Solution Manual for Beginning Algebra, 13th Edition

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SOLUTIONSMANUALEMILYKEATONBEGINNINGALGEBRATHIRTEENTHEDITIONMargaret L. LialAmerican River CollegeJohn HornsbyUniversity of New OrleansTerry McGinnis

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Table of ContentsRPrealgebra Review................................................................................................................... 1R.1R.2Fractions •.................................................................................................................Decimals and Percents • ...........................................................................................1161The Real Number System .......................................................................................................281.11.21.31.41.51.61.7Exponents, Order of Operations, and Inequality • ....................................................Variables, Expressions, and Equations • ...................................................................Real Numbers and the Number Line • ......................................................................Adding and Subtracting Real Numbers • ..................................................................Multiplying and Dividing Real Numbers • ...............................................................Summary Exercises: Performing Operations with Real Numbers•.........................Properties of Real Numbers •....................................................................................Simplifying Expressions •.........................................................................................Chapter 1 Review Exercises • ...................................................................................Chapter 1 Mixed Review Exercises •........................................................................Chapter 1 Test •.........................................................................................................2834424757676977838890Chapters R–1 Cumulative Review Exercises • .......................................................922Linear Equations and Inequalities in One Variable............................................................942.12.22.32.42.52.62.72.82.9The Addition Property of Equality • .......................................................................The Multiplication Property of Equality •...............................................................Solving Linear Equations Using Both Properties of Equality• ...............................Clearing Fractions and Decimals When Solving Linear Equations •......................Summary Exercises: Applying Methods for Solving Linear Equations• ...............Applications of Linear Equations • .........................................................................Formulas and Additional Applications from Geometry • .......................................Ratio, Proportion, and Percent •..............................................................................Further Applications of Linear Equations •.............................................................Solving Linear Inequalities •...................................................................................Chapter 2 Review Exercises • .................................................................................Chapter 2 Mixed Review Exercises •......................................................................Chapter 2 Test •.......................................................................................................Chapters R–2 Cumulative Review Exercises • .......................................................941041131211351391551681811932102192212243Linear Equations and Inequalities in Two Variables; Functions.........................................2283.13.23.33.43.53.63.7Linear Equations and Rectangular Coordinates •....................................................Graphing Linear Equations in Two Variables • ......................................................The Slope of a Line • ..............................................................................................Slope-Intercept Form of a Linear Equation • ..........................................................Point-Slope Form of a Linear Equation and Modeling •.........................................Summary Exercises: Applying Graphing and Equation-Writing Techniquesfor Lines• ........................................................................................................Graphing Linear Inequalities in Two Variables •....................................................Introduction to Functions •......................................................................................Chapter 3 Review Exercises • .................................................................................Chapter 3 Mixed Review Exercises •......................................................................228240258268282293298307314320

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Chapter 3 Test •.......................................................................................................Chapters R–3 Cumulative Review Exercises • .......................................................3233254Systems of Linear Equations and Inequalities .......................................................................3284.14.24.34.44.5Solving Systems of Linear Equations by Graphing • ..............................................Solving Systems of Linear Equations by Substitution •..........................................Solving Systems of Linear Equations by Elimination • ..........................................Summary Exercises: Applying Techniques for Solving Systems ofLinear Equations•...........................................................................................Applications of Linear Systems •............................................................................Solving Systems of Linear Inequalities • ................................................................Chapter 4 Review Exercises • .................................................................................Chapter 4 Mixed Review Exercises •......................................................................Chapter 4 Test •.......................................................................................................Chapters R–4 Cumulative Review Exercises • .......................................................3283413523653713893984054084125Exponents and Polynomials ....................................................................................................4165.15.25.35.45.55.65.7The Product Rule and Power Rules for Exponents •................................................Integer Exponents and the Quotient Rule • ..............................................................Summary Exercises: Applying the Rules for Exponents• .......................................Scientific Notation •.................................................................................................Adding, Subtracting, and Graphing Polynomials • ..................................................Multiplying Polynomials • .......................................................................................Special Products • ....................................................................................................Dividing Polynomials • ............................................................................................Chapter 5 Review Exercises • ..................................................................................Chapter 5 Mixed Review Exercises •.......................................................................Chapter 5 Test •........................................................................................................Chapters R–5 Cumulative Review Exercises • ........................................................4164244334374464594714784934995015046Factoring and Applications.......................................................................................................5086.16.26.36.46.56.6Greatest Common Factors; Factoring by Grouping • .............................................Factoring Trinomials • ............................................................................................More on Factoring Trinomials • .............................................................................Special Factoring Techniques • ..............................................................................Summary Exercises: Recognizing and Applying Factoring Strategies• ...............Solving Quadratic Equations Using the Zero-Factor Property •.............................Applications of Quadratic Equations •....................................................................Chapter 6 Review Exercises •.................................................................................Chapter 6 Mixed Review Exercises • .......................................................................Chapter 6 Test • ......................................................................................................Chapters R–6 Cumulative Review Exercises • .......................................................508517527539552556567579583585587iv

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7Rational Expressions and Applications ...................................................................................5917.17.27.37.47.57.67.77.8The Fundamental Property of Rational Expressions • ...........................................Multiplying and Dividing Rational Expressions • .................................................Least Common Denominators • .............................................................................Adding and Subtracting Rational Expressions • ....................................................Complex Fractions •...............................................................................................Solving Equations with Rational Expressions • .....................................................Summary Exercises: Simplifying Rational Expressions vs.Solving Rational Equations•............................................................................Applications of Rational Expressions •..................................................................Variation •..............................................................................................................Chapter 7 Review Exercises • ................................................................................Chapter 7 Mixed Review Exercises •.....................................................................Chapter 7 Test •......................................................................................................Chapters R–7 Cumulative Review Exercises • ......................................................5916056156256436576796826957027127157198Roots and Radicals ....................................................................................................................7248.18.28.38.48.58.6Evaluating Roots • .................................................................................................Multiplying, Dividing, and Simplifying Radicals • ...............................................Adding and Subtracting Radicals • ........................................................................Rationalizing the Denominator •............................................................................More Simplifying and Operations with Radicals •.................................................Summary Exercises: Applying Operations with Radicals• ..................................Solving Equations with Radicals • .........................................................................Chapter 8 Review Exercises • ................................................................................Chapter 8 Mixed Review Exercises •.....................................................................Chapter 8 Test •......................................................................................................Chapters R–8 Cumulative Review Exercises • ......................................................7247357437517597707737897957977999Quadratic Equations .................................................................................................................8059.19.29.39.4Solving Quadratic Equations by the Square Root Property •.................................Solving Quadratic Equations by Completing the Square • ....................................Solving Quadratic Equations by the Quadratic Formula • .....................................Summary Exercises: Applying Methods for Solving Quadratic Equations• ........Graphing Quadratic Equations • ............................................................................Chapter 9 Review Exercises • ................................................................................Chapter 9 Mixed Review Exercises •.....................................................................Chapter 9 Test •......................................................................................................Chapters R–9 Cumulative Review Exercises • ......................................................805814831847853868874876880v

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R.1 Fractions1Chapter RPrealgebra ReviewR.1 FractionsClassroom Examples, Now Try Exercises1.90 is composite and can be written asWriting 90 as the product of primes gives us902 3 3 5.N1.60 is composite and can be written asWriting 60 as the product of primes gives us602 2 3 5.2.(a)123 434331205 45455(b)8811486 86 16(c)905 185511629 1899N2.(a)305 656551427 67677(b)101011707 107 17 (c)723 243311205 24553.The fraction bar represents division. Divide thenumerator of the improper fraction by thedenominator.310 37307Thus, 3773.1010N3.The fraction bar represents division. Divide thenumerator of the improper fraction by thedenominator.185 92542402Thus,92218.554.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N4.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5.(a)To multiply two fractions, multiply theirnumerators and then multiply theirdenominators. Then simplify and write theanswer in lowest terms.5 185 189 2590229252552 455 45

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Chapter R Prealgebra Review2(b)To multiply two mixed numbers, first writethem as improper fractions. Multiply theirnumerators and then multiply theirdenominators. Then simplify and write theanswer as a mixed number in lowest terms.31073110 73 42 5 73 2 25, or 5134343566N5.(a)To multiply two fractions, multiply theirnumerators and then multiply theirdenominators. Then simplify and write theanswer in lowest terms.454 5787 820565 414 4514(b)To multiply two mixed numbers, first writethem as improper fractions. Multiply theirnumerators and then multiply theirdenominators. Then simplify and write theanswer as a mixed number in lowest terms.22172036535317 205 317 5 45 3682, or 22336.(a)To divide fractions, multiply by thereciprocal of the divisor.953 3 52 5 331, or93105101 232(b)Change both mixed numbers to improperfractions. Then multiply by the reciprocal ofthe second fraction.3111102344311340433310N6.(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7.To find the sum of two fractions having thesame denominator, add the numerators andkeep the same denominator.9961515933 39232

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R.1 Fractions3N7.To find the sum of two fractions having thesame denominator, add the numerators andkeep the same denominator.1313888481 42 4128.(a)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(b)Write each mixed number as an improperfraction.56312974263The least common denominator is 6, sowrite each fraction with a denominator of 6.296and772143326Now add.72914292963666436141, or 7 6N8.(a)Since122 2 3and82 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(b)Write each mixed number as an improperfraction.148513453548The least common denominator is 8, sowrite each fraction with a denominator of 8.458and13226413428Now add.13452645264548888717, or 8889.(a)Since102 5and42 2,the leastcommon denominator is2 2 520.Writeeach fraction with a denominator of 20.33261010220and115545420Now subtract.31651102020420(b)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 78N9.(a)Since1111and 93 3,the leastcommon denominator is3 3 1199.Writeeach fraction with a denominator of 99.559451111 999and22 112299 1199Now subtract.52452223119999999

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Chapter R Prealgebra Review4(b)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 210.To find out how many yards of fabric Jenshould buy, add the lengths needed for eachpiece to obtain the total length. The commondenominator is 12.12138617112112443212121212Because17511212, we have17554415121212. Jen should buy5512ydof fabric.N10.To find out how long each piece must be,divide the total length by the number of pieces.121421 121104221248 ,or52 8Each piece should be52 8feet long.11.(a)In the circle graph, the sector for Other isthe second largest, so Other had the secondlargest share of Internet users,23 .100(b) The total number of Internet users, 3900million, can be rounded to 4000 million (or4 billion). Multiply110by 4000.14000400 million10(c)Multiply the fraction from the graph forAfrica by the actual number of users.13900390 million10N11.(a)In the circle graph, the sector for Africa isthe smallest, so Africa had the least numberof Internet users.(b) The total number of Internet users, 3900million, can be rounded to 4000 million (or4 billion). Multiply12by 4000.1 40002000 million,2or 2 billion(c)Multiply the fraction from the graph forAsia by the actual number of users.1 39001950 million,2or 1.95 billionExercises1.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 number 1 is neither prime norcomposite.5.False; the fraction 1339 can be written in lowestterms as13since1313 11 .3913 336.False; the reciprocal of632is21 .637.False;productrefers to multiplication, so theproduct of 10 and 2 is 20. Thesumof 10 and 2is 12.8.False;differencerefers to subtraction, so thedifference between 10 and 2 is 8. Thequotientof 10 and 2 is 5.9.162 82243 83Therefore, Cis correct.

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R.1 Fractions510.Simplify each fraction to find which are equalto 59 .3 5531529796 5563059492 20202 3740743711 5511559999Therefore, Cis correct.11.A common denominator forpqandrsmust bea multiple of both denominators,qands. Sucha number is.q sTherefore, A is correct.12.We need to multiply 8 by 3 to get 24 in thedenominator, so we must multiply 5 by 3as well.55 31588 324Therefore, B is correct.13.Since 19 has only itself and 1 as factors, it is aprime number.14.Since 31 has only itself and 1 as factors, it is aprime number.15.2 152 3305Since 30 has factors other than itself and 1, it isa composite number.16.502 252 5 5,so 50 is a composite number.17.2 322 2 162 2 2 82 2 2 2 42 242 2262Since 64 has factors other than itself and 1, it isa composite number.18.273 3 983 3133 3Since 81 has factors other than itself and 1, it isa composite number.19.As stated in the text, the number 1 is neitherprime nor composite, by agreement.20.The number 0 is not a natural number, so it isneither prime nor composite.21.573 19,so 57 is a composite number.22.513 17,so 51 is a composite number.23.Since 79 has only itself and 1 as factors, it is aprime number.24.Since 83 has only itself and 1 as factors, it is aprime number.25.2 621242 2 31,so 124 is a composite number.26.2 691382 3 23,so 138 is a composite number.27.2 2502 2 1252 2 5 252 2 55 05 5,0so 500 is a composite number.28.2 3502 2 1752 2 5 352 2 57 05 7,0so 700 is a composite number.29.2 17292 7 2472 7 13 193458Since 3458 has factors other than itself and 1, itis a composite number.30.20515 5 401255Since 1025 has factors other than itself and 1, itis a composite number.31.81 818111162 8282232.1 414111123 43433433.153 535551183 63666

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Chapter R Prealgebra Review634.164 44424415 45054535.903 303 303 11505 3053055336.100514075 205205 17 20720737.181 181 18111905 185 185538.161 161 16111644 164441639.1446 246246611205 245245540.13212 1112 11121217 11711777741.17 1275Therefore, 1251.7742.19 1697Therefore, 1671.9943.612 77725Therefore,7756.121244.615 1019011Therefore,101116.151545.711 83776Therefore,8367.111146.513 67652Therefore,6725.131347.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48.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49.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.8 1080and80383The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,38310.8850.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

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R.1 Fractions751.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52.Multiply the denominator of the fraction by thenatural number and then add the numerator toobtain the numerator of the improper fraction.6 18108and1081109The denominator of the improper fraction is thesame as the denominator in the mixed number.Thus,110918.6653.464 624575 73554.25 21079 7596355.36115 8120222 31 615 820 6056.351512021423 51 1520 21280815257.1121 121 2 6610510 52 5 52558.1 101 101 2 558 72 4 7288759.15815 84254 253 5 4 24 5 53 2561,or 15560.21 421 478 73 7 44 2 731, or81 2261.321 321 71721 31 73 7 31 73 39162.436436136 41 94 9 41 94 41699163.Change both mixed numbers to improperfractions.1213 531434313 54 3655, or5121264.Change both mixed numbers to improperfractions.233338 821558 85644, or 4151565.Change both mixed numbers to improperfractions.3119 1623858519 168 519 2 88 5383, or 755

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Chapter R Prealgebra Review866.Change both mixed numbers to improperfractions.31184337565618 435 63 6 435 63 4351294, or 255567.Change both numbers to improper fractions.1215 210105 211 105 2512111 2 521, or 101 212268.Change both numbers to improper fractions.2383 4 993 381 93 38131381 33 3, o38233r 1269.To divide fractions, multiply by the reciprocalof the divisor.3377292914277 29 370.To divide fractions, multiply by the reciprocalof the divisor.6564161411 52415455171.To divide fractions, multiply by the reciprocalof the divisor.535848435 84 35 4 24 35 23101, or 33372.To divide fractions, multiply by the reciprocalof the divisor.37 101037 105 37 2 55 3755142, or 43373.To divide fractions, multiply by the reciprocalof the divisor.32832 155155832 155 88 4 3 51 5 84 3121

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R.1 Fractions974.To divide fractions, multiply by the reciprocalof the divisor.24242124 217 66721764 6 3 71 7 64 312175.To divide fractions, multiply by the reciprocalof the divisor.3311244 123 14 123 14 3 4114 41676.To divide fractions, multiply by the reciprocalof the divisor.22130302 15 302 15 2 15115 15557577.To divide fractions, multiply by the reciprocalof the divisor.36565136 51 32 3 51 32 510178.To divide fractions, multiply by the reciprocalof the divisor.489818 91 42 4 91 4299181479.Change the first number to an improperfraction, and then multiply by the reciprocal ofthe divisor.332736 48482784327 84 33 9 2 44 39 218180.Change the first number to an improperfraction, and then multiply by the reciprocal ofthe divisor.372875 528 1028 105 74105107 2 55 74 2815781.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.15512212727572 125 72 123511, or 12424

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Solution Manual for Beginning Algebra, 13th Edition - Page 15 preview image

Chapter R Prealgebra Review1082.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.2220721520 59 7951003952097637, or 1 6383.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.51521472121 3221 328 4721 8 48 4721 4478437, or8328328174747484.Change both mixed numbers to improperfractions, and then multiply by the reciprocal ofthe divisor.342392123510923 52 5 9235, or 181051051885.7474111515151586.25257999987.71711212128122 43 42388.53511616138616289.Since 93 3,and 3 is prime, the LCD (leastcommon denominator) is3 39.113333 39Now add the two fractions with the samedenominator.515389399990.To add415and1 ,5first find the LCD. Since153 5and 5 is prime, the LCD is 15.414135534315151515437151591.Since82 2 2and62 3,the LCD is2 2 224.333 3988324and 5542066424Now add fractions with the same denominator.35920298624244 ,2or51 2492.Since62 3and 93 3,the LCD is2 3 318.5533156618and222992418Now add fractions with the same denominator.515,691818241918or1118

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Solution Manual for Beginning Algebra, 13th Edition - Page 16 preview image

R.1 Fractions1193.Since93 3and164 4,the LCD is3 3 4 4144.168099 1556144and3392716169144Now add fractions with the same denominator.53802791614414410714494.Since42 2and 255 5,the LCD is2 2 5 5100.3325754425100and6642425254100Now add fractions with the same denominator.3675249942510001001 095.1124125338888811819224444411259328484Since82 2 2and42 2,the LCD is2 2 2or 8.1125923284842251888433, or 58896.2122144411121132223333366666Since 62 3,the LCD is 6.362114213462 63262813415,66or697.11121133344444445491155555Since42 2,and 5 is prime, the LCD is2 2 520.1413 59431454554653620201011, or 5202098.To add5 34and11,3first change to improperfractions then find the LCD, which is 12.3123513323 34443346916121, or 7 1244412851299.727259999100.8383111115111101.13313315151510152 523 53102.113113121212122 448233103.Since124 3(12 is a multiple of 3), the LCDis 12.1443412Now subtract fractions with the samedenominator.717431 311231212124 34

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