and 20 axa ..Therefore, 0 xaxa ,whichisacontradiction.
If xa ,then20 xaa and 0 xa .Therefore, 0 xaxa
Sothesolutionsetfor2xa is <or.xxaxa
91. 21
95. 216 1616 44 x x x
Thesolutionsetis 44xx .
96. 29 99 33 x x x
Thesolutionsetis 33xx .
97. 24 4or4 2or2 x xx xx
Thesolutionsetis 2or2xxx
98. 216 16or16 4or4 x xx xx
Thesolutionsetis 4or4xxx
99. 3214 3214or3214 xx xxxx
11 11 x x x
Thesolutionsetis 11xx .
92. 24 44 22 x x x
Thesolutionsetis 22xx
93. 29 9or9 3or3 x xx xx
Thesolutionsetis 3or3xxx
94. 21 1or1 1or1 x xx xx
Thesolutionsetis 1or1xxx
3214 3421 xx xx 2134or2134 5or2134 5or53 53 or 5 xxxx xxx xx xx or 3214 3421 xx xx 2134or2134 3or2134 3or55 3or1 xxxx xxx xx xx Thevalues3and3 5 areextraneous.
Thesolutionsetis 1,5.
0,1
101. 2513 2513 18
xx xx x x 432 36 2
xx xx x or25(13) 2513 38 8 3
y y y
y y y or 432 32 2 3
Thevalueof y x islargestusing x =18and y =2, so 21 189 y x
102. Since0 x forallrealnumbers,then 0 1 x x and11 1 x x .Thismeans 113 12 0111 12 111 12 111 12 112 01
x x x x x x x x Therefore11 x .Thesolutionsetin intervalnotationis 1,1
103 – 105. Answerswillvary.
Section 1.7
1. mathematicalmodeling
2. interest
3. uniformmotion
4. False;theamountchargedfortheuseof principalistheinterest.
5. True;thisistheuniformmotionformula.
6. a
7. b
8. c
9. Let A representtheareaofthecircleand r the radius.Theareaofacircleistheproductofπ timesthesquareoftheradius:2 Ar
10. Let C representthecircumferenceofacircle and r theradius.Thecircumferenceofacircle istheproductofπtimestwicetheradius: 2 Cr
11. Let A representtheareaofthesquareand s the lengthofaside.Theareaofthesquareisthe squareofthelengthofaside:2 As
12. Let P representtheperimeterofasquareand s thelengthofaside.Theperimeterofasquareis fourtimesthelengthofaside:4Ps
13. Let F representtheforce, m themass,and a theacceleration.Forceequalstheproductofthe masstimestheacceleration: Fma
14. Let P representthepressure, F theforce,and A thearea.Pressureistheforceperunitarea: PF A
15. Let W representthework, F theforce,and d thedistance.Workequalsforcetimesdistance: WFd
16. Let K representthekineticenergy, m themass, and v thevelocity.Kineticenergyisone-half theproductofthemassandthesquareofthe velocity:2 1 2 Kmv
17. C totalvariablecostindollars, x number ofdishwashersmanufactured:150Cx
18. R totalrevenueindollars, x numberof dishwasherssold:250Rx
19. Let x representtheamountofmoneyinvestedin bonds.Then50,000 x representstheamount ofmoneyinvestedinCD's.Sincethetotal interestistobe$6,000,wehave:
0.150.07(50,000)6,000
1000.150.07(50,000)6,000100
157(50,000)600,000
15350,0007600,000
0.150.07(50,000)7,000
1000.150.07(50,000)7,000100
157(50,000)700,000 15350,0007700,000 8350,000700,000 8350,000 43,750
$43,750shouldbeinvestedinbondsat15%and $6,250shouldbeinvestedinCD'sat7%.
21. Let x representtheamountofmoneyloanedat 8%.Then12,000 x representstheamountof moneyloanedat18%.Sincethetotalinterestis tobe$1,000,wehave:
0.080.18(12,000)1,000
1000.080.18(12,000)1,000100 818(12,000)100,000 8216,00018100,000 10216,000100,000 10116,000 11,600 xx xx
x x x
$11,600isloanedat8%and$400isat18%.
22. Let x representtheamountofmoneyloanedat 16%.Then1,000,000 x representstheamount ofmoneyloanedat19%.Sincethetotalinterest istobe$1,000,000(0.18),wehave:
0.16190,0000.19180,000 0.03190,000180,000 0.0310,000
0.160.19(1,000,000)1,000,000(0.18) 10,000 0.03 $333,333.33 xx x x x xx x
Wendycanlend$333,333.33at16%.
x x x
8350,000600,000
8250,000 31,250
$31,250shouldbeinvestedinbondsat15%and $18,750shouldbeinvestedinCD'sat7%.
20. Let x representtheamountofmoneyinvestedin bonds.Then50,000 x representstheamount ofmoneyinvestedinCD's.Sincethetotal interestistobe$7,000,wehave:
23. Let x representthenumberofpoundsofEarl Graytea.Then100 x representsthenumberof poundsofOrangePekoetea. 64(100)5.50(100) 64004550 2400550 2150 75
xx xx x x x 75poundsofEarlGrayteamustbeblendedwith 25poundsofOrangePekoe.
24. Let x representthenumberofpoundsofthe firstkindofcoffee.Then100 x representsthe numberofpoundsofthesecondkindofcoffee. 2.755(100)4.10(100)
2.755005410
2.25500410 2.2590
40poundsofthefirstkindofcoffeemustbe blendedwith60poundsofthesecondkindof coffee.
25. Let x representthenumberofpoundsof cashews.Then60 x representsthenumberof poundsinthemixture.
94.50(60)7.75(60) 92707.75465 1.25195
156poundsofcashewsmustbeaddedtothe60 poundsofalmonds.
26. Let x representthenumberofcaramelsinthe box.Then30 x representsthenumberof cremesinthebox.
RevenueCostProfit
12.500.250.45(30)3.00
12.500.2513.50.453.00
12.5013.50.203.00
12.5013.500.203.00
Theboxshouldcontain20caramelsand10 cremes.
27. Let r representthespeedofthecurrent. 2016 1 6033 1516 1 6044
Sincethedistanceisthesameineachdirection: 1616 34 4(16)3(16) 644483 167 162.286 7 rr
RateTimeDistance
Thespeedofthecurrentisapproximately2.286 milesperhour.
28. Let r representthespeedofthemotorboat.
RateTimeDistance Upstream3553
Downstream32.52.53 rr rr
Thedistanceisthesameineachdirection: 5(3)2.5(3) 5152.57.5 2.522.5 9 rr rr r r
Thespeedofthemotorboatis9milesperhour.
29. Let r representthespeedofthecurrent.
RateTimeDistance 10 Upstream1510 15 10 Downstream1510 15 r r r r
Sincethetotaltimeis1.5hours,wehave: 2 2 2 2 10101.5 1515 10(15)10(15)1.5(15)(15) 15010150101.5(225) 3001.5(225) 200225 250 (5)(5)0 5or5 rr rrrr rrr r r r rr rr
Speedmustbepositive,sodisregard5 r Thespeedofthecurrentis5milesperhour.
30. Let r representtherateoftheslowercar.Then 10 r representstherateofthefastercar.
RateTimeDistance
Slowercar3.53.5 Fastercar103310 rr rr
Chapter 1: Equations and Inequalities
3.53(10) 3.5330 0.530
Theslowercartravelsatarateof60milesper hour.Thefastercartravelsatarateof70miles perhour.Thedistanceis(70)(3)=210miles.
31. Let r representKaren’snormalwalkingspeed. RateTimeDistance
With50walkway2.550 2.5
Against50walkway2.550 2.5
Sincethetotaltimeis48seconds:
50(2.5)50(2.5)48(2.5)(2.5) 501255012548(6.25) 10048300 048100300 0122575
Speedmustbepositive,sodisregard1.67
Karen’normalwalkingspeedisapproximately 3.75feetpersecond.
32. Let r representthespeedoftheairportwalkway. RateTimeDistance
Walking280with1.5280
Standing280still280
Walkingwiththewalkwaytakes60secondsless timethanstandingstillonthewalkway:
270or20 7or2 2 rr rr
Speedmustbepositive,sodisregard7 2 r
Thespeedoftheairportwalkwayis2metersper second.
33. Let w representthewidthofaregulationdoubles tenniscourt.Then26 w representsthelength. Theareais2808squarefeet: 2 2 2 (26)2808 262808 2628080 314040 (39)(36)0 ww ww ww ww ww
390or360 39or36 ww ww
Thewidthmustbepositive,sodisregard39 w Thewidthofaregulationdoublestenniscourtis36 feetandthelengthis2(36)+6=78feet.
34. Let t representthetimeittakestheBrotherHLL8350CDWtocompletetheprintjobalone. Then9 t representsthetimeittakestheXerox VersaLinkC500tocompletetheprintjobalone. 1 1 9 1 20 TimePartofjobdone todojobinoneminute Brother Xerox9 Together20
t t t t 2 2 111 920 20(9)20(9) 20180209 031180 0(36)(5)
Section 1.7: Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications
360or50 36or5
Timemustbepositive,sodisregard5 t
TheBrotherHL-L8350CDWtakes36minutesto completethejobalone,printing144040 36 pagesperminute.XeroxVersaLinkC500takes 36+9=45minutestocompletethejobalone, printing144032 45 pagesperminute.
35. Let t representthetimeittakestodothejob together.
Workingtogether,thejobcanbedonein12 minutes.
36. Let t representthetimeittakesApriltodothe jobworkingalone.
37. l lengthofthegarden w widthofthegarden
a. Thelengthofthegardenistobetwiceits width.Thus,2lw
Thedimensionsofthefenceare4 l and 4 w
Theperimeteris46feet,so: 2(4)2(4)46 2(24)2(4)46 482846 61646 630 5 lw ww ww w w w
Thedimensionsofthegardenare5feetby 10feet.
b. Area51050 lw squarefeet
c. Ifthedimensionsofthegardenarethesame, thenthelengthandwidthofthefenceare alsothesame(4) l .Theperimeteris46 feet,so: 2(4)2(4)46 282846 41646 430 7.5 ll ll l l l
Thedimensionsofthegardenare7.5feetby 7.5feet.
d. Area7.5(7.5)56.25 lw squarefeet.
38. l lengthofthepond w widthofthepond
a. Thepondistobeasquare.Thus, lw . Thedimensionsofthefencedareaare6 w oneachside.Theperimeteris100feet,so: 4(6)100 424100 476 19 w w w w Thedimensionsofthepondare19feetby 19feet.
b. Thelengthofthepondistobethreetimes thewidth.Thus,3lw .Thedimensionsof thefencedareaare6and6 wl .The perimeteris100feet,so:
Chapter 1: Equations and Inequalities
2(6)2(6)100
2(6)2(36)100 212612100 824100 876 9.5 3(9.5)28.5
Thedimensionsofthepondare9.5feetby 28.5feet.
c. Ifthepondiscircular,thediameteris d and thediameterofthecirclewiththepondand thedeckis6 d
d 3 3
Theperimeteris100feet,so: (6)100 6100
Thediameterofthepondis25.83feet.
d. 2 square Area19(19)361ft lw
Thecircularpondhasthelargestarea.
39. Let t representthetimeittakesforthedefensive backtocatchthetightend. 1002525 1233 100 10 TimetorunDistance 100yards Def. Back TimeRate
Sincethedefensivebackhastorun5yards farther,wehave:
Thedefensivebackwillcatchthetightendatthe 45yardline(15+30=45).
40. Let x representthenumberofhighwaymiles traveled.Then30,000 x representsthenumber ofcitymilestraveled.
5240,0008180,000
Thereseisallowedtoclaim20,000milesasa businessexpense.
41. Let x representthenumberofgallonsofpure water.Then1 x representsthenumberof gallonsinthe60%solution.
%gallons%gallons%gallons 01(1)0.60(1) 10.60.6 0.40.6 42 63 xx x x x
2 3gallonofpurewatershouldbeadded.
42. Let x representthenumberofliterstobe drainedandreplacedwithpureantifreeze. %liters%liters%liters 10.40(15)0.60(15) 60.409 0.603 5 xx xx x x
5litersshouldbedrainedandreplacedwithpure antifreeze.
43. Let x representthenumberofouncesofwater tobeevaporated;theamountofsaltremainsthe same.Therefore,weget 2 3 0.04(32)0.06(32) 1.281.920.06 0.060.64 0.64643210 0.0663 x x x x 2 3 1010.67 ouncesofwaterneedtobe evaporated.
44. Let x representthenumberofgallonsofwater tobeevaporated;theamountofsaltremainsthe same.
0.03(240)0.05(240)
96gallonsofwaterneedtobeevaporated.
45. Let x representthenumberofgramsofpure gold.Then60 x representsthenumberof gramsof12karatgoldtobeused. 12(60)(60)23 300.540 0.510 20 xx
20gramsofpuregoldshouldbemixedwith40 gramsof12karatgold.
46. Let x representthenumberofatomsofoxygen. 2 x representsthenumberofatomsofhydrogen. 1 x representsthenumberofatomsofcarbon. 2145 444 11 xxx x
Thereare11atomsofoxygenand22atomsof hydrogeninthesugarmolecule.
47. Let t representthetimeittakesforMiketo catchupwithDan.Sincethedistancesarethe same,wehave: 11(1) 69
MikewillpassDanafter2minutes,whichisa distanceof1 3 mile.
48. Let t representthetimeofflightwiththewind. Thedistanceisthesameineachdirection: 330270(5) 3301350270 6001350
Thedistancetheplanecanflyandstillreturn safelyis330(2.25)=742.5miles.
49. Let t representthetimetheauxiliarypump needstorun.Sincethetwopumpsareemptying onetanker,wehave:
Theauxiliarypumpmustrunfor2.25hours.It mustbestartedat9:45a.m.
50. Let x representthenumberofpoundsofpure cement.Then20 x representsthenumberof poundsinthe40%mixture.
0.25(20)0.40(20) 50.48 0.63 305 6 xx xx x x
5poundsofpurecementshouldbeadded.
51. Let t representthetimeforthetubtofillwith thefaucetsonandthestopperremoved.Since onetubisbeingfilled,wehave: 1 1520 4360 60 tt tt t
60minutesisrequiredtofillthetub.
52. Let t bethetimethe5horsepowerpumpneeds toruntofinishemptyingthepool.Sincethetwo pumpsareemptyingonepool,wehave: 221 58 4(2)520 84520 47 1.75 t t t t t
The5horsepowerpumpmustrunforan additional1.75hoursor1hourand45minutesto emptythepool.
53. Let t representthetimespentrunning.Then 5 t representsthetimespentbiking. RateTimeDistance Run66 Bike25525(5) tt tt
Chapter 1: Equations and Inequalities
Thetotaldistanceis87miles: 625(5)87 61252587 1912587 1938 2 tt
Thetimespentrunningis2hours,sothe distanceoftherunis6(2)12 miles.The distanceofthebicycleraceis25(52)75 miles.
54. Let r representthespeedoftheeastbound cyclist.Then5 r representsthespeedofthe westboundcyclist.
RateTimeDistance Eastbound66 Westbound566(5) rr rr
Thetotaldistanceis246miles: 66(5)246 6630246 1230246 12216 18 rr rr r r r
Thespeedoftheeastboundcyclistis18miles perhour,andthespeedofthewestboundcyclist is18523 milesperhour.
55. Burke'srateis10012meters/sec.In9.81seconds, Burkewillrun100(9.81)81.75 12 meters.Bolt wouldwinby100-81.75=18.25meters.
56. 2 22 Arrh .Since58.9 A square inchesand6.4 h inches, 2 2 2 22(6.4)58.9 212.858.90 212.858.90
rr rr rr 2 12.8(12.8)4(2)(58.9) 2(2) 12.8635.04 4 3.1or9.5 r rr . Theradiusofthecoffeecanis3.1inches.
57.Lettheindividualtimestocompletetheproject beEforElaine,BforBrian,andDforeither daughter.Usingtherespectiveratesgives 1111111 , 22 EBEDD (or121 2 ED ), and111 4 BD .Fromthefirsttwoequations, 12 BD .Substitutingintothethirdequation gives211 4 DD or 3112hours. 4 D D Then1213hours 122 E E and 1116hours. 124
B B Thecombinedrate ofElaine,Brian,andoneoftheirdaughtersis 1117 361212
projectperhour,soitwilltake them127hourstocompletetheproject.
58. If x =litersoforiginalsolution,thentherewere originally0.2x litersofsaltand0.8litersofpure water.Overtime,thesolutionloses 0.25(0.8)0.2 xx litersofpurewater.Sheadds 20litersofsaltsothetotalamountofsaltis 0.220 x liters.Shealsoadds10litersofpure water,sothetotalamountofpurewateris 0.80.2100.610 xxx liters.Theresulting concentrationis331/3%whichmeans 0.22010.2201 or 0.2200.61030.8303 xx xxx or 0.6600.830150 xxx .Therewere initially150litersofsolutioninthevat.
59. Thespeedofthetrainrelativetothemanis30–4=26milesperhour.Thetimeis 5551 secminhh. 603600720
Thefreighttrainisabout190.67feetlong.
60. Answerswillvary.
61. Let x betheoriginalsellingpriceoftheshirt. ProfitRevenueCost 40.4020240.6040 xxxx
Theoriginalpriceshouldbe$40toensurea profitof$4afterthesale.
Ifthesaleis50%off,theprofitis: 400.50(40)204020200
At50%offtherewillbenoprofit.
62. Let12 and tt representthetimesforthetwo segmentsofthetrip.SinceAtlantaishalfway betweenChicagoandMiami,thedistancesare equal.
TheaveragespeedforthetripfromChicagoto Miamiis49.5milesperhour.
63. Thetimetraveledwiththetailwindwas: 9191.67091hours 550 t
Sincetheywere20minutes 1 3hourearly,the timeinstillairwouldhavebeen:
1.67091hrs20min1.670910.33333hrs 2.00424hrs
Thus,withnowind,thegroundspeedis 919458.53 2.00424 .Therefore,thetailwindis 550458.5391.47knots
64. Itisimpossibletomixtwosolutionswitha lowerconcentrationandendupwithanew solutionwithahigherconcentration.
AlgebraicSolution:
Let x =thenumberoflitersof25%solution.
%liters%liters%liters 0.250.48200.5820 0.259.610.60.58 0.331 3.03liters (notpossible) xx xx x x
Chapter 1 Review
1. 28 3 624 18 x x x
Thesolutionsetis{18}.
2. 2(53)845 106845 6245 6 xx xx xx x
Thesolutionsetis{6}.
Chapter 1: Equations and Inequalities
3. 6 15 566 6 x x xx x
Since x =6doesnotcauseadenominatorto equalzero,thesolutionsetis{6}.
4. 2 (27)20 2720 2725 2725 725 2
x x x x x
Thesolutionsetis725725 , 22
5. 2 2 (1)6 6 06 xx xx xx
2241416 12423 bac
Therefore,therearenorealsolutions.
6. 2 2 (1)6 6 60 (3)(2)0 xx xx xx xx
Thesolutionsetis 3,2. 7.
113 2346 113 1212 2346 6292 811 11 8 x x x x xx x x
Thesolutionsetis 11 8 8.
1361 432 1361 1212 432 3(13)4(6)6 394246 1327 27 13 xx xx xx xx x x
Thesolutionsetis
2 2 (1)(23)3 233 260 (23)(2)0 xx xx xx xx
3or2 2 xx
27 13
Thesolutionsetis 2,3 2 10. 2 2 2 234 0423 (2)(2)4(4)(3) 2(4) 2522213113 884 xx xx x
Thesolutionsetis113113 , 44 11.
32 333 2 2 2 12 12 18 9 3 x x x x x
Check3: x Check3: x 32 3 3 (3)12 912 82 22
32 3 3 (3)12 912 82 22
Thesolutionsetis 3,3
12.
3 2 32 3 3 3 13 1(3) 19 8 82 x x x x x
Check2: x 3 123 93 33
Thesolutionsetis 2
13. 2 2 (1)20 20 1(1)4(1)(2)17 2(1)2 xx xx x
Norealsolution.
14. 42 22 22 540 410 40or10 2or1 xx xx xx xx
Thesolutionsetis 2,1,1,2
15. 2 2 233 233 2396 8120 (2)(6)0 xx xx xxx xx xx
2or6xx
Check2:2(2)32123
Check13: 2 13 23133162 2 x
Thesolutionsetis 13 2 . 17.
16.
Check6:2(6)369693
x
4 444 232 232 2316 213 13 2 x x x x x
444
makestheradicandnegative.
Thesolutionsetis
Thesolutionsetis 1,1 2 21.
222 2222 2222 222 2 2 20 120 xmmxnx xmmxnx xnxmxm nxmxm
222 2 2222 2 22 22 22 (2)(2)41 21 2444 21 2422 2121 211 211 mmnm x n mmmmn n mmnmmn nn mnmn nn
2 2 11 1111 or 11 1111 mnmnm x nnnn mnmnm x nnnn
Thesolutionsetis ,,1,1.11 mm nn nn
22. 222 222 102360 5180 5920 axabxb axabxb axbaxb 590 59 9 5 axb axb b x a
or20 2 2 axb axb b x a
Thesolutionsetis 92 5,,0. bb a aa
Thesolutionsetis 9 . 5
24. 237 x 237or237 24or210 2or5 xx xx xx
Thesolutionsetis{–5,2}.
25. 2329 237 x x 237or237 35or39 5or3 3 xx xx xx
Thesolutionsetis 5,3 3
26. 32 32 2 23 230 230 xx xx xx
20or230 03 or 2 xx xx
Thesolutionsetis 0,3 2 .
27. 32 2 2 258200 254250 2540 xxx xxx xx
2 2 250or40 25or4 5or2 2 xx xx xx
Thesolutionsetis 5,2,2 2
28. 23 2 52 2(23)10(2)5 46205 14 14 xx xx xx x x 14or14, xx
Chapter 1: Equations and Inequalities
29. 23 97 4 362328 33231 3331 22 3133 22 x x x x x
30. 33 26 12 243372 21369 723 x x x x
31. 1 34 2 11 34 22 97 3 22 37 26 x x x x
32. 259 x 259or259 24or214 2or7 xx xx xx
33. 2234 232 2232 430 4 0 3 x x x x x
34. 1234 235 235 235or235 73or33 7 or1 3 7 1or 3 x x x xx xx xx xx
77 1oror,1, 33 xxx
6324623447 iiii
36. 4335212415632 iiiii
37. 2 33393 333933 9393 101010 ii iiiiii i i
12 504824212111 iiiii
32 2 2 232323 412923 51223 10152436 469 iii iii ii iii i
40. 210xx
Thesolutionsetis1313 , 2222 ii
41. 2 220 xx
42.
44. 50,00095 cx
0.080.05(70,000)5000
1000.080.05(70,000)5000100
8350,0005500,000 3350,000500,000
3150,000 50,000 xx xx xx x x x
$50,000shouldbeinvestedinbondsat8%and $20,000shouldbeinvestedinCD'sat5%.
46. Using svt ,wehave3and1100 tv . Findingthedistance s infeet: 1100(3)3300 s
Thestormis3300feetaway.
47. 16003600 I 2 2 2 900 16003600 11 16009003600 91 164 31 42 x x x x
Therangeofdistancesisfrom0.5metersto0.75 meters,inclusive.
48. Let s representthedistancetheplanecantravel. 22 WithwindAgainstwind Rate2503028025030220 Time(/2)(/2) 280220 Dist. ss ss
Sincethetotaltimeisatmost5hours,wehave: /2/2 5 280220 5 560440 11145(6160) 2530,800 1232 ss ss ss s s
Theplanecantravelatmost1232milesor616 milesonewayandreturn616miles.
45. Let x representtheamountofmoneyinvestedin bonds.Then70,000 x representstheamount ofmoneyinvestedinCD's. Sincethetotalinterestistobe$5000,wehave:
Chapter 1: Equations and Inequalities
49. Let t representthetimeittakesthehelicopterto reachtheraft.
RaftHelicopter Rate590
Time Dist.590 tt tt
Sincethetotaldistanceis150miles,wehave: 590150 95150 1.58hours1hourand35minutes tt t t
Thehelicopterwillreachtheraftinabout1hour and35minutes.
50. Giventhat2 12803216 stt ,
a. Theobjecthitsthegroundwhen0 s .
2 2 012803216 2800 1080 tt tt tt
10,8tt
Theobjecthitsthegroundafter8seconds.
b. After4seconds,theobject’sheightis
2 1280324164896 s feet.
51. Let t representthetimeittakesClarissato completethejobbyherself.
ClarissaShawna Timetodo5 jobalone Partofjob11
done5 in1day Timeonjob66 (days) Partofjob66 donebyeach5 person tt tt tt
Sincethetwopeoplepaintonehouse,wehave: 2 2 661 5 6(5)6(5) 63065 7300 (10)(3)0 tt tttt tttt tt tt
10or3tt IttakesClarissa10daystopaintthehousewhen workingbyherself.
52. Let t representthetimeittakesthesmaller pumptoemptythetank.
SmallPumpLargePump Timetodo4 jobalone Partofjob11 done4 in1hr Timeonjob55 (hrs)
Partofjob55 donebyeach4 pump tt tt tt
Sincethetwopumpsemptyonetank,wehave: 2 2 551 4 5(4)5(4) 52054 14200 tt tttt tttt tt
Wecansolvethisequationfor t byusingthe quadraticformula: 2 (14)(14)4(1)(20) 2(1) 1411614229 22 72975.385 12.385or1.615(notfeasible) t tt
Ittakesthesmallpumpapproximately12.385 hours(12hr23min)toemptythetank.
53. Let x representtheamountofwateradded.
%saltTot.amt.amt.ofsalt 10%640.1064 0%0.00
2%640.0264 xx xx
256 xx x x x
0.10640.000.0264
6.41.280.02
5.120.02
256ouncesofwatermustbeadded.
54. Considerthediagram w w+ 2 10
BythePythagoreanTheoremwehave
222 22 2 2 210 44100 24960 2480 860 ww www ww ww ww
8or6ww
Thewidthis6inchesandthelengthis6+2=8 inches.
55. Let x representtheamountofthe15%solution added.
%acidtot.amt.amt.ofacid 40%600.4060 15%0.15 25%600.2560 xx xx
90cubiccentimetersofthe15%solutionmustbe added,producing150cubiccentimetersofthe 25%solution.
56. a. Considerthefollowingdiagram:
4650 42450 426 6.5 s s s s
Thepaintingis6.5inchesby6.5inches. 612.5 s ,sotheframeis12.5inchesby 12.5inches.
b. Considerthefollowingdiagram: w 2w
2262650 ww 1 3 2 3 41221250 626 264 6 28 ww w
Thepaintingis283inchesby143inches. Theframeis2143inchesby1103inches.
57. Let x representtheamountScottreceives.Then 3 4 x representstheamountAlicereceivesand 1 2 x representstheamountTriciareceives.The totalamountis$900,000,sowehave: 31900,000 42 31 44900,000 42 4323,600,000 93,600,000 400,000 xxx xxx xxx x x
So, 33400,000300,000 44 x and 11400,000200,000 22 x Scottreceives$400,000,Alicereceives $300,000,andTriciareceives$200,000.
58. Let t representthetimeittakestheolder machinetocompletethejobbyitself.
OldcopierNewcopier
Timetodo1 jobalone
Partofjob11 done1 in1hr
Timeonjob1.21.2 (hrs)
Partofjob1.21.2 donebyeach1 copier tt tt tt
Sincethetwocopierscompleteonejob,wehave:
Chapter 1: Equations and Inequalities
2 2 2 1.21.21 1 1.2(1)1.2(1) 1.21.21.2 3.41.20 51760 (52)(3)0 tt tttt
0.4or3tt
Ittakestheoldcopier3hourstodothejobby itself.(0.4hourisimpossiblesincetogetherit takes1.2hours.)
59. Let Sr representScott'srateandlet Tr represent Todd'srate.ThetimeforScotttorun95meters isthesameasforToddtorun100meters.
95100
0.95 0.950.95 ST ST SsTT rr rr dtrtrd
IfToddstartsfrom5metersbehindthestart: 105
0.950.95(105)99.75 T ST d dd
a. Theracedoesnotendinatie.
b. Toddwinstherace.
c. Toddwinsby0.25meters.
d. Toendinatie: 1000.95(100) 100950.95 50.95 5.26meters x x x x
e. 95=0.95(100)Therefore,theraceendsina tie.
60. Wewillusetheformulaforinterest, Iprt . Sincesheowed27060attheendoftheloanshe adaccumulated3060ininterestandtheprincipal is24000.
3060(24000)(3) 3060
3(24000)
0.0425 Iprt r r r Theinterestrateis4.25%.
Chapter 1 Test 1. 25 3212 251212 3212 865 25 5 2 xx xx xx x x
Thesolutionsetis 5 2.
2. 2 2 (1)6 6 60 (3)(2)0 xx xx xx xx
30or20 3or2 xx xx
Thesolutionsetis{2,3}.
3.
42 22 340 410 xx xx 22 22 40or10 4or1 2orNotreal xx xx x
Thesolutionsetis{2,2}.
4. 22 2524 252 252 254 29 9 2 x x x x x x
Check:92524 2 9524 424 224 44
Thesolutionsetis 9 2
5.
6.
10. 2259 257 x x
257or257 22or212 1or6 xx xx xx
1or6xxx
or
,16,.
7.
8.
Thisequationhasnorealsolutions.
11. 2 2236262 3339(1) 933 62331 10555 iii iiiiii ii i
12. 2 4450 xx 2 (4)(4)4(4)(5) 2(4) 464481 882 x i i
Thissolutionsetis 11 ,.22 ii
Chapter 1: Equations and Inequalities
13. Let x representtheamountofthe$8-per-pound coffee.
Add263poundsof$8/lbcoffeetoget2 263 poundsof$5/lbcoffee.
Chapter 1 Projects
Project I
Internet-based Project
Project II
1.
2. Allofthetimesgiveninproblem1werein seconds,so T =0.1boardpersecondneedsto usedasthevaluefor T intheequationfoundin problem1.
1 0.1 0.22 0.220.11
40partsperboard
3. T =0.15boardpersecond
1 0.15 0.22 0.220.151
0.030.31 0.030.7 23.3partsperboard p p p p p
Thus,only23partsperboardwillwork.
Forproblems4–6, C isrequested,sosolvefor C first:
n T CnpLM CnpLMTn CnpTLTMTn CnpTnLTMT nLTMT C npT
4. T =0.06, n =3, p =100, M =1, L =5
350.0610.06 0.147sec 31000.06 C
5. T =0.06, n =3, p =150, M =1, L =5
350.0610.06 0.098sec 31500.06 C
6. T =0.06, n =3, p =200, M =1, L =5
350.0610.06 0.073sec 32000.06 C
7. Asthenumberofpartsperboardincreases,the tacttimedecreases,ifalltheotherfactorsremain constant.