Chapter 1: Algebra and Equations
Section 1.1 The Real Numbers
1. True.Thisstatementistrue,sinceeveryinteger canbewrittenastheratiooftheintegerand1.
Forexample,5 5 1
2. False.Forexample,5isarealnumber,and 10 5 2 whichisnotanirrationalnumber.
3. Answersvarywiththecalculator,but 2,508,429,787 798,458,000 isthebest.
4. 0(7)(7)0
Thisillustratesthecommutativepropertyof addition.
5. 6(4)664 tt
Thisillustratesthedistributiveproperty.
6. 3+(–3)=(–3)+3
Thisillustratesthecommutativepropertyof addition.
7. (–5)+0=–5
Thisillustratestheidentitypropertyofaddition.
8. 1 4 (4)()1
Thisillustratesthemultiplicativeinverse property.
9. 8+(12+6)=(8+12)+6
Thisillustratestheassociativepropertyof addition.
10. 1(20)20
Thisillustratestheidentitypropertyof multiplication.
11. Answersvary.Onepossibleanswer:Thesumof anumberanditsadditiveinverseistheadditive identity.Theproductofanumberandits multiplicativeinverseisthemultiplicative identity.
12. Answersvary.Onepossibleanswer:Whenusing thecommutativeproperty,theorderofthe addendsormultipliersarechanged,whilethe groupingoftheaddendsormultipliersis changedwhenusingtheassociativeproperty.
ForExercises13–16,let p =–2, q =3and r =–5.
13. () [][] () 35325(3)3215 31339 pq −+=−−+=−−+ =−=−
14. ()()() 22352816 qr−=+==
15. qr qp 3(5)22 3(2)1 +−− ===− +−
16. 33(3)99 323(2)2(5)6104 q pr === −−−−−+
17. Let r =3.8. APR1212(3.8)45.6% r
18. Let r =0.8. APR1212(0.8)9.6% r
19. Let APR =11. 12 1112 11 12 .9167% APRr r r r
20. Let APR =13.2. 12 13.212 13.2 12 1.1% APRr r r r
21. 3455320517512
22. 2 8(4)(12) Takepowersfirst. 8–16–(–12) Thenaddandsubtractinorderfromlefttoright. 8–16+12=–8+12=4
23. (45)66166660
24. 2(37)4(8) 4(3)(3)(2)
Workaboveandbelowfractionbar.Do multiplicationsandworkinsideparentheses. 2(4)32832244 1261266
25. 2 84(12)
Takepowersfirst. 8–16–(–12)
Thenaddandsubtractinorderfromlefttoright. 8–16+12=–8+12=4
26. 2 (35)2313
Takepowersfirst. –(3–5)–[2–(9–13)]
Workinsidebracketsandparentheses. –(–2)–[2–(–4)]=2–[2+4] =2–6=–4
27.
32 2(3)(2)16 641
Workaboveandbelowfractionbar.Takeroots. 32 2(3)(2)(4) 81
Domultiplicationsanddivisions. 31 622 81
Addandsubtract. 3 12114 222271 777
28. 2 2 6325 613
Takepowersandroots. 363(5)3615213 3613497
29. 20401894587 ,,27,,6.735,4752337691
30. 187385 ,2.9884,85,,10, 63117
31. 12islessthan18.5. 12<18.5
32. –2isgreaterthan–20. –2>–20
33. x isgreaterthanorequalto5.7. 5.7 x
34. y islessthanorequalto–5. 5 y
35. z isatmost7.5. 7.5 z
36. w isnegative. 0 w
37. 62 38. 34.75
39. 3.14
40. 13.33
41. a liestotherightof b orisequalto b
42. b + c = a
43. c < a < b
44. a liestotherightof0
45. (–8,–1)
Thisrepresentsallrealnumbersbetween–8and –1,notincluding–8and–1.Drawparenthesesat –8and–1andaheavylinebetweenthem.The parenthesesat–8and–1showthatneitherof thesepointsbelongstothegraph.
46. [–1,10]
Thisrepresentsallrealnumbersbetween–1and 10,including–1and10.Drawbracketsat –1and10andaheavylinebetweenthem.
47. 2,3
Allrealnumbers x suchthat–2< x ≤ 3 Drawaheavylinefrom–2to3.Usea parenthesisat–2sinceitisnotpartofthegraph. Useabracketat3sinceitispartofthegraph.
48. [–2,2)
Thisrepresentsallrealnumbersbetween–2and 2,including–2,butnotincluding2. Drawabracketat–2,aparenthesisat2,anda heavylinebetweenthem.
49. 2,
Allrealnumbers x suchthat x >–2 Startat–2anddrawaheavylinetotheright. Useaparenthesisat–2sinceitisnotpartofthe graph.
50. (–∞,–2]
Thisrepresentsallrealnumberslessthanor equalto–2.Drawabracketat–2andaheavy linetotheleft.
51. 9129(12)3
52. 848(4)4
53.
54. 6124(6)166(16)22
55. 55 5__5
58. 6(4)46
59. 2828 66 66 66
60. 353(5) 3–515 3515 1515
61. 3535 235 22
63. When a <7, a –7isnegative. So7(7)7aaa
64. When b ≥ c, b – c ispositive. So bcbc
Answerswillvaryforexercises65–67.Sampleanswers aregiven.
65. No,itisnotalwaystruethat. abab For example,let a =1and b =–1.Then, 1(1)00 ab ,but 1(1)112 ab
66. Yes,if a and b areanytworealnumbers,itis alwaystruethat. abba Ingeneral, a – b =–(b – a).Whenwetaketheabsolute valueofeachside,weget () abbaba
67. 22bb onlywhen b =0.Theneachside oftheequationisequalto2.If b isanyother value,subtractingitfrom2andaddingitto2 willproducetwodifferentvalues.
68. Forfemales:|63.5|8.4 x ;formales: |68.9|9.3 x
69. 1;30062007
70. 8;2008,2009,2010,2011,2012,2013,2014, 2015
71. 9;2006,2008,2009,2010,2011,2012,2013, 2014,2015
72. 4;2008,2010,2013,2015
73. 4;2006,2007,2009,2011
74. 8;2006,2007,2009,2010,2011,2012,2013, 2014,2015
75. |10.614.9||4.3|4.3 −=−=
76. |63.1(8.0)||71.1|71.1 −−==
77. |1.063.1||64.1|64.1 −−=−=
78. |10.6(5.7)||16.3|16.3 −−==
79. |5.7(8.0)||2.3|2.3 −−−==
80. |1.0(5.7)||4.7|4.7 −−−==
81. 3;30062010,2015,2016
82. 7;2010,2011,2012,2013,2014,2015,2016
83. 6;2010,2011,2012,2013,2014,2015
84. 3;2014,2015,2016
Section 1.2 Polynomials
1. 6 11.21,973,822.685
2. 11 (6.54)936,171,103.1
3. 186 289.0991339 7
4. 57 .0163339967 9
5. 2 3isnegative,whereas2(3)ispositive.Both 3 3and3(3)arenegative.
6. Tomultiply34and54,addtheexponentssince thebasesarethesame.Theproductof34and 34cannotbefoundinthesamewaysincethe basesaredifferent.Toevaluatetheproduct,first dothepowers,andthenmultiplytheresults.
7. 23235 4444 8. 464610 4444
9. 25257 (6)(6)(6)(6)
565611 (2)(2)(2)(2) zzzz
7 44728555 uuu
4 35320 23 6666 (6) yyyy y
13. degree4;coefficients:6.2,–5,4,–3,3.7; constantterm3.7.
14. degree7;coefficients:6,4,0,0,–1,0,1,0; constantterm0.
15. Sincethehighestpowerof x is3,thedegree is3.
16. Sincethehighestpowerof x is5,thedegree is5.
17. 3232 32548 x xxxxx 3322 342(58) x xxxxx 3213 x xx
18. 32 257482 pppp 32 24(58)(72) pppp 32 2439 ppp
Copyright © 2019 Pearson Education, Inc.
19. 22 438262 yyyy
22 438262 yyyy 22 438262 yyyy
22 42(36)(82) yyyy
2 636 yy
20.
22 725326 bbbb
22 725326 bbbb
22 732256 bbbb
2 41 b
21.
3232 2243281 xxxxx
3232 2243281 xxxxx 3232 2243281 xxxxx
3322 2228(4)(31) xxxxx 2 644 xx
22.
322 391184106 yyyyy
322 391184106 yyyyy
322 394(1110)(86) yyyyy 32 3132114 yyy
23. 2 9261 mmm
2 (9)29(6)(9)(1) mmmmm 32 18549mmm
24. 2 2468 aaa 2 242(6)2(8) aaaaa 32 81216 aaa
25. 2 (35)421 zzz
22 322 32 (3)421(5)421 126320105 121475 zzzzz zzzzz zzz
322 122061035 zzzzz 32 121475 zzz
26. 32(23)43kkkk 3232 243343 kkkkkkk 43232 8621293 kkkkkk 432 8673 kkkk
27. (6k –1)(2k +3) (6)(23)(1)(23) kkk 2 121823 kkk 2 12163 kk
28. (8r +3)(r –1) UseFOIL.
2 8833 rrr 2 853 rr
29. (3y +5)(2y +1) UseFOIL.
2 63105 yyy 2 6135 yy
30. (5r –3s)(5r –4s) 2225201512 rrsrss 22253512 rrss
31. (9k + q)(2k – q) 221892kkqkqq 22187kkqq
32. (.012x –.17)(.3x +.54) =(.012x)(.3x)+(.012x)(.54) +(–.17)(.3x)+(–.17)(.54) 2 .0036.00648.051.0918 xxx 2 .0036.04452.0918 xx
33. (6.2m –3.4)(.7m +1.3) 2 4.348.062.384.42 mmm 2 4.345.684.42 mm
34. 2p –3[4p –(8p +1)]
=2p –3(4p –8p –1) =2p –3(–4p –1) =2p +12p +3 =14p +3
35. 5k –[k +(–3+5k)]
=5k –[6k –3] =5k –6k +3 =–k +3
36. (31)(2)(25)2 xxx
22 35242025 xxxx 22 35242025 xxxx
37. R =5(1000x)=5000x C =200,000+1800x P =(5000x)–(200,000+1800x) =3200x –200,000
38. R =8.50(1000x)=8500x C =225,000+4200x P =(8500x)–(225,000+4200x) =4300x –225,000
39. R =9.75(1000x)=9750x 2 2 260,000(33480325) 33480259,675 Cxx xx
2 2 (9750)(33480259,675) 36270259,675
40. R =23.50(1000x)=23,500x 2 2 145,000(4.23220425) 4.23220144,575 Cxx xx
2 2 (23,500)(4.23220144,575) 4.220,280144,575.
41. a. Accordingtothebargraph,thenetearnings in2007were$673million.
b. Let x =7.
32 32 4.79122.511042863
505.47 xxx −+− =−+− =
4.797122.57110472863
Accordingtothepolynomial,thenetearnings in2007wereapproximately$505million.
42. a. Accordingtothebargraph,thenetearnings in2015were$2757million.
b. Let x =15.
4.7915122.5151104152863 2300.75
Accordingtothepolynomial,thenet earningsin2015wereapproximately$2301 million.
43. a. Accordingtothebargraph,thenetearnings in2012were$1384million.
b. Let x =12.
4.7912122.5121104122863 1022.12 xxx −+− =−+−
Accordingtothepolynomial,thenet earningsin2012wereapproximately$1022 million.
44. a. Accordingtothebargraph,thenetearnings in2013were$8million.
b. Let x =13. ()()() 32 32 4.79122.511042863
4.7913122.5131104132863 1310.13 xxx −+− =−+−
Accordingtothepolynomial,thenet earningsin2013wereapproximately$1,310 million.
45. Let x =17. ()()() 32 32 4.79122.511042863
=−+−
4.7917122.5171104172863 4035.77 xxx
Accordingtothepolynomial,thenetearningsin 2017willbeapproximately$4036million.
46. Let x =18.
32 32 4.79122.511042863
52. Let x =11.
()()()
4.7918122.5181104182863
5254.28
Accordingtothepolynomial,thenetearningsin 2018willbeapproximately$5254million.
47. Let x =19. ()()()
32 9.5(11)401.6(11)6122(11)25,598 5794.9 Let x =16.
32 9.5(16)401.6(16)6122(16)25,598 8456.4
Thus,thecostswere$5794.9millionin2011 and$8456.4millionin2016.Thestatementis true.
4.79122.511042863
32 32
4.7919122.5191104192863
6745.11
Accordingtothepolynomial,thenetearningsin 2019willbeapproximately$6745million.
48. Thefiguresfor2013–2015seemhigh,but plausible.Toseehowaccuratetheseconclusions are,searchStarbucks.comforlaterannual reports.
Forexercises49–52,weusethepolynomial
32 9.5401.6612225,598. xxx
49. Let x =10.
32 9.5(10)401.6(10)6122(10)25,598 4962
Thus,thecostswereapproximately$4962 millionin2010.Thestatementisfalse.
50. Let x =15.
32 9.5(15)401.6(15)6122(15)25,598 7934.5
Thus,thecostswereapproximately$7934.5 millionin2015.Thestatementistrue.
51. Let x =12.
32
9.5(12)401.6(12)6122(12)25,598 6451.6 Let x =15.
32 9.5(15)401.6(15)6122(15)25,598 7934.5
Thus,thecostswere$6451.6millionin2012 and$7934.5millionin2015.Thestatementis false.
Forexercises53–58,weusethepolynomial 32 72.85208216,53259,357. xxx −+−+
53. Let x =7.
32 72.85208216,53259,357
72.85(7)2082(7)16,532(7)59,357 20,663.45 xxx −+−+ =−+−+ =
Thus,theprofitforPepsiCoIncin2007was approximately$20,663million.
54. Let x =10.
32
32 72.85208216,53259,357 72.85(10)2082(10)16,532(10) 59,357 29,387 xxx −+−+ =−+− + =
Thus,theprofitforPepsiCoIncin2010was $29,387million.
55. Let x =12.
32
xxx −+−+ =−+− + =
32
72.85208216,53259,357 72.85(12)2082(12)16,532(12) 59,357 34,896.2
Thus,theprofitforPepsiCoIncin2012was approximately$34,896million.
56. Let x =15.
32
32
72.85208216,53259,357 72.85(15)2082(15)16,532(15) 59,357 33,958.25 xxx −+−+ =−+− + =
Thus,theprofitforPepsiCoIncin2015was approximately$33,958million.
Copyright © 2019 Pearson Education, Inc.
57. Let x =13.
32 32
72.85208216,53259,357
xxx
72.85(13)2082(13)16,532(13) 59,357
36,247.55
Thus,theprofitforPepsiCoIncin2013was approximately$36,248million.
Let x =9.
60. Inorderforthecompanytomakeaprofit, 2 7.25005230,0000Pxx Settheprofitfunctionequalto0andsolve for x. 2 84450215,0000 xx Bythequadraticformula, x ≈ 45or x ≈ –601. Since x representsapositivenumber, x =45.
32 32
72.85(9)2082(9)16,532(9) 59,357 26,103.35 xxx
72.85208216,53259,357
Thus,theprofitforPepsiCoIncin2009was approximately$26,104million.Therefore,the profitwashigherin2013.
58. Bycomparingtheanswerstoproblems55and 56,theprofitwashigherin2012.
59. 2 7.25005230,000Pxx .Hereispartof thescreencapture.
For25,000,thelosswillbe$100,375;
Therefore,between40,000and45,000 calculatorsmustbesoldforthecompanyto makeaprofit.
61. Let x =100(inthousands)
2 7.2(100)5005(100)230,000342,500
d. Theprofitforselling100,000calculatorsis $342,500.
62. Let x =150(inthousands) 2 7.2(150)5005(150)230,000682,750
d. Theprofitforselling150,000 calculatorsis$682,750.
Section 1.3 Factoring
12241212212(2)
2. 5655(1)5(13)5(113) yxyyyxyx
For60,000,thereprofitwillbe$96,220.
Thereisalossatthebeginningbecauseof largefixedcosts.Whenmoreitemsare made,thesecostsbecomeasmallerportion ofthetotalcosts.
Copyright © 2019 Pearson Education, Inc.
5. 32 61218 zzz 2 66(2)6(3) zzzzz
2 623 zzz
6. 32 55510 x xx 2 55(11)5(2) x xxxx
2 5112 xxx
7. 23 3(21)7(21) yy 22 (21)(3)(21)7(21) yyy 2 (21)[37(21)] yy
(21)(3147) yy
(21)(144)
8. 53 (37)4(37) xx 323 (37)(37)(37)(4) xxx
32 (37)(37)4 xx
32 (37)942494 xxx
32 (37)94245 xxx
9. 46 3(5)(5) xx 442 (5)3(5)(5) xxx 42(5)35 xx
42 (5)31025 xxx
42 (5)1028 xxx
10. 24 3(6)6(6) xx 222 22 22 22 22 3(6)(1)3(6)2(6) 3(6)12(6) 36121236 36122472 3622473 xxx xx xxx xxx xxx
11. 254(1)(4) xxxx
12. 276(1)(6) uuuu
13. 2712(3)(4) xxxx
14. 2812(2)(6) yyyy 15.
2632 xxxx 16.
17.
18.
24551 xxxx
22331 xxxx
21243 yyyy
23414 xxxx
20.
22824 uuuu
21.
291427 zzzz
261628 wwww
23. 21024(4)(6) zzzz
24. 21660(6)(10) rrrr
25. 2 294(21)(4) xxxx
26. 2 384(32)(2) wwww
27. 2 15234(34)(51) pppp
28. 2 8143(41)(23) xxxx
29. 2 41615(25)(23) zzzz
30. 2 122915(35)(43) yyyy
31. 2 654(21)(34) xxxx
32. 2 121(41)(31) zzzz
33. 2 102110(52)(25) yyyy
34. 2 1544(52)(32) uuuu
35. 2 654(21)(34) xxxx
36. 2 12710(32)(45) yyyy
37. 2 325351 aaaa
38. 22 6481206820 6(10)(2) aaaa aa
39. 22281(9)(9)(9)xxxx
40. 221772(8)(9) x xyyxyxy
41. 222 2 9124(3)2(3)(2)2 (32) pppp p
42. 2 32(32)(1) rrrr
43. 22310(2)(5) rrttrtrt
44. 22 26(23)(2). aabbabab
45. 2222 2 816()2()(4)(4) (4) mmnnmmnn mn
46. 22 816102485 2(21)(25) kkkk kk
47. 22 412923 uuu
48. 2 22 916343434 pppp
49. 2 25104 pp Thispolynomialcannotbefactored
50.
2 101735123 xxxx
51.
52.
53.
54.
22 492323 rvrvrv
2232874 x xyyxyxy
222 442 x xyyxy
22 1612182869 24323 uuuu uu
55. 2 31330(35)(6) aaaa
56. 2 328(34)(2) kkkk
57. 22 21132(72)(3) mmnnmnmn
58. 2 81100(910)(910) yyy
59. 22421(7)(3) yyzzyzyz
60. 2 499 a Thispolynomialcannotbefactored.
61. 2 12164(118)(118) xxx
62.
22 22 22 2 41449 42()(7)(7) 456196 4(7) zzyy zzyy zzyy zy
63. 333264(4)(4)416aaaaa
64.
33322166(6)636bbbbb
66.
33 33 22 22 827 (2)(3) (23)2233 (23)469 rs rs rsrrss rsrrss
33 33 22 100027 (10)(3) (103)100309 pq pq pqppqq
67. 3 64125 m 33(4)(5) m
22(45)4455 mmm
2 (45)162025 mmm
68. 3 216343 y 33(6)(7) y
2 (67)364249
69. 100033 yz 33(10)() yz 22 (10)(10)(10)()()
22(10)10010yzyyzz
70.
422271025
442222 22 16814949 232349 ababab ababab
81.
4222 6332133 xxxx
isnotthe correctcompletefactorizationbecause233 x containsacommonfactorof3.Thiscommon factorshouldbefactoredoutasthefirststep. Thiswillrevealadifferenceoftwosquares, whichrequiresfurtherfactorization.Thecorrect factorizationis
82. Thesumoftwosquarescanbefactoredwhenthe termshaveacommonfactor.Anexampleis 2222 (3)3999(1) xxx
83. 32 2 322 32 (2)(2)(2) (2)(44) 42848 6128, xxx xxx xxxxx xxx
whichisnotequalto38 x .Thecorrect factorizationis328(2)(24).xxxx
84. Factoringandmultiplicationareinverse operations.Ifwefactorapolynomialandthen multiplythefactors,wegettheoriginal polynomialback.Forexample,wecanfactor 26xx toget(3)(2) xx .Thenifwe multiplythefactors,weget 22 (3)(2)2366 xxxxxxx
Section 1.4 Rational Expressions
1. 2 88 56787 x xxx xx
2. 322 27271 812733 mm mmmm
3. 22 32 25555 35757 pp p ppp
4. 4222 22 18633 24644 yyyy yy
5. 5155(3)5 4124(3)4 mm mm
6. 1055(21)1 20105(21)22 zx zx
7. 4(3)4 (3)(6)6 w www
8. 6(2)6 (4)(2)4 x xxx or 6 4 x
9.
2 32 2 3123(4) 933 4 3 yyyy y yy y y
10. 2 2 154515(3) 933 35(3) 33 5(3) 3 kkkk kk k kk kk k k
11. 2 2 44(2)(2)2 6(3)(2)3 mmmmm mmm mm
12. 2 2 6(3)(2)2 12(4)(3)4 rrrrr rrr rr
13.
2 2 31 233 1111 xx xxx xxx x
14. 22 2 2 442 4222 z zzz zzz z
15. 22 33 38383 6416 2642 aa a aa
16. 2323 44 2102105 918889 uuuu u uuu
17. 3 332 7147667663 116611141114 x xxyxyy y x xx
18. 226216 22217 x yxyxyyy xyxxy
19. 215(2)15155 34(2)(2)12124 abab cababccc
20. 22 4(2)34(2)33 8(2)4(2)22 x wwxw wxwx
21. 1531021533 636102 3(51)3 322(51) 3(51)33 3(51)224
22. 28312283 636312 2(4)3 63(4) 6(4)61 18(4)183
23. 918369(2)3(2) 61215306(2)15(2) 27(2)(2)273 90(2)(2)9010
24. 122461212(2)6(2) 36368836(1)8(1) 23(2) 3(1)4(1) 24(1)4 3(1)3(2)9
25. 22 22 412941220 210210209 4(3)(5)(4) 2(5)(3)(3) 4(3)(5)(4) 2(5)(3)(3) 2(4) 3 aaaaa
26.
2 22 6181216 9624412 6(3)4(34)2(34) 4(3) 3328328 2(34)2 (34)(2)2
27. 22 22 634 1223 (3)(2)(4)(1) (4)(3)(3)(1) (3)(2)(4)(1)2 (4)(3)(3)(1)3 kkkk kkkk kkkk kkkk kkkkk kkkkk
28. 22 22 69 28712 (3)(2)(3)(3) (4)(2)(3)(4) 33344 44434 nnn nnnn nnnn nnnn nnnnn nnnnn
Answerswillvaryforexercises29and30.Sample answersaregiven.
29. Tofindtheleastcommondenominatorfortwo fractions,factoreachdenominatorintoprime factors,multiplyalluniqueprimefactorsraising eachfactortothehighestfrequencyitoccurred.
30. Toaddthreerationalexpressions,firstfactor eachdenominatorcompletely.Then,findthe lowestcommondenominatorandrewriteeach expressionwiththatdenominator.Next,addthe numeratorsandplaceoverthecommon denominator.Finally,simplifytheresulting expressionandwriteitinlowestterms.
31. Thecommondenominatoris35z. 2125171073 757557353535 zzzzzzz
32. Thecommondenominatoris12z. 45445316151 343443121212 zzzzzzz
33. 22(2)(2) 333 224 33 rrrr
34. 3131(31)(31)21 88884 yyyy
35. Thecommondenominatoris5x 414512020 555555 x xx x xxxxx
36. Thecommondenominatoris4r 63643243 44444 2433(8) 44 rr rrrrr rr
37. Thecommondenominatoris m(m –1). 121(1)2 1(1)(1) 2(1) (1)(1) 2(1)22 (1)(1) 32 (1) mm
38. Thecommondenominatoris(y +2)y 8383(2)83(2) 2(2)(2)(2) 8365656 or (2)(2)(2) yyyy yyyyyyyy yyyy yyyyyy
39. Thecommondenominatoris5(b +2).
40. Thecommondenominatoris3(k +1).
41. Thecommondenominatoris20(k –2). 25825 5(2)4(2)20(2)20(2) 82533 20(2)20(2) kkkk kk
42. Thecommondenominatoris6(p +4). 115225 3(4)6(4)6(4)6(4) 22517 6(4)6(4) pppp pp
43. Firstfactorthedenominatorsinordertofindthe commondenominator.
2 2 4331 632 xxxx xxxx
Thecommondenominatoris(x –3)(x–1)(x +2).
22 25 436 25 (3)(1)(3)(2) 2(2)5(1) (3)(1)(2)(3)(2)(1) 2(2)5(1)2455 (3)(2)(1)(3)(1)(2) 71 (3)(1)(2) xxxx xxxx xx xxxxxx xxxx xxxxxx x xxx
44. Firstfactorthedenominatorsinordertofindthe commondenominator.
2 2 31052 2054 mmmm mmmm
Thecommondenominatoris (5)(2)(4) mmm . 22 37 31020 37 (5)(2)(5)(4) 3(4)7(2) (5)(2)(4)(5)(4)(2) 3127141026 (5)(2)(4)(5)(2)(4) mmmm mmmm mm mmmmmm mmm mmmmmm
45. Firstfactorthedenominatorsinordertofindthe commondenominator.
2 2 71234 5632 yyyy yyyy
Thecommondenominatoris (y +4)(y +3)(y +2). 22 22 2 2 71256 2 (4)(3)(3)(2) 2(2)(4) (4)(3)(2)(4)(3)(2) 2(2)(4)244 (4)(3)(2)(4)(3)(2) (4)(3)(2) yy yyyy yy yyyy yyyy yyyyyy
46. Firstfactorthedenominatorsinordertofindthe commondenominator.
2 2 101682 2842 rrrr rrrr
Thecommondenominatoris(8)(2)(4) rrr
22 2222 2 3 101628 3 (8)(2)(4)(2) (4)3(8) (8)(2)(4)(4)(2)(8) 43244324 (8)(2)(4)(8)(2)(4) 420 (8)(2)(4)
Multiplybothnumeratoranddenominatorofthis complexfractionbythecommondenominator, x.
49. 11 x hx h
Thecommondenominatorinthenumeratoris x(x + h) 11() ()()() 1 ()() 11 or ()() xxh x xhh x xhxxhxxh xhx hhhh hh h x xhxxhh xxhxxh
50. 22 11 () x hx h Thecommondenominatorofthenumeratoris ()22 x hx
22 222222 11() ()()() 1 22 22 222 22 2 2222 22 ()1 () 21 () 2(2) ()() 2 () xh x xhxxhxxhx h h xxh h xhx xxxhh h xhx xhhhxh x hxhxhxh xh xhx
51. Thelengthofeachsideofthedartboardis2x,so theareaofthedartboardis24. x Theareaofthe shadedregionis2 x
a. Theprobabilitythatadartwilllandinthe shadedregionis 2 42 x x b. 2 424 x x
52. Theradiusofthedartboardis x +2x +3x =6x, sotheareaofthedartboardis 22 636. x x Theareaoftheshadedregionis2. x
Multiplybothnumeratoranddenominatorbythe commondenominator, y
a. Theprobabilitythatadartwilllandinthe shadedregionis 2 362 x x
b. 2 2 1 3636 x x
53. Thelengthofeachsideofthedartboardis5x,so theareaofthedartboardis225. x Theareaof theshadedregionis2 x
a. Theprobabilitythatadartwilllandinthe shadedregionis 2 252 x x
b. 2 2 1 2525 x x
54. Thelengthofeachsideofthedartboardis3x,so theareaofthedartboardis29. x Theareaofthe shadedregionis2 1 2 x
a. Theprobabilitythatadartwilllandinthe shadedregionis 122 2 22918 x x x x
b. 2 2 1 1818 x x
55. Averagecost=totalcost C dividedbythe numberofcalculatorsproduced.
2 7.26995230,000 1000 xx x
56. Let x =20(inthousands).
2 7.2(20)6995(20)230,000$18.35 1000(20)
Let x =50(inthousands).
2 7.2(50)6995(50)230,000$11.24 1000(50)
Let x =125(inthousands).
2 7.2(125)6995(125)230,000$7.94 1000(125)
57. Let x =13.Then
()() 2 .505134.5871327.6 3.8 131 −+ ≈ +
Theadcostapproximately$3.8millionin 2013
58. Let x =16.Then
()() 2 .505164.5871627.6 4.9 161 −+ ≈ +
Theadcostapproximately$4.9millionin 2016.
59. Let x =45.Then 2 .07245.744451.2 3.84 452 Let x =20.Then ()() 2 .505204.5872027.6 6.6 201 −+ ≈ + Thecostofanadwillnotreach$7millionin 2020.
60. Let x =45.Then 2 .07245.744451.2 3.84 452
Let x =22.Then ()() 2 .505224.5872227.6 7.4 221 −+ ≈ + Thecostofanadwillnotreach$8millionin 2022.
61. Let x =11.Then
()() 2 .049112.4011.83 2.55 112 ++ ≈ + Thehourlyinsurancecostin2011was$2.55.
62. Let x =17.Then ()() 2 .049172.4017.83 2.94 172 ++ ≈ + Thehourlyinsurancecostin2017is$2.94.
63. Let x =20.Then ()() 2 .049202.4020.83 3.11 202 ++ ≈ +
Thehourlyinsurancecostin2020willbe$3.11. Theannualcostwillbe3.11(2100)=$6531.
64. Let x =23.Then ()() 2 .049232.4023.83 3.28 232 ++ ≈ +
Thehourlyinsurancecostin2023willbe$3.28. Theannualcostwillbe3.28(2100)=$6888.
No;theannualcostwillnotreach$10,000by 2023.
Section 1.5 Exponents and Radicals
1.
2.
3.
4.
22224416 ccc
6. 33 3333 55125 xy x
9. 1
14. 22 162636 61
16. 2224 22 x yy x yx
17. 1133 3 abb aa b
18. 4 4 11 2 216
,but 4 4 11 (2)16 (2)
19. 12 497 because2749
20. 13 82 because328 .
21. 14.25 (5.71)(5.71)1.55 Useacalculator.
22. 52125 1212498.83
23. 2 23132 6464(4)16 24.
3 32123 64648512
25. 4 434134 134 8273381 27216 82
26. 1313 27644 64273
27. 32 23 54 45
28. 4 431 3 71 777 77
29. 363444 30. 9101 9999
31. 106 10648 4 44 4444 4
33 24 32343234 2133223413 31494134 5555 55 kkkk kkk
2313231323 53 252(5) 25 ppppppp pp
32323232323232 0623 32323 6363 x xxxxxx x xx
1323 222343 232232 24323 33 11 33 xy xy xyxy yy
312 1233232 1433434314 (32)(34)(32)(34) 3434 cdcd cd cd cd cd 47.
3 2 3 2 2322 223232 3243232444 2 12 52 7575 57577 49 ab aba ab ab b a b
48.
12121212121212 322322322 123221232 1232 444 22 2 xxyxxy x xy xyxyxy xyxyxy xy
49. 12234312231243 76116 x xxxxxx x x
50. 12321212321212 2 3232 32 xxxxxxx x
51. Thisisadifferenceoftwosquares.
22121212121212 xyxyxy xy
52. UseFOIL.
13121332 131313321312 1232 23131213322 2 22 22 xyxy x xxyxy yy x xyxyy
53. 133 (3)3 x x ,(f)
54. 13333 x x ,(b)
55. 13 133 11 (3) (3)3 x x x ,(h)
56. 13 133 33 3 x x x ,(d)
57. 133 (3)3 x x ,(g)
58. 13 133 33 3 x x x ,(a)
59. 13 133 11 (3) (3)3 x x x ,(c)
60. 13333 x x ,(e)
61. 313 1251255
62. 61664642
63. 4146256255
64. 717128(128)2
65. 6373773721
66. 3338197299
67. 81477
68. 491633
69. 5157525325353
70. 89688128812843 843823163
71. 507252622
72 751925383133
73. 52045280 52535245 1053585155
22 323232341
22 525252 523
76. 33444 .Acorrectstatementwouldbe 3334416
22 3312312 121212(1)2 312312 121 312332
2215215 15151515 21515151 4212
22 9393332793333 33333333 24632463 43 936
3131323132 32323234 3233213 13 11
81. 323232 323232 927 96221162
82. 171717 232317 17 227337 6 227321
83. kM x f
Notethatbecause x representsthenumberof unitstoorder,thevalueof x shouldberounded tothenearestinteger.
a. k =$1, f =$500, M =100,000 1100,00020014.1
500 x
Thenumberofunitstoorderis14.
b. k =$3, f =$7, M =16,700 316,70084.6
7 x
Thenumberofunitstoorderis85.
c. k =$1, f =$5, M =16,800 116,800336058.0
5 x
Thenumberofunitstoorderis58.
84. 12.313hT
If T =216,find h 13 12.3(216)73.8 h
Aheightof73.8in.correspondstoathreshold weightof216lb.
Forexercises85–88,weusethemodel .188 revenue6.67,5, xx =≥ x =5correspondsto2005.
85. Let x =15.Then ().188 6.671511.1 ≈
Thedomesticrevenuefor2015wereabout$11.1 billion.
86 Let x =18.Then ().188 6.671811.5 ≈
Thedomesticrevenuefor2018willbeabout $11.5billion.
87. Let x =20.Then ().188 6.672011.7 ≈
Thedomesticrevenuefor2020willbeabout $11.7billion.
88. Let x =23.Then ().188 6.672312.0 ≈
Thedomesticrevenuefor2023willbeabout $12.0billion.
Forexercises89–92,weusethemodel .527 revenue1.08,1, xx =≥ x =1correspondstothefirst quarteroftheyear2013.
89. Let x =13.Then ().527 1.08134.2 ≈
Theadvertisingrevenueinthefirstquarterof 2016wasapproximately$4.2billion.
90. Let x =18.Then ().527 1.08185.0 ≈
Theadvertisingrevenueinthesecondquarterof 2017wasapproximately$5.0billion.
91. Let x =24.Then ().527 1.08245.8 ≈
Theadvertisingrevenueinthefourthquarterof 2018willbeapproximately$5.8billion.
92. Let x =27.Then ().527 1.08276.1 ≈
Theadvertisingrevenueinthethirdquarterof 2019willbeapproximately$6.1billion.
Forexercises93–96,weusethemodel Part-time.238Students2.72;10, xx =≥ x =10 correspondsto1990.
93. Let x =29.Then.2382.72(29)6.1 ≈
Accordingtothemodel,therewere approximately6.1millionpart-timestudents attendingcollegeoruniversityin2009.
94. Let x =34.Then.2382.72(34)6.3 ≈
Accordingtothemodel,therewere approximately6.3millionpart-timestudents attendingcollegeoruniversityin2014.
Copyright © 2019 Pearson Education, Inc.
95. Let x =38.Then.2382.72(38)6.5 ≈
Accordingtothemodel,therewillbe approximately6.5millionpart-timestudents attendingcollegeoruniversityin2018
96. Let x =43.Then.2382.72(43)6.7 ≈
Accordingtothemodel,therewillbe approximately6.7millionpart-timestudents attendingcollegeoruniversityin2023
Section 1.6 First-Degree Equations
1. 3820 388208 312 11 (3)(12)33 4 x x x x x
2. 4–5y =19
Add–4tobothsides. 45(4)19(4) 515 y y
Multiplybothsidesby 1 5 11 (5)(15)55 3 y y
3. .6.3.5.4 .6.5.3.5.5.4 .1.3.4 .1.3.3.4.3 .1.7 .1.7 7 .1.1 kk kkkk k k k k k
4. 2.55.048.5.06
2.55.04.068.5.06.06
2.55.18.5
2.55.1(2.5)8.5(2.5) 5.16.0 5.16.0 5.15.1 6.0 1.18 5.1 mm mmmm m m m m mm
5. 214(1)75 214475 21119 2211129 199 19999 109 10910 999 aaa aaa aa aaaa a a a a a
326431 366431 3121 312()1() 2121 21212112 213 21313 222 kkk kkk kk kkkk k k k k k
7. 2[(32)9]38 2(329)38 2(6)38 21238 1258 2054 xxx xxx xx xx x x x
8.
24231142 2(4833)142 2(5)142 210142 21021422 10144 101414414 244 244 6 44 kkk kkk kk kk kkkk k k k k k
9. 343(1)2(34)5510 x xx
Multiplybothsidesbythecommon denominator,10. 34 1010(1) 55 x x
3 (10)(2)(10)(34) 10 x
2(3)8(1)203(34) 68820912 28329 29328 740 1140 (7)(40) 777 xxx xxx xx xx x xx
10. 413 (2)21 324 4813 2 3322 4193 2 362 419434 2 36323 191 2 66 191 222 66
Multiplybothsidesbythecommon denominator,10,toeliminatethefractions. 3 103101 25 530610 53056105 3010 30101010 40 xx xx x xxx x x x
13. 22 165 212 165 1212 212 1 (12)(12)(6)(5) 2 61265 125 1112 (12)(5) 555 mm m mm mm m m mmmmm m mmm m mm
14. 395118 26 kkk k Multiplybothsidesbythecommon denominator,6k toeliminatethefractions. 2 22 395118 66 26 395118 6666 26 9(95)6(11)6(8) 9956648 56648 566664866 7148 714848 717171 kkk kk k kkk kkkk kk kkkk kkkk kk kkkk k k k
15. 483 0 3253 438 0 3325 78 0 325 xxx xxx xx
Multiplyeachsidebythecommondenominator, (x –3)(2x +5). 78 (3)(25)(3)(25)325 (3)(25)(0) xxxx xx xx
7(25)8(3)0 14358240 6590 59 659 6 xx xx x xx
16. 534 23223 53545 232232323 31 223 ppp ppppp pp
Multiplybothsidesbythecommon denominator,(2p +3)(p –2). 53 (23)(2)232 pp pp
3 223 2 1 223 23 32312 692 11 511 5 pp p pp p pp pp pp
2523324 5106948 1948 11 115 5 ppp ppp pp pp
17. 31 2 242 31 2 2(2)2 mm mm
3 2(2)2(2) 1 2(2)2(2)(2) 2 m m mm m
1 2(2)2(2)(2) 2 mm m
324(2) 3248 9 36494 4 m m mmm
18. 85 4 393 kk
Multiplybothsidesbythecommon denominator,3k –9. 85 (39)(39)4 393 kk kk
85 (39)3(3)1236 393 8151236 71236 29 2912 12 kkk kk k k kk
19. 9.063.59(85)12.07.5612 9.0628.7217.9512.07.5612
9.0628.7212.0717.95.5612
25.7118.5112 18.5112 .72 25.71
20. 5.74(3.12.7)1.095.2588 17.79415.4981.095.2588 15.4981.095.258817.794
14.40823.0528
23.0528 1.6 14.408 pp pp pp p p
21. 2.638.993.901.77 1.252.45 rr r Multiplybythecommondenominator (1.25)(2.45)toeliminatethefractions.
2.452.638.991.253.901.77 2.451.25
6.443522.02554.8752.21253.0625
1.568519.8133.0625
19.8131.494
19.8131.494 1.4941.494 13.26
22.
8.192.558.179.94 4 4.341.04 1.048.192.554.348.179.94
23. 4()2 442 42 52 52 22 55 or 22 axbax axbax abax abx abx abba xx
24. (3)(2) 32 abbxax abbxaxa
Isolatetermswith x ontheright. 32 5 5()5 abaxabx abaxbx ab ababxx ab
25. 5(b – x)=2b + ax First,usethedistributiveproperty. 5b –5x =2b + ax 525 35 3(5) 3(5)3 555 bbaxx baxx bax baxb x aaa Nowusethedistributivepropertyontheright. 3(5) 3(5) 55 3 5 bax bax aa b x a
26. bx –2b =2a – ax Isolatetermswith x ontheleft. 22 22 ()2() 2()2 bxaxab axbxab abxab ab xx ab
27. for 11 ()() PVkV PVk PP k V P
28. for iprtp i p rt
29. 0 0 0 0 for VVgtg VVgt VV gt tt VV g t
30. 2 0 2 0 2 00 222 SSgtk SSkgt SSkSSk gt g ttt
31. 1 2()for 11 22 2Multiplyby2. 2 21 Multiplyby. 22 ABbhB ABhbh ABhbh AbhBh AbhBh hhh AbhA bB hh
32. 5(32)for 9 99 3232 55 CFF
33. 215 h 215or215 26or24 3or2 hh hh hh
34. 4312 m 4312or4312 415or49 159 or 44 mm mm mm
35. 6210 p 6210or6210 24or216 2or8 pp pp pp
36. 5715 x 5715or5715 58or522 822 or 55 xx xx xx
5 10 3 55 10or10 33 5103or5103 51030or51030 3510or2510 357255 or 102102 r rr rr rr rr rr
38. 3 4 21 33 4or4 2121 3421or3421 384or384 78or18 71 or 88 h hh hh hh hh hh
5 532 9 995 532 559 93223 F F F F
Thetemperature–5°C=23°F.
40.
5 1532 9 995 1532 559 27325 F F F F
Thetemperature–15°C=5°F.
41.
5 2232 9 995 2232 559
39.63271.6 F F F F
42.
5 3632 9 995 3632 559 64.83296.8 F F F F
43. Let x =10.
1.151.62 1.15(10)1.62 13.12 yx y y =+ =+ =
Therefore,thegrossfederaldebtin2010was $13.12trillion.
44. Let x =15.
1.151.62 1.15(15)1.62 18.87 yx y y =+ =+ =
Therefore,thegrossfederaldebtin2015was $18.87trillion.
45. 1.151.62yx=+
Substitute22.32for y 22.321.151.62 20.71.1518 x x x =+ =⇒= Therefore,thefederaldeficitwillbe$22.32 trillionin2018.
46. 1.151.62yx=+
Substitute24.62for y. 24.621.151.62 231.1520 x x x =+ =⇒= Therefore,thefederaldeficitwillbe$24.628.83 trillionin2020.
47. 1.151.62yx=+
Substitute25.77for y 25.771.151.62 24.151.1521 x x x =+ =⇒= Therefore,thefederaldeficitwillbe$825.77 trillionin2021.
48. 1.151.62yx=+
Substitute30.37for y. 30.371.151.62
28.751.1525 x x x =+ =⇒= Therefore,thefederaldeficitwillbe$30.378 trillionin2025.
49. E =.108x +1.517
Substitute$2.7052422infor E. 2.705.1081.517
1.188.10811 x x x =+ =⇒= Thehealthcareexpenditureswereat$2.7052250 trillionin2011.
50. E =.108x +1.517
Substitute$3.029infor E. 3.029.1081.517 1.512.10814 x x x =+ =⇒= Thehealthcareexpenditureswereat$3.029 trillionin2014.
51. E =.108x +1.517
Substitute$3.461infor E 3.461.1081.517
1.944.10818 x x x =+ =⇒=
Thehealthcareexpenditureswillbe$3.4612422 trillionin2018.
52. E =.108x +1.517
Substitute$3.893infor E. 3.893.1081.517
2.376.10822 x x x =+ =⇒= Thehealthcareexpenditureswillbe$3.893 trillionin2022.
53. () 114.8201053390.5 xy−=−
Substitute746.98for y andsolvefor x ()() 114.820105746.983390.5 114.8230,748344.4
114.8231,092.4 2013 x x x x −=− −= = =
Theamountofincomewas$746.98billionin 2013.
54. () 114.8201053390.5 xy−=−
Substitute815.86for y andsolvefor x. ()() 114.820105815.863390.5 114.8230,748688.8
114.8231,436.8 2016 x x x x −=− −= = =
Theamountofincomewas$815.86billionin 2016.
55. () 114.8201053390.5 xy−=−
Substitute907.7for y andsolvefor x ()() 114.820105907.73390.5
114.8230,7481148
114.8231,896 2020 x x x x −=− −= = =
Theamountofincomewillbe$907.7billionin 2020.
56. () 114.8201053390.5 xy−=−
Substitute1022.5for y andsolvefor x. ()() 114.8201051022.53390.5 114.8230,7481722 114.8232,470 2025 x x x x −=− −= = = Theamountofincomewillbe$1022.5billionin 2025.
57. f =800, n =18, q =36 (1)18(19) 800 (1)36(37) 342 800205.41 1332 nn uf qq
Theamountofunearnedinterestis$205.41.
58. f =1400, q =48, n =12 (1)12(121)1400$92.86 (1)48(481) nn uf qq
Theamountofunearnedinterestis$92.86.
59. Let x =thenumberinvestedat5%. .05.04(52,000)2290 .052080.042290 .0120802290 .012080208022902080 .01210 .01210 .01.01 21,000 xx xx x x x x x x
Joeinvested$21,000at5%.
60. Let x representtheamountinvestedat4%.Then 20,000– x istheamountinvestedat6%.Since thetotalinterestis$1040, .04.06(20,000)1040 .041200.061040 .0212001040 .02160 8000 xx xx x x x
Sheinvested$8000at4%.
61. Let x =priceoffirstplot. Then120,000– x =priceofsecondplot. .15x =profitfromfirstplot –.10(120,000– x)=lossfromsecondplot. .15.10(120,000)5500 .1512,000.105500 .2517,500 70,000 xx xx x x
Shepaid$70,000forthefirstplotand 120,000–70,000,or$50,000forthesecond plot.
62. Let x representtheamountinvestedat4%. $20,000investedat5%(or.05)plus x dollars investedat4%(or.04)mustequal4.8%(or .048)ofthetotalinvestment(or20,000+ x). Solvethisequation.
.05(20,000).04.048(20,000) 1000.04960.048 1000960.008 40.0085000 x x xx x xx
$5000shouldbeinvestedat4%.
63. Let x =averagerateofgrowthofTwitter. Then5.7+ x =averagerateofgrowthof Instagram.
35x =visitorstoTwitter 21(5.7+ x)=visitorstoInstagram.
3521(5.7)
35119.721
14119.7 8.6 x x x x x x =+ =+ = ≈
Since x representstheaveragerateofgrowthof Twitter,therewasanaveragegrowthof approximately8.6millionvisitorsShe.
64. TheaveragerateofgrowthofInstagramwas 5.7+ x orapproximately14.3millionvisitors.
65. ThenumberofvisitorstoTwitterwas 35x =35(8.6)=301million.
66. ThenumberofvisitorstoInstagramwas 21(5.7+ x)=21(5.7+8.6)=300.3million (Note:Duetoroundingerror,thevaluesin65 and66aredifferent)
67. Let x =thenumberoflitersof94octanegas; 200=thenumberoflitersof99octanegas; 200+ x =thenumberoflitersof97octanegas. 9499(200)97(200) 9419,80019,40097
4003
400 3 x x x x x x
Thus,4003litersof94octanegasareneeded.
68. Let x betheamountof92octanegasoline.Then 12– x istheamountof98octanegasoline.A mixtureofthetwomustyield12Lof96octane gasoline,so 9298(12)96(12) 921176981152 624 4 xx xx x x
Then12– x =12–4=8. Mix4Lof92octanegasolinewith8Lof 98octanegasoline.
Section 1.7 Quadratic Equations
1. (x +4)(x –14)=0 40or140 4or14 xx xx
Thesolutionsare–4and14.
2. (p –16)(p –5)=0 160or50 16or5 pp pp
Thesolutionsare16and5.
3. x(x +6)=0
0or60 6 xx x
Thesolutionsare0and–6.
4. 220xx x(x –2)=0 0or20 2 xx x Thesolutionsare0and2.
5. 2 2 24 240 220 zz zz zz 20or20 0or2 zz zz Thesolutionsare0and2.
6. 2640 x (x –8)(x +8)=0 80or80 8or8 xx xx Thesolutionsare8and–8.
7. 215560yy (y +7)(y +8)=0 70or80yy 7or8yy
Thesolutionsare–7and–8.
8. 2450 (1)(5)0 kk kk
10or50 1or5 kk kk Thesolutionsare–1and5.
9. 2 2 273 2730 (21)(3)0 xx xx xx 210or30 1 or3 2 xx xx
Thesolutionsare 1 2 and3.
10. 2 2 215 0152 03152 zz zz zz 310or520 12 or 35 zz zz
Thesolutionsare 1 3 and 2 5
11. 2 2 61 610 (31)(21)0 rr rr rr 310or210 11 or 32 rr rr
Thesolutionsare 1 3 and 1 2
12. 2 2 3165 31650 (31)(5)0 yy yy yy
310or50 1 or5 3 yy yy
Thesolutionsare 1 3 and5.
13. 2 2 22013 213200 (25)(4)0 mm mm mm
250or40 5 or4 2 mm mm
Thesolutionsare5 2 and4.
14. 2 617120 23340 aa aa 230or340 34 or 23 aa aa
Thesolutionsare3 2 and 4 3
15. 2 (7)10 7100 (5)(2)0 mm mm mm 50or20 5or2 mm mm
Thesolutionsare–5and–2.
16. 2 (27)4 2740 (21)(4)0 zz zz zz 210or40 1 or4 2 zz zz
Thesolutionsare 1 2 and–4.
17. 2 9160 (34)(34)0 x xx 340or340 3434 44 or 33 xx xx xx
Thesolutionsare 4 3 and 4 3
18. 2 36490 67670 y yy 670or670 77 or 66 yy yy
Thesolutionsare7 6 and7 6
19. 2 16160 16(1)0 xx xx 160or10 0or1 xx xx Thesolutionsare0and1.
20. 2 12480 12(4)0 yy yy 0or40 0or4 yy yy Thesolutionsare0and4.
21. 2 (2)7 r 27or27 27or27 rr rr
Weabbreviatethesolutionsas27
Copyright © 2019 Pearson Education, Inc.
22. 2 (4)27 b 427or427 433or433 433or433 bb bb bb
Weabbreviatethesolutionsas433
23. 2 (41)20 x Usethesquarerootproperty. 4120or4120 4125or4125 4125or4125 xx xx xx
Thesolutionsare125 4
24. 2 (35)11 t 3511or3511 3511or3511 511511 or 33 tt tt tt
Thesolutionsare511 3
25. 2 2710 xx
Usethequadraticformulawith a =2, b =7,and c =1.
bbac x a
2 2 4 2 774(2)(1) 2(2) 7498741 44
Thesolutionsare741 4 and7414, whichareapproximately–.1492and–3.3508.
26. 2 370 xx
bbac x a x
Usethequadraticformulawith a =3, b =–1, and c =–7. 2 2 4 2 (1)(1)4(3)(7) 2(3) 1184185 66
Thesolutions185 6 areapproximately 1.7033and–1.3699.
27. 2 421 kk
Rewritetheequationinstandardform.
2 4210 kk
Usethequadraticformulawith a =4, b =2,and c =–1.
15 4 k k
2 224(4)(1)2416 2(4)8 220225215 8824
Thesolutionsare15 4 and154,which areapproximately.3090and–.8090.
28. 2 2 35 350 rr rr
Usethequadraticformulawith a =1, b =–3, and c =–5.
bbac r a r
2 2 4 2 (3)(3)4(1)(5) 2(1) 3920329 22
Thesolutions329 2 areapproximately 4.1926and–1.1926.
29. 2 552 yy 2 5520 yy a =5, b =5, c =–2 2 554(5)(2)52540 2(5)10 565565 1010 y
Thesolutionsare565 10 and56510, whichareapproximately.3062and–1.3062.
30. 2 2 238 2830 zz zz
a =2, b =–8,and c =3.
2 2 4 2 (8)(8)4(2)(3) 2(2) 864248408210 444 2410410 42 bbac z a
Thesolutions410 2 ,areapproximately 3.5811and.4189.
31. 2 6640 xx a =6, b =6, c =4 2 664(6)(4)63696 2(6)12 660 12 x
Because60isnotarealnumber,thegiven equationhasnorealnumbersolutions.
32. 2 3220 aa a =3, b =–2,and c =2. 2 (2)(2)4(3)(2) 2(3) 2424220 66 a
Since20isnotarealnumber,thegiven equationhasnorealnumbersolutions.
33. 2 2350 rr a =2, b =3, c =–5 (3)94(2)(5)3940 2(2)4 34937 44 37437105 1or 44442 r rr
Thesolutionsare5 2 and1.
34. 2 2 883 8830 xx xx
bbac x a
a =8, b =–8,and c =3. 224(8)(8)4(8)(3) 22(8) 86496832 1616
Because32isnotarealnumber,thereareno realnumbersolutions.
35. 2 27300 xx a =2, b =–7, c =30 (7)494(2)(30)7191 2(2)4 x
Since191isnotarealnumber,thereareno realsolutions.
36. 2 2 36 360 kk kk
bbac k a k k
a =3, b =1,and c =–6. 2 2 4 2 114(3)(6)1172 2(3)6 173 6
Thesolutions173 6 areapproximately 1.2573and–1.5907.
37. 2 715 1 22 a a
Toeliminatefractions,multiplybothsidesby thecommondenominator,2 2 a . 22 271527150 aaaa a =2, b =7, c =–15 2 774(2)(15)749120 2(2)4 716971363 or 4442 71320 5 44 a a a
Thesolutionsare3 2 and–5.
38. 2 41 50 k k
Multiplybothsidesby2 k
2 5410 kk a =5, b =–4, c =–1
2 44451 25 4162043646 101010 46104621 1or 101010105 k kk
Thesolutionsare 1 5 and1.
39. 2 254970 tt 2 2570490 tt 224(70)4(25)(49) 49004900 0 bac
Thediscriminantis0. Thereisonerealsolutiontotheequation.
40. 2 9121 zz 2 91210 zz 224(12)4(9)(1) 14436 180 bac
Thediscriminantispositive. Therearetworealsolutionstotheequation.
41. 2 132450 xx 224(24)4(13)(5) 576260836 bac
Thediscriminantispositive. Therearetworealsolutionstotheequation.
42. 2 201950 xx 224(19)4(20)(5) 36140039 bac
Thediscriminantisnegative. Therearenorealsolutionstotheequation.
ForExercises43–46usethequadraticformula:
43. 2 4.4210.143.790 xx 2 (10.14)(10.14)4(4.42)(3.79) 2(4.42) 10.145.9843 .4701or1.8240 8.84 x
44. 2 382.74570.49230 xx 2 (82.74)(82.74)4(3)(570.4923) 2(3) 82.74 13.79 6 x
45. 2 7.632.795.32 xx 2 7.632.795.320 xx 2 2.79(2.79)4(7.63)(5.32) 2(7.63) 2.7913.0442 1.0376or.6720 15.26 x
46. 2 8.0625.872625.047256 xx 2 8.0625.872625.0472560 xx 25.8726(25.8726)2 4(8.06)(25.047256) 2(8.06) 25.872638.4307 3.9890or.7790 16.12 x
47. Let x =10. 2 2 .3967.0974.9 .396(10)7.09(10)74.9 43.6 Pxx P P =−+ =−+ =
Theaverageannualpayfortruckersin2010was $43.6thousand.
48. Let x =15. 2 2 .3967.0974.9 .396(15)7.09(15)74.9 57.65 Pxx P P =−+ =−+
Theaverageannualpayfortruckersin2015was $57.65thousand.
49. Let P =48.
() 2 2 2 2 .3967.0974.9 48.3967.0974.9 0.3967.0926.9 7.097.094.39626.9 2.396 5.46or12.45 Pxx xx
Thefirstfullyeartheaverageannualpayfor truckerswasbelow$48,000was2006.
50. Usingtheanswerfrom49,thefirstfullyearthe averagepayfortruckersrecoveredtobeabove $48,000was2013.
51. Let P =55.
Thefirstfullyeartheaverageannualpayfor truckersrecoveredtobeabove$55,000was 2015.
52. Let P =60. ()()() () 2 2 2 2 .3967.0974.9 60.3967.0974.9 0.3967.0914.9 7.097.094.39614.9 2.396 2.43or15.47 Pxx xx xx x =−+ =−+ =−+ ±−− = ≈
Thefirstfullyeartheaverageannualpayfor truckersrecoveredtobeabove$60,000was 2016.
53. Let x =3. 2 2 .8679.3163.32 .867(3)9.31(3)63.32 43.19 Oxx O O =−+ =−+ = Thepriceofcrudeoilforthethirdquarterof 2015was$43.19.
54. Let x =6. 2 2 .8679.3163.32 .867(6)9.31(6)63.32 38.67 Oxx O O =−+ =−+ = Thepriceofcrudeoilforthesecondquarterof 2016was$38.67.
55. Let O =45. ()()() () 2 2 2 2 .8679.3163.32
45.8679.3163.32
0.8679.3118.32 9.319.314.86718.32 2.867 2.59or8.14 Oxx xx xx x =−+ =−+ =−+ ±−− = ≈
Themostrecentquarterwherethepriceofcrude oilwasatleast$45wasthefirstquarterof2017.
56. Let O =55. ()()() () 2 2 2 2 .8679.3163.32
55.8679.3163.32
0.8679.318.32
0.98or9.75 Oxx xx xx x =−+ =−+ =−+ ±−− = ≈
9.319.314.8678.32 2.867
Themostrecentquarterwherethepriceofcrude oilwasatleast$55wasthesecondquarterof 2017.
Copyright © 2019 Pearson Education, Inc.
57. Triangle ABC representstheoriginalpositionof theladder,whiletriangle DEC representsthe positionoftheladderafteritwasmoved.
UsethePythagoreantheoremtofindthedistance fromthetopoftheladdertotheground.
Intriangle ABC, 2222 2 13516925 14412 BCBC BCBC Thus,thetopoftheladderwasoriginally12feet fromtheground.
Intriangle DEC, 2222 2 13716949 120120 ECEC ECEC
Thetopoftheladderwas120feetfromthe groundaftertheladderwasmoved.Therefore, theladdermoveddown121201.046 feet.
58. Triangle ABC showstheoriginalpositionofthe ladderagainstthewall,andtriangle DEC isthe positionoftheladderafteritwasmoved.
Intriangle ABC, 222222 159159144 12 BCBC BC
Intriangle DEC 222 22 2 15129 225144248118 062230 200or303 xx xxxx xxxx xxxx Thebottomoftheladdershouldbepulled3feet awayfromthewall.
59. a. Theeastboundtraintravelsataspeedof x +20.
b. Thenorthboundtraintravelsadistanceof 5x in5hours. Theeastboundtraintravelsadistanceof 5(x +20)= 5x +100in5hours.
c.
BythePythagoreantheorem, 222 (5)(5100)300 xx
d. Expandandcombineliketerms.
22 2 2525100010,00090,000 50100080,0000 xxx xx Factoroutthecommonfactor,50,and dividebothsidesby50.
2 2 50(201600)0 2016000 xx xx Nowusethequadraticformulatosolvefor x
2 2020411600 21 51.23or31.23 x
Since x cannotbenegative,thespeedofthe northboundtrainis x ≈ 31.23mph,andthe speedoftheeastboundtrainis x +20 ≈ 51.23mph.
60. Let x representthelengthoftimetheywillbe abletotalktoeachother.
Since d = rt,Tyrone’distanceis2.5x and Miguel’sdistanceis3x.
Thisisarighttriangle,so 22222 22 (2.5)(3)46.25916
Sincetimemustbenonnegative,theywillbe abletotalktoeachotherforabout1.024hror approximately61min.
61. a. Let x representthelength.Then,3002 2 x or150– x representsthewidth.
b. Usetheformulafortheareaofarectangle. (150)5000LWAxx
c. 2 1505000 xx
Writethisquadraticequationinstandard formandsolvebyfactoring.
2 01505000 xx
215050000xx (x –50)(x –100)=0 500or1000 50or100 xx xx
Choose x =100becausethelengthisthe largerdimension.Thelengthis100mand thewidthis150–100=50m.
62. Let x representthewidthoftheborder.
Theareaoftheflowerbedis9·5=45.Thearea ofthecenteris(9–2x)(5–2x).Therefore,
2 2 45(92)(52)24 454528424 454528424 xx xx xx
2 2 042824 0476 04(1)(6) xx xx xx 10or60 1or6 xx xx
Thesolution x =6isimpossiblesinceboth 9–2x and5–2x wouldbenegative.Therefore, thewidthoftheborderis1ft.
63. Let x =thewidthoftheuniformstriparoundthe rug.
Thedimensionsoftherugare15–2x and 12–2x.Thearea,108,isthelengthtimesthe width.
Discard x =12sinceboth12–2x and15–2x wouldbenegative. If3 2 x ,then
Forexercises65–70,usetheformula 2 00 16,htvth where h0istheheightofthe objectwhen t =0,and v0istheinitialvelocityattime t =0.
65. 000,625,0vhh
22 0160625166250 25 42542506.25or 4 25 6.25 4 ttt ttt t
Thenegativesolutionisnotapplicable.Ittakes 6.25secondsforthebaseballtoreachthe ground.
66. Whentheballhasfallen196feet,itis 625–196=429feetabovetheground.
000,625,429vhh
Thedimensionsoftherugshouldbe 9ftby12ft.
64. Let x =Haround’sspeed.Then x +92= Franchitti’sspeed.Fromtheformula d = rt,we haveHaround’stime=500 x andFranchitti’s time= 500 92 x Then,wehave
50050046,0003.72342.24 03.72342.2446,000 xx xxxx x xxx xx
Usethequadraticformulatosolvefor x.
166.3or74.3 x Thenegativevalueisnotapplicable. Haround’sspeedwas74.3mphandFranchitti’s speedwas74.3+92=166.3mph.
22 429160625161960 14 41441403.5or 4 14 3.5 4 ttt ttt t
Thenegativesolutionisnotapplicable.Ittakes 3.5secondsfortheballtofall196feet.
67. a. 000,200,0vhh 22 2 016020016200 200200 3.54 164 ttt tt
Thenegativesolutionisnotapplicable.It willtakeabout3.54secondsfortherockto reachthegroundifitisdropped.
b. 0040,200,0vhh 2 01640200 tt Usingthequadraticformula,wehave 2 4040416200 216 5or2.5 t
Thenegativesolutionisnotapplicable.It willtakeabout2.5secondsfortherockto reachthegroundifitisthrownwithan initialvelocityof40ft/sec.
c. 0040,200,2vht
2 16240220056 h After2seconds,therockis56feetabove theground.Thismeansithasfallen 200–56=144feet.
68. a. 00800,0,3200vhh 2 2 2 3200168000 1680032000 502000 tt tt tt
Usingthequadraticformula,wehave
2 505041200 21 4.38or45.62 t tt
Therocketrisesto3200feetafterabout 4.38seconds.Italsoreaches3200feetasit fallsbacktothegroundafterabout45.62 seconds.Becauseweareaskedhowlongto riseto3200feet,theanswerwillbeabout 4.38sec.
b. 00800,0,0vhh
2 2 0168000 16800016500 0or50 tt tttt tt
Therockethitsthegroundafter50seconds.
69. a. 0064,0,64vhh
2 2 22 6416640 1664640 440202 tt tt tttt
Theballwillreach64feetafter2seconds.
b. 0064,0,39vhh
2 2 3916640 166439041343 133 3.25or.75 44 tt tttt tt
Theballwillreach39feetafter.75seconds andafter3.25seconds.
c. Twoanswersarepossiblebecausetheball reachesthegivenheighttwice,onceonthe wayupandonceonthewaydown.
70. a. 00100,0,50vhh 2 2 2 50161000 16100500 850250.55or5.7 tt tt ttt
Theballwillreach50feetonthewayup afterapproximately0.55seconds.
b. 00100,0,35vhh 2 2 35161000 16100350 .37or5.88 tt tt t
Theballwillreach35feetonthewayup afterapproximately0.37seconds.
Inexercises71–76,wediscardnegativerootssinceall variablesrepresentpositiverealnumbers.
71. 2 2 2 1 for 2 2 2 2 2 Sgtt Sgt S t g g S t g g Sg t g
72. Solvefor r. 2 2 (0) ar a r a rr aa r
73. 4 2 24 4 2 44 2 for dk Lh h Lhdk dk h L dkLdkL h LL L dkL h L
74. Solvefor v 2 2 (0) kMv F r Fr v kM FrkM vv kM kM FrFrkM v kMkM
Solvefor R byusingthequadraticformulawith a = P, 2Pr2bE ,and2 Pr c .
76. Solvefor r 222Srhr Writeasaquadraticequationin r. 2 (2)(2)0 rhrS Solvefor r usingthequadraticformulawith a =2π, b =2πh,and c =–S. 2 2 22 22 22 22 4 2 224(2)() 2(2) 248 4 222 4 22 4 2 2 bbac r a hhS r hhS hhS hhS hhS r
22 2Pr4Pr 2 EEE R P
Chapter 1 Review Exercises
1. 0and6arewholenumbers.
2. –12,–6,4,0,and6areintegers.
3. –12,–6, 9 10,4,0, 1 8,and6arerational numbers.
4. 7,,11 4 areirrationalnumbers.
5. 9[(–3)4]=9[4(–3)] Commutativepropertyofmultiplication
6. 7(4+5)=(4+5)7 Commutativepropertyofmultiplication
7. 6(x + y –3)=6x +6y +6(–3) Distributiveproperty
8. 11+(5+3)=(11+5)+3 Associativepropertyofaddition
9. x isatleast9. x ≥ 9
10. x isnegative. x <0
11. 642,22 ,8199 , 3(2)3255 Since–5,–2,2,9areinorder,then 3(2),2,64,81 areinorder.
12. 164,8,7,1212
13. 78781
14. 39633330
15. x ≥ –3
Startat–3anddrawaraytotheright.Usea bracketat–3toshowthat–3isapartofthe graph.
16. –4< x ≤ 6 Putaparenthesisat–4andabracketat6.Draw alinesegmentbetweenthesetwoendpoints.
17. 9(6)(3)99189927 6(3)6399
2042515251 9(3)1239(3)4 1051211 9346410
3232 3232 3322 32 8852410 8852410 8284510 104510 yyyyy yyyyy yyyyy yyy
21. 2222525252254 khkhkhkh
22. 22 22 (25)(25)(2)(5) 425 ryryry ry
23. 222 22 3432344 92416 x yxxyy xxyy
24. 222 22 (25)(2)2(2)(5)(5) 42025 abaabb aabb
25. 22 245245 khkhkkhh
26.
27.
222222 2616238 mnmnnnmm
43222 2 51245124 522 aaaaaa aaa
28.
322 2444461 4(31)(21) xxxxxx xxx
Copyright © 2019 Pearson Education, Inc.
29.
2222 144169(12)(13) 12131213 pqpq pqpq
30. 2222 8125(9)(5) (95)(95) zxzx zxzx
3 33 22 2 27131 313311 31931 yy
32.
333 22 2 125216(5)(6) (56)(5)5(6)6 (56)253036
36.
2322(1)(2)8 2(1)82(1)(2) 8(1)(2)84 2(1)(2)2
322 22 2844 3169 ppppp p pp
22 289 3(4)(4)(2)(2) ppp p ppppp
(4)(2)92 3(4)(4)(2)(2) pppp ppppp
3(4)(2)32 3(4)(2)(4)(2) pppp ppppp
32 (4)(2) p pp
37. 2 22 242618 123 mmm mmm 22 22123 (1)(1)618 mm mm mmm 2 2(1)(3)(1) (1)(1)6(3) mmm mmm 2(1)(3)(1)2 2(1)(3)3(1) mmm mmm (1)2 3(1) m m
38. 2 22 652(1) 4125 xxxx x x 2 (5)(1)2(1) 41(5)(5) xxxx xxx
2 2 2(5)(1) 2(5)21(5) xxx xxx
2 2 (1) 21(5) xx xx
39. 3 3 11 5or 5125
40. 2 2 11 10or 10100
00 881
42. 22 5636 6525
43. 636(3)3 4444
44. 525(2)7 7 1 7777 7
45. 5 5(4)541 4 81 888 88
46. 3 347 47 61 66 66
47. 1111752 5210
48. 21 2 11116 55 5525525
49. 1312 13123223 3223 551 55 55
50. 341214 1414 222 1 22
51.
52.
1232 22123327252 33333 aaaaa
32 23323233292
53. 3273
54. 664isnotarealnumber.
55. 3533233 3322333 54272 27232 pqpqq
56. 4444534336416424 abaabaab
57. 331212331243331223 33243213
58. 8726387297 8767147
59.
61. 34229 34829 829 1 13 3 xxx xxx xx xx
62. 493(12)5 49365 4926 7 27 2 yy yy yy yy
63. 26 4 33 264(3) 26412 246 623 m mm mm mm mm mm
Because m =3wouldmakethedenominatorsof thefractionsequalto0,makingthefractions undefined,thegivenequationhasnosolution.
64. 153 4 55 k kk Multiplybothsidesoftheequationbythe commondenominator k +5.
15453 1542035 kk kkk
If k =–5,thefractionswouldbeundefined,so thegivenequationhasnosolution.
65. 832 823 823 3 82 axx axx xa x a
66. 22 22 2 2 24 24 4 2 bxxb bxb b x b
67. 2 8 5 22 8or8 55 22 55(8)or55(8) 55 240or240 38or42 38or42
Thesolutionsare–38and42.
68. 4163 kk 4163or41(63) 4163or4163 24or102 1 2or 5 kkkk kkkk kk kk
Thesolutionsare2and 1 5 .
69. 2 (7)5 b
Usethesquarerootpropertytosolvethis quadraticequation.
75or75 75or75 bb bb
Thesolutionsare75 and75,which weabbreviateas75
70. 2 (21)7 p
Solvebythesquarerootproperty. 217or217 217or217 1717 or 22 pp pp pp
Thesolutionsare17 2 .
71. 2 232 pp
Writetheequationinstandardformandsolveby factoring.
2 2320 (21)(2)0 pp pp 210or20 1 or2 2 pp pp
Thesolutionsare 1 2 and–2.
72. 2 215yy Writetheequationinstandardformandsolveby factoring.
2 2150 (3)(25)0 yy yy y =3or 5 2 y
Thesolutionsare3and5 2
73. 22 21121211210 (23)(7)0 qqqq qq 230or70 3 or7 2 qq qq
Thesolutionsare3 2 and7.
74. 22 321632160 (38)(2)0 xxxx xx 380or20 8 or2 3 xx xx
Thesolutionsare8 3 and2.
75. 4242 61610 kkkk Let2 pk ,so 24pk 2 610 (31)(21)0 pp pp 310or210 11 or 32 pp pp
2 1113
If,3333pkk
If211 , 22pk hasnorealnumber solution.
Thesolutionsare3 3
76. 4242 2122120 pppp
Let2 up ;then24 up
2 2120 (31)(72)0 uu uu
22 310or720 12 or 37 12 or 37 uu xx pp
If222 , 77xp hasnorealnumber solution.
13 33 p
Thesolutionsare3 3
77. 2 ()2 ER p rR for r 22 222 222 222 () 2 2 20 prRER prrRRER prrpRRpER prrpRRpER
Usethequadraticformulatosolvefor r 2222 2 244 2 24 2 pRpRpRpER r p pRpERpREpR r pp
78. 2 ()2for. ER pE rR 2 222 2 ()() ()() prR prRERE R rRpR prR E RR
79. K = s(s – a)for s 220KsassasK Usethequadraticformula. 224()4 22 aaK aaK s
80. 20kzhzt for z Usethequadraticformulawith a = k, b =–h,and c =–t 2 2 ()()4()() 2 4 2 hhkt z k hhkt k
81. 27(55)8282% −−==
82. 22.5(2.5)2020%
83. Let x =theoriginalprice .21229 .81229 1536.25 xx x x −= = = Theoriginalpriceofthelaptopwas$1536.25.
84. Let x =theoriginalprice .27329.83 1.27329.83 23.43 xx x x += = = Theoriginalpriceofthestockwas$23.43.
85. Let O =3.63 3.63.112.2 1.43.11 13 x x x =+ = =
Theamountofoutlaysreached$3.63trillionin theyear2013.
86. Let O =3.96 3.96.112.2 1.76.11 16 x x x =+ = =
Theamountofoutlaysreached$3.96trillionin theyear2016.
87. Let O =10.8 10.8.28 2.8.2 14 x x x =+ = =
Thecrudeoiloutputreached10.8millionbarrels perdayintheyear2014.
Copyright © 2019 Pearson Education, Inc.
88. Let O =11.4 11.4.28 3.4.2 17 x x x =+ = =
Thecrudeoiloutputreached11.4millionbarrels perdayintheyear2017.
89. Let x =7 2 2 .04.4718.38 .04(7).47(7)18.38 17.05
17.05million.
90. Let x =10
was17.68million.
91. Let V =17.5.
Thefirstfullmonthinwhichsaleswerebelow 17.5millionvehicleswasMarch2016.
92. Usingthesolutionto91,thesecondmonthofthe yearinwhichsaleswas17.5millionwas October2016.
93. Let x =13 2 .061(13)6.09(13)38.89.16 131 F ++ =≈ +
Thetotalnumberofflightsthatdepartedin2013 wasapproximately9.16million.
94. Let x =20 2 .061(20)6.09(20)38.88.81 201 F ++ =≈ +
Thetotalnumberofflightsthatwilldepartin 2020willbeapproximately8.81million.
95. Let F =9.2.
2 2 2 2 2 .0616.0938.8 1 .0616.0938.8 9.2 1 9.2(1).0616.0938.8 0.0613.1129.6 3.113.114.06129.6 2.061 12.67or38.32 xx F x xx x xxx xx x
Thenumberofflightdeparturesfellbelow9.2 millionin2013.
96. Let F =8.9.
2 2 2 2 2 .0616.0938.8 1 .0616.0938.8 8.9 1 8.9(1).0616.0938.8 0.0612.8129.9 2.812.814.06129.9 2.061 16.70or29.38 xx F x xx x xxx xx x
Thenumberofflightdeparturesfellbelow8.9 millionin2017.
97. Let x =12 .26 16.2(12)30.91 R =≈
Thetotalamountspentonbasicresearchbythe U.S.governmentin2012wasapproximately $30.91billion.
98. Let x =17 .26 16.2(17)33.84 R =≈
Thetotalamountspentonbasicresearchbythe U.S.governmentin2017wasapproximately $33.84billion.
99.Let x =thesingleinterestrate 2000(.12)500(.07)2500 2752500 .11 x x x
Therefore,asingleinterestrateof11%willyield thesameresults.
100.Let x =theamountofbeef.Then30– x =the amountofpork.
Thereforethebutchershoulduse10poundsof beefand30–10=20poundsofpork.
101. Let x =112 2 2 .0477.69341.6 .047(112)7.69(112)341.6 69.9 Dxx D D
= TheratiooffederaldebttoGDPin2012was approximately69.9%.
102. Let D =55%.
Themostrecenttimetheratiooffederaldebtto GDPwas55%wasin2006.
103. Let x =thewidthofthewalk.
Touseallofthecement,solve 2 2 200450 0450200 xx xx
Usingthequadraticformula,wehave
Thewidthcannotbenegative,sothesolutionis approximately3.2feet.
104. Let x =thelengthoftheyard.Then160– x =the widthoftheyard.Areaisequaltolengthtimes width. 2 Thearea 4000()(160) 01604000 lw xx xx
Usingthequadraticformula,wehave
2 160(160)414000
Thelengthislongerthanthewidth,sothelength isapproximately129feetandthewidthis approximately160–129=31feet.
105. 00150,0,200vhh 2 2 2 200161500 161502000 87510001.61or7.77 tt tt ttt
Theballwillreach200feetonitsdownwardtrip afterapproximately7.77seconds.
106. 0055,700,0vhh 2 2 01655700 165570008.55or5.12 tt ttt
Theballwillreachthegroundafter approximately5.12seconds
Case 1 Consumers Often Defy Common Sense
1. Thetotalcosttobuyandrunthiselectrichot watertankfor x yearsis E =218+508x
2. Thetotalcosttobuyandrunthisgashotwater tankfor x yearsis G =328+309x
3. Over10years,theelectrichotwatertankcosts 218+508(10)=5298or$5298,andthegashot watertankcosts328+309(10)=3418or$3418. Thegashotwatertankcosts$1880lessover10 years.
4. Thetotalcostsforthetwohotwatertankswill beequalwhen218508328309 x x 1991100.55. xx Thecostswillbeequalwithinthefirstyear.
5. ThetotalcosttobuyandrunthisMaytag refrigeratorfor x yearsis M =1529.10+50x
6. ThetotalcosttobuyandrunthisLGrefrigerator for x yearsis L =1618.20+44x.
7. Over10years,theMaytagrefrigeratorcosts 1529.10+50(10)=2029.10or$2029.10, andtheLGrefrigeratorcosts 1618.20+44(10)=2058.20or$2058.20. TheLGrefrigeratorcosts$29.10moreover 10years.
8. Thetotalcostsforthetworefrigeratorswillbe equalwhen1529.10501618.2044 x x 689.1014.85. xx
Thecostswillbeequalinthe14th year.
Copyright © 2019 Pearson Education, Inc.
