Solution Manual for Introductory and Intermediate Algebra for College Students, 6th Edition

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’SSOLUTIONSMANUALDANIELS.MILLERNiagara County Community CollegeINTRODUCTORY&INTERMEDIATEALGEBRA FORCOLLEGESTUDENTSSIXTHEDITIONRobert BlitzerMiami Dade College

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iiiTABLE OF CONTENTS forSOLUTIONSINTRODUCTORY ANDINTERMEDIATEALGEBRA FORCOLLEGESTUDENTS6EChapter 1Variables, Real Numbers, and Mathematical Models ................................................1Chapter 2Linear Equations and Inequalities in One Variable..................................................59Chapter 3Linear Equations in Two Variables ........................................................................153Chapter 4Systems of Linear Equations ..................................................................................221Chapter 5Exponents and Polynomials....................................................................................325Chapter 6Factoring Polynomials ............................................................................................407Chapter 7Rational Expressions ..............................................................................................469Mid-Textbook Check Point ....................................................................................605Chapter 8Basics of Functions.................................................................................................611Chapter 9Inequalities and Problem Solving...........................................................................653Chapter 10Radicals, Radical Functions, and Rational Exponents ...........................................719Chapter 11Quadratic Equations and Functions ........................................................................809Chapter 12Exponential and Logarithmic Functions.................................................................973Chapter 13Conic Sections and Systems of Nonlinear Equations...........................................1055Chapter 14Sequences, Series, and the Binomial Theorem.....................................................1159Appendices..............................................................................................................................1235

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Chapter 1Variables, Real Numbers, and Mathematical Models11.1 Check Points1.a.6262(10)26x+=+=b.2(6)2(106)32x+=+=2.a.727 32 8211637xy+=⋅+⋅=+=b.66 38102282 8385xyyx−⋅−===−−⋅−−3.a.6xb.4x+c.35x+d.122x−e.15x4.a.93429(6)342543425142, falsex−=−=−==6 is not a solution.b.2(3)532(33)5(3)32(6)1531212, trueyy+=−+=−=−=3 is a solution.5.a.56x=b.721x−=6.a.454(15)565dnd=+=+=65% of marriages end in divorce after 15 yearswhen the wife is under 18 at the time of marriage.b.According to the line graph,60% of marriagesend in divorce after 15 years when the wife isunder 18 at the time of marriage.c.The mathematical model overestimates theactual percentage shown in the graph by 5%.1.1 Concept and Vocabulary Check1.variable2.expression3.substituting; evaluating4.equation; solution5.formula; modeling; models1.1 Exercise Set1.84812x+=+=2.1041014x+=+=3.121248x−=−=4.1616412x−=−=5.55 420x=⋅=6.66 424x=⋅=7.282874x==8.363694x==9.5353 451217x+=+⋅=+=10.3535 432023x+=+⋅=+=11.2(5)2(45)2(9)18x+=+==12.5(3)5(43)5(7)35x+=+==13.12812 4848840522 488xx−⋅−−====⋅14.5525 452205272633 41212xx+⋅++====⋅15.22 7514519xy+=⋅+=+=16.33 7521526xy+=⋅+=+=17.2()2(75)2(12)24xy+=+==

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Chapter 1Variables, Real Numbers, and Mathematical Models218.3()3(75)3(12)36xy+=+==19.434 73 5281513xy−=⋅−⋅=−=20.545 74 5352015xy−=⋅−⋅=−=21.21352135371075xy+=+=+=22.50145014102857yx−=−=−=23.262 756145615522 571073xyyx−+⋅−+−+====−⋅−−24.2242 57241072427322 751459yxxy−+⋅−+−+====−⋅−−25.4x+26.6x+27.4x−28.6x−29.4x+30.6x+31.9x−32.3x−33.9x−34.3x−35.35x−36.53x−37.121x−38.133x−39.1010xx+40.2020xx+41.630x+42.304x+43.1420614202020, truex+=+==The number is a solution.44.1722517222222, truex+=+==The number is a solution.45.30103020101010, truey−=−==The number is a solution.46.50205030202020, truey−=−==The number is a solution.47.4204(10)204020, falsez===The number is not a solution.48.5305(8)304030, falsez===The number is not a solution.49.86488688, truer===The number is a solution.50.79637977, truer===The number is a solution.

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Section 1.1Introduction to Algebra: Variables and Mathematical Models351.43234(6)323243232723, falsem+=+=+==The number is not a solution.52.34193(6)419184192219, falsem+=+=+==The number is not a solution.53.54255(3)42(3)5154651111, trueaa−=+−=+−=+=The number is a solution.54.53265(3)32(3)6153661212, trueaa−=+−=+−=+=The number is a solution.55.6(4)36(84)3(8)6(4)242424, truepp−=−===The number is a solution.56.4(3)64(63)6(6)4(9)363636, truepp+=+===The number is a solution.57.2(1)3(1)2(71)3(71)2(8)3(6)1618, falseww+=−+=−==The number is not a solution.58.3(2)4(3)3(102)4(103)3(12)4(7)3628, falseww+=−+=−==The number is not a solution.59.428x=60.535x=61.1412x=62.184x=63.205x−=64.4010x−=65.2616x+=66.2929x+=67.357x−=68.4329x−=69.4533x+=70.6333x+=71.4(5)33x+=72.6(3)33x+=73.524xx=−74.425xx=−75.First findx.727(5)237xyx=+=+=Evaluate the expression.375328444xy−−===76.First findx.525(4)222xyx=+=+=Evaluate the expression.224186333xy−−===

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Chapter 1Variables, Real Numbers, and Mathematical Models477.First findx.141213124yxx=−=−=−=Evaluate the expression.43(5)4(2)3(125)83(17)85159xy++=++=+=+=78.First findx.131515143yxx=−=−=−=Evaluate the expression.34(6)3(4)4(156)124(21)128496xy++=++=+=+=79.a.2(3 )2(43 1)2(7)14xy+=+⋅==b.530405(14)30407030404040, truez−=−=−==Yes, it is a solution.80.a.3(2)3(2 15)3(7)21xy+=⋅+==b.430544(21)30548430545454, truez−=−=−==Yes, it is a solution.81.a.626 32 618126xy−=⋅−⋅=−=b.74527(6)452(6)4245124233, falseww=−=−=−=No, it is not a solution.82.a.15145 31415782xy−=⋅−⋅=−=b.45454(8)545(8)3254403214, falseww=−=−=−=No, it is not a solution.83.a.0.162.830.16(10)2.834.43TnT=+=+=According to the formula, the average price in1990 was $4.43.The model overestimates the actual amountshown in the bar graph by $0.20.b.0.162.830.16(30)2.837.63TnT=+=+=According to the formula, the average price in2010 was $7.63.The bar graph shows a price in 2010 of $7.89.$7.89$7.63$0.26−=The model underestimates the actual amountshown in the bar graph by $0.26.84.a.0.162.830.16(5)2.833.63TnT=+=+=According to the formula, the average price in1985 was $3.63.The model overestimates the actual amountshown in the bar graph by $0.08.b.0.162.830.16(25)2.836.83TnT=+=+=According to the formula, the average price in2005 was $6.83.The bar graph shows a price in 2005 of $6.41.$6.83$6.41$0.42−=The model overestimates the actual amountshown in the bar graph by $0.42.85.520.3520.3(24)44.8pap=−=−=According to the formula, 44.8% of 24-year-oldssay that marriage is obsolete.The model overestimates the actual percentageshown in the bar graph by 0.8%.

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Section 1.2Fractions in Algebra586.520.3520.3(66)32.2pap=−=−=According to the formula, 32.2% of 66-year-oldssay that marriage is obsolete.The model overestimates the actual percentageshown in the bar graph by 0.2%.87.a.0.8(200)0.8(200145)44HAH=−=−=A bowler with an average score of 145 will havea handicap of 44.b.The bowler’s final score will be12044,+or164.88.a.0.8(200)0.8(200165)28HAH=−=−=A bowler with an average score of 165 will havea handicap of 28.b.The bowler’s final score will be14028,+or168.89. – 97.Answers will vary.98.0.8(200)0.8(200200)0HAH=−=−=A bowler with an average score of 200 will have nohandicap.99.makes sense100.does not make sense; Explanations will vary.Sample explanation:47x+is not an equation.101.makes sense102.makes sense103.true104.false; Changes to make the statement true will vary.A sample change is: Equations contain the equalitysymbol, =.105.true106.false; Changes to make the statement true will vary.A sample change is: Hard work combined with awillingness to use available learning resources cangreatly improve your chances of learning thematerial in this book.107.Choices of variables may vary.Leth=hours worked.Lets=salary.20sh=108.Choices of variables may vary.Letw=number of workers.Lett=number of televisions assembled.10tw=109.323 26757 535⋅⋅==⋅110.27252 51035373 721⋅÷=⋅==⋅111.9595417171717−−==1.2 Check Points1.52 85165212 8888⋅++===2.5 divided by 3 is 1 with a remainder of 2, so52133=.3.Begin by selecting any two numbers whose productis 36.Here is one possibility:364 9=⋅Because the factors 4 and 9 are not prime, factoreach of these composite numbers.364 92 2 3 3=⋅=⋅⋅⋅Notice that 2 and 3 are both prime. The primefactorization of 36 is2 2 3 3.⋅⋅⋅4.a.102515⋅=3 5⋅23=b.42224=3⋅72 22⋅⋅⋅3⋅74=When reducing fractions, it may not benecessary to write prime factorizations.We can use the greatest common factor toreduce this fraction.427624⋅=46⋅74=

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Chapter 1Variables, Real Numbers, and Mathematical Models6c.1315; Because 13 and 15 share no commonfactors (other than 1),1315is already reduced toits lowest terms.d.91 945⋅=59⋅15=5.a.424 2811311 333⋅⋅==⋅b.3631836351555⋅=⋅==c.323 2623737 321⋅⋅⋅===⋅73⋅27=Remember that you can divide numerators anddenominators by common factors beforeperforming multiplication.32373⋅=273⋅27=d.2117351131552521010⎛⎞ ⎛⎞=⋅==⎜⎟ ⎜⎟⎝⎠ ⎝⎠6.a.5358548434÷=⋅=4⋅21013333⋅==b.2232123331339÷=÷=⋅=c.31279274932848489÷=÷=⋅=34⋅42⋅⋅931122==7.a.235111111+=b.51426663−==c.3127918913128888844−=−===8.22 71433 721⋅==⋅9.a.131 53 25611252 55 2101010⋅⋅+=+=+=⋅⋅or1110b.434 43 31697343 44 3121212⋅⋅−=−=−=⋅⋅c.111192319 223316126126 21238231512121251or 144⋅−=−=−⋅=−==10.102 5=⋅122 2 3=⋅⋅LCD2 2 3 560=⋅⋅⋅=373 67 5183553101210 612 5606060⋅⋅+=+=+=⋅⋅11.a.219xx−=22211179792917979217771711, true⎛⎞−=⎜⎟⎝⎠⎛⎞−=⎜⎟⎝⎠−===The given fraction is a solution.b.1153131352032043120202011 , true2020ww−=⎛⎞−=⎜⎟⎝⎠−==The given fraction is a solution.12.a.2 (6)3x−b.31245xx−=

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Section 1.2Fractions in Algebra713.5 (32)955(7732)(45)2599CFC=−=−==77 F°is equivalent to25 C.°1.2 Concept and Vocabulary Check1.numerator; denominator2.mixed; improper3.5; 3; 2; 54.natural5.prime6.factors; product7.ab8.a cb d⋅⋅9.reciprocals10.dc11.acb+12.least common denominator1.2 Exercise Set1.32 83163192 8888⋅++===2.72 97187252 9999⋅++===3.37 53353387 5555⋅++===4.26 52302326 5555⋅++===5.78 16712871358 16161616⋅++===6.59 16514451499 16161616⋅++===7.23 divided by 5 is 4 with a remainder of 3, so233455=.8.47 divided by 8 is 5 with a remainder of 7, so477588=.9.76 divided by 9 is 8 with a remainder of 4, so764899=.10.59 divided by 9 is 6 with a remainder of 5, so595699=.11.711 divided by 20 is 35 with a remainder of 11, so71111352020=.12.788 divided by 25 is 31 with a remainder of 13, so78813312525=.13.composite;222 11=⋅14.composite;153 5=⋅15.composite;204 52 2 5=⋅=⋅⋅16.composite;753 253 5 5=⋅=⋅⋅17.37 has no factors other than 1 and 37, so 37 isprime.18.23 has no factors other than 1 and 23, so 23 isprime.19.composite;364 92 2 3 3=⋅=⋅⋅⋅20.composite;1004 252 2 5 5=⋅=⋅⋅⋅21.composite;14010 142 5 2 72 2 5 7=⋅=⋅⋅⋅=⋅⋅⋅22.composite;11010 112 5 11=⋅=⋅⋅23.79 has no factors other than 1 and 79, so 79 isprime.

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Chapter 1Variables, Real Numbers, and Mathematical Models824.83 has no factors other than 1 and 83, so 83 isprime.25.composite;819 93 3 3 3=⋅=⋅⋅⋅26.composite;648 82 4 2 42 2 2 2 2 2=⋅=⋅⋅⋅=⋅⋅⋅⋅⋅27.composite;24010 242 5 2 122 5 2 3 42 5 2 3 2 22 2 2 2 3 5=⋅=⋅⋅⋅=⋅⋅⋅⋅=⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅28.composite; 36010 362 5 6 62 5 2 3 2 32 2 2 3 3 5=⋅=⋅⋅⋅=⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅29.10255162 88⋅==⋅30.824414277⋅==⋅31.153 55183 66⋅==⋅32.18945=29⋅255=⋅33.355 77505 1010⋅==⋅34.455 99505 1010⋅==⋅35.3216228016 55⋅==⋅36.75580=155⋅151616=⋅37.44222225022525⋅==⋅38.382 19195022525⋅==⋅39.120260608624343⋅==⋅40.116258588624343⋅==⋅41.212 12535 315⋅⋅==⋅42.313 13747 428⋅⋅==⋅43.373 7218 118 1188⋅⋅==⋅44.535 3158 118 1188⋅⋅==⋅45.4949 43619or 57171 777⋅⋅=⋅==⋅46.3838 32438or 37171 777⋅⋅=⋅==⋅47.151 555 1110610 6605 1212⋅⋅⋅====⋅⋅48.121 222838 324⋅⋅===⋅12⋅11212=⋅49.565 6302 15151or 1474 7282 141414⋅⋅⋅====⋅⋅50.767 64224 114 1144⋅⋅===⋅212⋅212222=⋅51.331581202031454520⎛⎞ ⎛⎞=⋅==⎜⎟ ⎜⎟⎝⎠ ⎝⎠620⋅61=⋅52.41145701021545420⎛⎞ ⎛⎞=⋅==⎜⎟ ⎜⎟⎝⎠ ⎝⎠710⋅271or 322⋅=53.54535 31543444 416⋅÷=⋅==⋅54.72737 3215or 183828 21616⋅÷=⋅==⋅

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Section 1.2Fractions in Algebra955.18181255218 1182994or 15 2102555÷=⋅⋅⋅====⋅⋅56.12121377312 1123 447 3213 77÷=⋅⋅⋅/====⋅⋅/57.1825102552511818299⋅÷=⋅===⋅58.1237213 7733or 171 12123 444⋅/÷=⋅===⋅/59.31343 412344414 14⋅÷=⋅===⋅60.31373 721377717 17⋅÷=⋅===⋅61.75737 3213 7763656 5303 1010⋅⋅÷=⋅====⋅⋅62.73787 8564 1448434 3124 3142or 433⋅/⋅÷=⋅===⋅/⋅=63.111777 1114714114722⋅÷=⋅===⋅64.111444 11848184 22/⋅÷=⋅===/⋅65.3133116151051033 1011511÷=÷=⋅=35⋅5⋅211⋅661==66.3572112484878562842184÷=÷=⋅==228⋅233=⋅67.2424611111111++==68.5252713131313++==69.718422121212433⋅+===⋅70.5162161616+==32⋅388=⋅71.55102551or 18882444⋅+===⋅72.336233888244⋅+===⋅73.7522121212−==12⋅166=⋅74.13582181818−==42⋅499=⋅75.162147777−==27⋅21=⋅76.172153555−==77.1115122525525252710101010+=⋅+⋅+=+==78.1115133535535353815151515+=⋅+⋅+=+==79.333534204520153202018220+=⋅+=+==92⋅91010=⋅80.222325155315628151515+=⋅+=+=

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Chapter 1Variables, Real Numbers, and Mathematical Models1081.3533528128312291019242424+=⋅+⋅=+=82.32332210151031529413303030+=⋅+⋅=+=83.112112211471891892181818−=−⋅=−=84.1741742178918918921818189−=−⋅=−==19⋅122=⋅85.4344333434431697121212−=⋅−⋅=−=86.323322232332945666−=⋅−⋅=−=87.7378351016108165561541808080−=⋅−⋅=−=88.7574553024304245282531201201203−=⋅−⋅=−==13⋅14040=⋅89.311573243431537443344528175or 112121212−=−=⋅−⋅=−=90.21115323232112533223221571or 16666−=−=⋅−⋅=−=91.72827 828278282172x=⋅=⋅=42⋅⋅2812828, true==The given number is a solution.92.53035 183035 18303153x=⋅=⋅=63⋅⋅3013030, true==The given number is a solution.93.2334w−=52311234172312341783121249312433 , true44−=−=−===The given number is a solution.

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Section 1.2Fractions in Algebra1194.3744w−=1372 244537244103744477 , true44−=−=−==The given number is a solution.95.11203211201212321 12112203121204121612, falsezz−=−⋅=⋅−⋅=⋅−==The given number is not a solution.96.111242111220204212012012412112510710, falsezz−=−⋅=⋅−⋅=⋅−==The given number is not a solution.97.213937yy+=2271273935335729⋅+⋅=9⋅31353⋅+3⋅93357693353571533573 5⋅=+==⋅75⋅3733 , true77==The given number is a solution.98.25236yy+=215111233632454233638202918810299182922, true⎛⎞⎛⎞+=⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞+=⎜⎟⎜⎟⎝⎠⎝⎠+=+===The given number is a solution.99.11(2)(4)3511(262)(264)3511(24)(30)3586, falsexx−=+−=+==The given number is not a solution.100.13(2)3(34)2813(42)3(3 44)2813(2)3(124)28313(8)843, falsexx−+=−−+=⋅−+=−+==The given number is not a solution.

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Chapter 1Variables, Real Numbers, and Mathematical Models12101.27(6)(2)39yy÷+=÷−12174642333913621327313319131213173633291321371836913123914181818182525 , true1818⎛⎞⎛⎞÷+=÷−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞÷+=÷−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞⋅+=⋅−⎜⎟⎜⎟⎝⎠⎝⎠+=−+=−=The given number is a solution.102.15(6)(2)39yy÷+=÷−212526223339861825313319811815363329818518369862410181818181414 , true1818⎛⎞⎛⎞÷+=÷−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞÷+=÷−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞⋅+=⋅−⎜⎟⎜⎟⎝⎠⎝⎠+=−+=−=The given number is a solution.103.15x104.16x105.14xx−106.13xx−107.1142xx−=108.1132xx−=109.111278xx+=110.1115910xx+=111.2 (6)3x+112.3 (9)4x+113.2633xx+=−114.3924xx+=−115.333454 520aaa⋅⋅==⋅116.2272 71437333aaaa⋅÷=⋅==⋅117.11911920xxxx++==118.1061064yyyy−−==119.115325238668156818826530⎛⎞⎛⎞−÷=−÷⎜⎟⎜⎟⎝⎠⎝⎠=÷=⋅==42⋅41515=⋅120.11112132242344663546364518220⎛⎞⎛⎞⎛⎞⎛⎞+÷+=+÷+⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠=÷=⋅==92⋅91010=⋅

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Section 1.2Fractions in Algebra13121.()1112525xx⎛⎞+=−⎜⎟⎝⎠15151258285152151581285151612585882404012111758240211740804217 , false8080⎛⎞⎛⎞+=−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞+=−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞+=−⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠==The given number is not a solution.122.()()123243xxx−−=−+111123 324 3332221237247731121122112374221122⎛⎞⎛⎞−−=⋅−+⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞−−=⋅−+⎜⎟⎜⎟⎝⎠⎝⎠⋅⎛⎞−−=⎜⎟⎝⎠712⋅76221233141311212249281322221515 , true22⎛⎞−+⎜⎟⎝⎠⎛⎞−=−⎜⎟⎝⎠−=−=The given number is a solution.123.5 (32)955(6832)(36)2099CFC=−=−==68 F°is equivalent to20 C.°124.5 (32)955(4132)(9)599CFC=−=−==41 F°is equivalent to5 C.°125.a.()()()72201072202010720010140HaH=−=−==The lower limit of the heart rate for a 20-year-old with this exercise goal is 140 beats perminute.b.()()()4 22054 2202054 2005160HaH=−=−==The upper limit of the heart rate for a 20-year-old with this exercise goal is 160 beats perminute.126.a.()()()1 22021 2203021 190295HaH=−=−==The lower limit of the heart rate for a 30-year-old with this exercise goal is 95 beats perminute.b.()()()3 22053 2203053 1905114HaH=−=−==The upper limit of the heart rate for a 30-year-old with this exercise goal is 114 beats perminute.

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