Solution Manual for Algebra for College Students, 9th Edition

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SOLUTIONSMANUALEMILYKEATONALGEBRA FORCOLLEGESTUDENTSNINTHEDITIONMargaret L. LialAmerican River CollegeJohn HornsbyUniversity of New OrleansTerry McGinnis

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Table of ContentsRReview of the Real Number System.......................................................................................................1R.1R.2R.3R.4R.5Fractions, Decimals, and Percents •...................................................................................Basic Concepts from Algebra • .........................................................................................Operations on Real Numbers • ..........................................................................................Exponents, Roots, and Order of Operations •....................................................................Properties of Real Numbers • ............................................................................................Chapter R Test • ................................................................................................................119243038421Linear Equations, Inequalities, and Applications ............................................................................. 451.11.21.31.41.51.61.7Linear Equations in One Variable •...................................................................................Formulas and Percent •......................................................................................................Applications of Linear Equations •....................................................................................Further Applications of Linear Equations • .......................................................................Summary Exercises Applying Problem-Solving Techniques• ...........................................Linear Inequalities in One Variable • ................................................................................Set Operations and Compound Inequalities • ....................................................................Absolute Value Equations and Inequalities •.....................................................................Summary Exercises Solving Linear and Absolute Value Equationsand Inequalities•....................................................................................................Chapter 1 Review Exercises •............................................................................................Chapter 1 Mixed Review Exercises • ................................................................................Chapter 1 Test • .................................................................................................................Chapters R and 1 Cumulative Review Exercises• .............................................................4555667988921051131281321381411442Linear Equations, Graphs, and Functions........................................................................................1472.12.22.32.42.52.6Linear Equations in Two Variables •.................................................................................The Slope of a Line • .........................................................................................................Writing Equations of Lines• .............................................................................................Summary Exercises Finding Slopes and Equations of Lines•...........................................Linear Inequalities in Two Variables • ..............................................................................Introduction to Relations and Functions • .........................................................................Function Notation and Linear Functions •.........................................................................Chapter 2 Review Exercises •............................................................................................Chapter 2 Mixed Review Exercises • ................................................................................Chapter 2 Test • .................................................................................................................Chapters R2 Cumulative Review Exercises •..................................................................147161175189192202209217223224227

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3Systems of Linear Equations..............................................................................................................2313.13.23.3Systems of Linear Equations in Two Variables • ..............................................................Systems of Linear Equations in Three Variables • ............................................................Applications of Systems of Linear Equations • .................................................................Chapter 3 Review Exercises •............................................................................................Chapter 3 Mixed Review Exercises • ................................................................................Chapter 3 Test • .................................................................................................................Chapters R3 Cumulative Review Exercises •..................................................................2312522722902972993034Exponents, Polynomials, and Polynomial Functions........................................................................3074.14.24.34.44.54.6Integer Exponents • ...........................................................................................................Scientific Notation • ..........................................................................................................Adding and Subtracting Polynomials •..............................................................................Polynomial Functions, Graphs, and Composition •...........................................................Multiplying Polynomials •.................................................................................................Dividing Polynomials •......................................................................................................Chapter 4 Review Exercises •............................................................................................Chapter 4 Mixed Review Exercises • ................................................................................Chapter 4 Test • .................................................................................................................Chapters R4 Cumulative Review Exercises •..................................................................3073163213283413523643703713745Factoring..............................................................................................................................................3795.15.25.35.45.5Greatest Common Factors and Factoring by Grouping • ....................................................Factoring Trinomials •.........................................................................................................Special Factoring • ..............................................................................................................A General Approach to Factoring •.....................................................................................Solving Quadratic Equations Using the Zero-Factor Property • .........................................Chapter 5 Review Exercises • .............................................................................................Chapter 5 Mixed Review Exercises •..................................................................................Chapter 5 Test •...................................................................................................................Chapters R5 Cumulative Review Exercises • ...................................................................379385394402410422427428430

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6Rational Expressions and Functions .................................................................................................4346.16.26.36.46.56.6Rational Expressions and Functions; Multiplying and Dividing • ......................................Adding and Subtracting Rational Expressions •..................................................................Complex Fractions •............................................................................................................Equations with Rational Expressions and Graphs • ............................................................Summary Exercises Simplifying Rational Expressions vs. SolvingRational Equations• ................................................................................................Applications of Rational Expressions •...............................................................................Variation • ...........................................................................................................................Chapter 6 Review Exercises • .............................................................................................Chapter 6 Mixed Review Exercises •..................................................................................Chapter 6 Test •...................................................................................................................Chapters R6 Cumulative Review Exercises • ...................................................................4344434574684834875045145205235277Roots, Radicals, and Root Functions.................................................................................................5327.17.27.37.47.57.67.7Radical Expressions and Graphs •.......................................................................................Rational Exponents • ...........................................................................................................Simplifying Radicals, the Distance Formula, and Circles • ................................................Adding and Subtracting Radical Expressions • ...................................................................Multiplying and Dividing Radical Expressions • ................................................................Summary Exercises Performing Operations with Radicals andRational Exponents• ................................................................................................Solving Equations with Radicals • ......................................................................................Complex Numbers • ............................................................................................................Chapter 7 Review Exercises • .............................................................................................Chapter 7 Mixed Review Exercises •..................................................................................Chapter 7 Test •...................................................................................................................Chapters R7 Cumulative Review Exercises • ...................................................................532539547559567579581594602609611614

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8Quadratic Equations and Inequalities...............................................................................................6198.18.28.38.48.5The Square Root Property and Completing the Square • .....................................................The Quadratic Formula •......................................................................................................Equations That Lead to Quadratic Methods • ......................................................................Summary Exercises Applying Methods for Solving Quadratic Equations• .........................Formulas and Further Applications • ...................................................................................Polynomial and Rational Inequalities • ................................................................................Chapter 8 Review Exercises • ..............................................................................................Chapter 8 Mixed Review Exercises •...................................................................................Chapter 8 Test •....................................................................................................................Chapters R8 Cumulative Review Exercises • ....................................................................6196346496746786927137217247289Additional Graphs of Functions and Relations ................................................................................7359.19.29.39.49.5Review of Operations and Composition • .........................................................................Graphs of Quadratic Functions • .......................................................................................More About Parabolas and Their Appications •................................................................Symmetry; Increasing and Decreasing Functions •...........................................................Piecewise Linear Functions • ............................................................................................Chapter 9 Review Exercises • ...........................................................................................Chapter 9 Mixed Review Exercises •................................................................................Chapter 9 Test •.................................................................................................................Chapters R9 Cumulative Review Exercises • .................................................................73574875877378379580180380610Inverse, Exponential, and Logarithmic Functions ...........................................................................81210.110.210.310.410.510.6Inverse Functions •............................................................................................................Exponential Functions • ....................................................................................................Logarithmic Functions •....................................................................................................Properties of Logarithms • ................................................................................................Common and Natural Logarithms • ..................................................................................Exponential and Logarithmic Equations; Further Applications •......................................Chapter 10 Review Exercises • .........................................................................................Chapter 10 Mixed Review Exercises • ..............................................................................Chapter 10 Test •...............................................................................................................Chapters R10 Cumulative Review Exercises • ...............................................................812823833844850856867874876879

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11Polynomial and Rational Functions...................................................................................................88611.111.211.311.4Zeros of Polynomial Functions (I) •..................................................................................Zeros of Polynomial Functions (II) •.................................................................................Graphs and Applications of Polynomial Functions • ........................................................Summary Exercises Examining Polynomial Functions and Graphs•...............................Graphs and Applications of Rational Functions • .............................................................Chapter 11 Review Exercises • .........................................................................................Chapter 11 Mixed Review Exercises • ..............................................................................Chapter 11 Test •...............................................................................................................Chapters R11 Cumulative Review Exercises • ...............................................................88689290492593194995795896112Conic Sections and Nonlinear Systems..............................................................................................96812.112.212.312.4Circles Revisited and Ellipses • ........................................................................................Hyperbolas and Functions Defined by Radicals •.............................................................Nonlinear Systems of Equations • ....................................................................................Second-Degree Inequalities, Systems of Inequalities, and Linear Programming •...........Chapter 12 Review Exercises •.........................................................................................Chapter 12 Mixed Review Exercises • .............................................................................Chapter 12 Test • ..............................................................................................................Chapters R12 Cumulative Review Exercises • ...............................................................9689779861000102110281030103413Further Topics in Algebra................................................................................................................104113.113.213.313.413.513.613.7Sequences and Series • .....................................................................................................Arithmetic Sequences • ....................................................................................................Geometric Sequences •.....................................................................................................The Binomial Theorem •..................................................................................................Mathematical Induction • .................................................................................................Counting Theory •............................................................................................................Basics of Probability •......................................................................................................Chapter 13 Review Exercises • ........................................................................................Chapter 13 Mixed Review Exercises • .............................................................................Chapter 13 Test •..............................................................................................................Chapters R13 Cumulative Review Exercises • ..............................................................10411047105710691075108610931100110611081112

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R.1 Fractions, Decimals, and Percents1Chapter RReview of the Real Number SystemR.1 Fractions, Decimals, and PercentsClassroom Examples, Now Try Exercises1.(a)123 434331205 45455(b)8811486 86 16(c)905 185511629 1899N1.(a)305 656551427 67677(b)101011707 107 17 (c)723 243311205 24552.The fraction bar represents division. Divide thenumerator of the improper fraction by thedenominator.310 37307Thus,3773.1010N2.The fraction bar represents division. Divide thenumerator of the improper fraction by thedenominator.185 92542402Thus,92218.553.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.5 315and15419The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,4193.55N3.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.3 1133and33235The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,23511.334.To multiply two fractions, multiply theirnumerators and then multiply theirdenominators. Then simplify and write theanswer in lowest terms.5 185 189 2590229252552 455 45N4.To multiply two fractions, multiply theirnumerators and then multiply theirdenominators. Then simplify and write theanswer in lowest terms.454 5787 820565 414 45145.(a)To divide fractions, multiply by thereciprocal of the divisor.953 3 52 5 331, or93105101 232

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Chapter R Review of the Real Number System2(b)Change both mixed numbers to improperfractions. Then multiply by the reciprocal ofthe second fraction.3111102344311340433310N5.(a)To divide fractions, multiply by thereciprocal of the divisor.287978292 3 37 2 4289(b)To divide fractions, multiply by thereciprocal of the divisor.32153034474715743015 74 2 15786.(a)To find the sum of two fractions having thesame denominator, add the numerators andkeep the same denominator.9961515933 39232(b)Since302 3 5and453 3 5,the leastcommon denominator must have one factorof 2 (from 30), two factors of 3 (from 45),and one factor of 5 (from either 30 or 45),so it is2 3 3 590.Write each fraction with a denominatorof 90.732130739300and22244594520Now add.7221421425304590909090Write2590in lowest terms.255 559018 518(c)Since102 5and42 2,the leastcommon denominator is2 2 520.Writeeach fraction with a denominator of 20.33261010220and115545420Now subtract.31651102020420(d)Write each mixed number as an improperfraction.82312733182The least common denominator is 8. Writeeach fraction with a denominator of 8.278remains unchanged, and33412 .2248Now subtract.3271227121528,888827or1 78N6.(a)To find the sum of two fractions having thesame denominator, add the numerators andkeep the same denominator.1313888481 42 412(b)Since122 2 3and 82 2 2,the leastcommon denominator must have threefactors of 2 (from 8) and one factor of 3(from 12), so it is2 2 2 324.Write each fraction with a denominatorof 24.552101212224and3 39832384Now add.531091091912824242424

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R.1 Fractions, Decimals, and Percents3(c)Since1111and93 3,the leastcommon denominator is3 3 1199.Writeeach fraction with a denominator of 99.559451111 999and22 112299 1199Now subtract.52452223119999999(d)Write each mixed number as an improperfraction.151317423636The least common denominator is 6. Writeeach fraction with a denominator of 6.176remains unchanged, and 1313226 .3326Now subtract.1317261726179366666Now reduce.93 33622 ,3or11 27.(a)150.15100(b)90.0091000(c)5252.521010N7.(a)80.810(b)4310.4311000(c)582582.5821001008.(a)42.83071.0003.074116.904(b)32.5021.7210.78N8.(a)68.90042.7208.973120.593(b)351.8002.706349.0949.(a)30.21 decimal place0.0523 decimal places60415101341.57044 decimal places(b)0.062 decimal places0.122 decimal places1262240.00724 decimal places(c)To change the divisor 0.37 into a wholenumber, move each decimal point twoplaces to the right. Move the decimal pointstraight up and divide as with wholenumbers.14.837 547.6371771482962960Therefore,5.4760.3714.8.(d)To change the divisor 3.1 into a wholenumber, move each decimal point one placeto the right. Move the decimal point straightup and divide as with whole numbers.1.2131 37.6031666240319

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Chapter R Review of the Real Number System4We carried out the division to 2 decimalplaces so that we could round to 1 decimalplace. Therefore,3.763.11.2.N9.(a)9.322 decimal places1.41 decimal place372893221313.0483 decimal places(b)0.61 decimal place0.0043 decimal places241340.00244 decimal places(c)To change the divisor 1.3 into a wholenumber, move each decimal point one placeto the right. Move the decimal point straightup and divide as with whole numbers.5.64113 73.340658378545220137We carried out the division to 3 decimalplaces so that we could round to 2 decimalplaces. Therefore,7.3341.35.64.10.(a)Move the decimal point three places to theright.19.5100019,500(b)Move the decimal point one place to theleft.960.11096.01N10.(a)Move the decimal point one place to theright.294.72102947.2(b)Move the decimal point two places to theleft. Insert a 0 in front of the 4 to do this.4.7931000.0479311.(a)Divide 3 by 50. Add a decimal point and asmany 0s as necessary.0.0650 3.003 000Therefore,30.06.50(b)Divide 11 by 1. Add a decimal point and asmany 0s as necessary.0.090909...11 1.000000...9910099100991Note that the pattern repeats. Therefore,10.09,11or about0.091.N11.(a)Divide 20 by 17. Add a decimal point andas many 0s as necessary.0.8520 17.001601001000Therefore,170.85.20(b)Divide 2 by 9. Add a decimal point and asmany 0s as necessary.0.222...9 2.000...18201820182Note that the pattern repeats. Therefore,20.2,9or0.222.

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R.1 Fractions, Decimals, and Percents512.(a)85%0.85(b)110%1.10,or 1.1(c)0.3030%(d)0.16516.5%N12.(a)52%0.52(b)2%02%0.02(c)0.4545%(d)3.53.50350%13.(a)6565%100In lowest terms,6513 51310020 520(b)1.51.5101531.5%100100 101000200N13.(a)2020%100In lowest terms,201 2011005 205(b)160160%100In lowest terms,1608 2083, or 11005 205514.(a)33100%50503100 %5013 50 2 %506%(b)11 100%331 100 %31100 %3133%, or 33.3%3N14.(a)66100%25256100 %2516 25 4 %2524%(b)77 100%997 100 %91700 %9777%, or 77.7%9Exercises1.True; the number above the fraction bar iscalled the numerator and the number below thefraction bar is called the denominator.2.True; 5 divides the 31 six times with aremainder of one, so3116.553.False; this is an improper fraction. Its valueis 1.4.False; the reciprocal of632is21 .635.162 82243 83Therefore, Cis correct.6.Simplify each fraction to find which are equalto 59 .3 5531529796 5563059492 20202 3740743711 5511559999Therefore, Cis correct.7.81 818111162 82 822

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Chapter R Review of the Real Number System68.1 414111123 4343349.153 53 5551183 63 66610.164 44424415 45054511.903 303303 11505 3053055312.100514075 205205 17 20720713.181 181 18111905 185 185514.161 161 16111644 164441615.1446 246246611205 245245516.13212 1112 11121217 11711777717.17 1275Therefore,1251.7718.19 1697Therefore, 1671.9919.612 77725Therefore,7756.121220.513 67652Therefore,6725.131321.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.5 210and10313The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,3132.5522.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.7 535and35641The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,6415.7723.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.3 1236and36238The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,23812.3324.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.5 1050and50151The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,15110.5525.464 624575 73526.25 21079 7596327.36115 8120222 31 615 820 6028.351512021423 51 1520 212808152

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R.1 Fractions, Decimals, and Percents729.1121 121 2 6610510 52 5 52530.1 101 101 2 558 72 4 7288731.15815 84254 253 5 4 24 5 53 2561,or 15532.21 421 478 73 7 44 2 731, or81 2233.321 321 71721 31 73 7 31 73 39134.436436136 41 94 9 41 94 41699135.Change both mixed numbers to improperfractions.1213 531434313 54 3655, or5121236.Change both mixed numbers to improperfractions.233338 821558 85644, or 4151537.To divide fractions, multiply by the reciprocalof the divisor.3377292914277 29 338.To divide fractions, multiply by the reciprocalof the divisor.6564161411 52415455139.To divide fractions, multiply by the reciprocalof the divisor.535848435 84 35 4 24 35 23101, or 33340.To divide fractions, multiply by the reciprocalof the divisor.37 101037 105 37 2 55 3755142, or 433

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Chapter R Review of the Real Number System841.To divide fractions, multiply by the reciprocalof the divisor.32832 155155832 155 88 4 3 51 5 84 312142.To divide fractions, multiply by the reciprocalof the divisor.24242124 217 66721764 6 3 71 7 64 312143.To divide fractions, multiply by the reciprocalof the divisor.3311244 123 14 123 14 3 4114 41644.To divide fractions, multiply by the reciprocalof the divisor.22130302 15 302 15 2 15115 15557545.To divide fractions, multiply by the reciprocalof the divisor.36565136 51 32 3 51 32 510146.To divide fractions, multiply by the reciprocalof the divisor.489818 91 42 4 91 4299181447.Change the first number to an improperfraction, and then multiply by the reciprocal ofthe divisor.332736 48482784327 84 33 9 2 44 39 218148.Change the first number to an improperfraction, and then multiply by the reciprocal ofthe divisor.372875 528 1028 105 74105107 2 55 74 28157

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R.1 Fractions, Decimals, and Percents949.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.15512212727572 125 72 123511, or 1242450.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.2220721520 59 7951003952097637, or 1 6351.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.51521472121 3221 328 4721 8 48 4721 4478437, or8328328174747452.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.342392123510923 52 5 9235, or 181051051853.7474111515151554.25257999955.71711212128122 43 42356.53511616138616257.Since93 3,and 3 is prime, the LCD (leastcommon denominator) is3 39.113333 39Now add the two fractions with the samedenominator.515389399958.To add415and1 ,5first find the LCD. Since153 5and 5 is prime, the LCD is 15.414135534315151515437151559.Since82 2 2and62 3,the LCD is2 2 224.333 3988324and5542066424Now add fractions with the same denominator.35920298624244 ,2or51 2460.Since62 3and 93 3,the LCD is2 3 318.5533156618and222992418

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