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

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’SRESOURCEMANUALINTRODUCTORYALGEBRAFORCOLLEGESTUDENTSEIGHTHEDITIONRobert BlitzerMiami Dade College

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’s Resource Manual with TestsIntroductory Algebra for College Students, Eighth EditionRobert BlitzerTABLE OF CONTENTSMINI-LECTURES(per section)ML-1Chapter 1ML-1Chapter 2ML-9Chapter 3ML-17Chapter 4ML-29Chapter 5ML-34Chapter 6ML-43Chapter 7ML-49Chapter 8ML-57Chapter 9ML-63Mini-Lectures AnswersIncluded at end of sectionADDITIONALEXERCISES(per section)AE-1Chapter 1AE-1Chapter 2AE-49Chapter 3AE-88Chapter 4AE-133Chapter 5AE-170Chapter 6AE-215Chapter 7AE-251Chapter 8AE-302Chapter 9AE-338Additional Exercises AnswersAE-377GROUP ACTIVITIES(per chapter)A-1Chapter 1A-1Chapter 2A-2Chapter 3A-3Chapter 4A-4Chapter 5A-5Chapter 6A-6Chapter 7A-7Chapter 8A-8Chapter 9A-9Group Activities AnswersA-10

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ML-1Mini Lecture 1.1Introduction to Algebra: Variables and Mathematical ModelsLearning Objectives:1.Evaluate algebraic expressions.2.Translate English phrases into algebraic expressions.3.Determine whether a number is a solution of an equation.4.Translate English sentences into algebraic equations.5.Evaluate formulas.Examples:1.Evaluate each expression forx= 5.a.)3(4xb.xx31562.Evaluate each expression forx= 3 andy= 6.a.)(5yxb.yyx2323.Write each English phrase as an algebraic expression. Letxrepresent the number.a. the difference of a number and sixb. eight more than four times a numberc. four less than the quotient of a number and twelve4.Determine whether the given number is a solution of the equation.a.x– 8 = 12; 20b. 4x– 7 = 9; 3c. 3(y– 5) = 6; 75.Write each English sentence as an equation. Letxrepresent the number.a. The product of a number and seven is twenty-one.b. The difference of twice a number and three is equal to twenty-seven.c. Six less than three times a number is the same as the number increased by twelve.Teaching Notes:It may be helpful to draw students’ attention to the word “evaluate.” Help them seethe letters v – a – l – u . This will help them remember that evaluate means to find thevalue of an expression.Students often make mistakes with the phrase “less than” so they should be cautionedabout the order of the subtraction.Translating from English to algebra is an important skill that will be used often.Answers: 1a. 8 b. 12a. 45 b. 2 3a.x–6 b. 4x+ 8 c.412x4a. yes b. not a solutionc. yes 5a. 7x= 21 b. 2x– 3 = 27 c. 3x– 6 =x+ 12

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ML-2Mini Lecture 1.2Fractions in AlgebraLearning Objectives:1.Convert between mixed numbers and improper fractions.2.Write the prime factorization of a composite number.3.Reduce or simplify fractions.4.Multiply fractions.5.Divide fractions.6.Add and subtract fractions with identical denominators.7.Add and subtract fractions with unlike denominators.8.Solve problems involving fractions in algebra.Examples:1. Convert each mixed number to an improper fraction.a.1073b.738c.325d.4192. Convert each improper fraction to a mixed number.a.813b.1112c.325d.7373. Give the prime factorization of each of the following composite numbers.a. 24b. 48c. 90d. 1084. What makes a number a prime?5. Reduce the following fractions to lowest terms by factoring each numerator and denominatorand dividing out common factors.a.1210b.4832c.5024d.98776. Perform the indicated operation. Always reduce answer, if possible.a.43+61b.818+313c.107-83d.121110-414e.191897f.412326g.87÷43h.835÷412Teaching Notes:When teaching factorization, it is often helpful to review divisibility rules.To add or subtract fractions, you must have a LCD.To divide fractions, multiply by the reciprocal of the divisor.To multiply or divide mixed numbers, change to improper fractions first.Answers:1. a. 37/10 b. 59/7 c. 17/3 d. 37/4 2. a. 1 5/8 b. 1 1/11 c. 8 1/3 d. 5 2/73. a. 2 · 2 · 2 · 3b. 2 · 2 · 2 · 2 · 3 c. 2 · 3 · 3 · 5 d. 2 · 2 · 3 · 3 · 34. a number whose only factors are 1 and itself 5. a. 5/6 b. 2/3 c. 12/25 d. 11/14 6. a. 11/12b.24275or 11 11/24 c. 13/40 d.320or 6 2/3 e. 14/19 f. 15 g. 7/6 or 1 1/6 h. 43/18 or 2 7/18

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ML-3Mini Lecture 1.3The Real NumbersLearning Objectives:1.Define the sets that make up the set of real numbers.2.Graph numbers on a number line.3.Express rational numbers as decimals.4.Classify numbers as belonging to one or more sets of the real numbers.5.Understand and use inequality symbols.6.Find the absolute value of a real number.Examples:1.Answer the following questions about each number:Is it a natural number?Is it rational?Is it a whole number?Is it irrational?Is it an integer?Is it a real number?a. 18b. –3.5c.5d. 0e.43f.πg. –5h. 0.452.Graph each number on the number line.a. 5.5b.416c.412d. –3.23. Express each rational number as a decimal.a.87b.119c.35d.414. Use> or < to compare the numbers.a. 18□–20b. –16□–13c. –4.3□–6.2d.74□118e.53□235. Give the absolute value.a.8b.5c.2.3d.22Teaching Notes:Make sure the students have minimal understanding of square roots.Absolute value is ALWAYS POSITIVE because it measures distance from zero.Remind students that a number cannot be rational and irrational.To change a rational number to a decimal, divide the numerator by the denominator.Answers: 1. a. natural, whole, integer, rational , realb. rational, realc. irrational, reald. whole, integer, rational, reale. rational, realf. irrational, realg., integer, rational, realh. rational, real 2. See below 3. a. 0.875b. 0.81c. 0.6d. 0.254. a. >b. <c. >d. <e. <5. a. 8b. 5c. 3.2d. 22

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ML-4Mini Lecture 1.4Basic Rules of AlgebraLearning Objectives:1.Understand and use the vocabulary of algebraic expressions.2.Use commutative properties.3.Use associative properties.4.Use the distributive property.5.Combine like terms.6.Simplify algebraic expressions.Examples:1.Fill in the blanks.Algebraic Expression# of termscoefficientslike termsa. 6y– 3x– 4y +8__________________________b. 5x2+ 2y– 2x2+ 9 – 3y__________________________c. 6x2– 9y+ 4x+ 8 –y+ 5__________________________2. Name the property being illustrated and then simplify if possible.a. 6(x+ 2) = 6x+ 12________________________________b. (9 • 12)5 = 9(12 • 5)________________________________c. (x+ 4) + 8 =x+ (4 + 8)________________________________d. (2)(3.14)(5) = 2(5)(3.14)________________________________3. Simplify.a. 6x–x+ 2x= ____________b. 3a – 8 + 2a + 10 = _______________c. 6(x+ 3) – 5 = ___________d. 2 (x– 4) – (x– 2) = ______________e. 5 (y– 2) + 3(4 –y) = ____________Teaching Notes:A coefficient is the number factor of a term.Like terms have the very same variables raised to the same exponents.When applying the commutative property, only the order changes.The commutative property holds for addition and multiplication only.When applying the associative property the grouping changes.The associative property holds for addition and multiplication only.When combining like terms, add or subtract the coefficients, the variable part remains thesame.Always use parentheses when substituting a value for a variable.Answers: 1 a. 4; 6, –3, –4, 8; 6yand –4yb. 5; 5, 2, –2, 9, –3; 5x2and –2x2; 2yand –3yc. 6; 6, –9, 4, 8, –1, 5; 9yand –y; 8 and 52. a. distributiveb. associative of multiplicationc. associative of additiond. commutative of multiplication3. a. 7xb. 5a +2c. 6x + 13d.x– 6e. 2y+2

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ML-5Mini Lecture 1.5Addition of Real NumbersLearning Objectives:1.Add numbers with a number line.2.Find sums using identity and inverse properties.3.Add numbers without a number line.4.Use addition rules to simplify algebraic expressions.5.Solve applied problems using a series of additions.Examples:1.Find each sum using a number line.a. 3 + –5b. –4 + –6c. –1 + 2d. 5 + 42.Add without using a number line.a. –7 + –11b. –0.4 + –3.2c.103–54–d. –15 + 4e. 7.1 + 8.5f. –8 + 25g. –6.4 + 6.1h.43–853.Simplify the following.a. –30x+ 5xb. –2y+ 5x+ 8x+ 3yc. –2(3x+ 5y) + 6(x+ 2y)4.Write a sum of signed numbers that represents the following situation. Then, add to findthe overall change.If the stock you purchased last week rose 2 points, then fell 4, rose 1, fell 2, and rose 1,what was the overall change for the week?Teaching Notes:When adding numbers with like signs, add and take the sign.When adding numbers with unlike signs, subtract the smaller absolute value from thelarger absolute value, and the answer will have the sign of the number with the largerabsolute value.Answers: 1. a. –2b. –10c. 1d. 92. a. –18b. –3.6c.1011–or1011–d. –11e. 15.6f. 17 g. –0.3h.81–3. a. –25xb. 13x+yc. 2y4. 2 + (–4) + 1 + (–2) + 1 = –2; fell 2 points

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ML-6Mini Lecture 1.6Subtraction of Real NumbersLearning Objectives:1.Subtract real numbers.2.Simplify a series of additions and subtractions.3.Use the definition of subtraction to identify terms.4.Use the subtraction definition to simplify algebraic expressions.5.Solve problems involving subtraction.Examples:1.Subtract by changing each subtraction to addition of the opposite first.a. 6 – 12b. –15 – 15c. 13 – 21d.65–52e. 4.2 – 6.8f. 25 – (–25)g. –51 – (–13)h. 14 – (–13)2.Simplify.a. –16 – 14 – (–10)b. –20.3 – (–40.1) – 18c. 15 – (–3) – 10 – 18d. –11 – 21 – 31 – 413.Identify the number of terms in each expression; then name the terms.a. 4x– 6y+ 12 – 3yb. 16 – 2x– 15c. 15a– 2ab+ 3b– 6a+ 18d. 5y–x+ 3y– 14xy4.Simplify each algebraic expression.a. 8x+ 7 –xb. –11y– 14 + 2y– 10c. 15a– 10 – 12a+ 12d. 25 – (–3x) – 15 – (–2x)5.Applications.a. The temperature at dawn was –7 degrees but fortunately the sun came out and by 4:00p.m. the temperature had reached 38 degrees. What was the difference in the temperatureat dawn and 4:00 p.m.?b. Express 214 feet below sea level as a negative integer. Express 10,510 above sea levelas a positive integer. What is the difference between the two elevations?Teaching Notes:Say the problem to yourself. When you hear the word “minus”, immediately make a“change-change”. That means to “change” the subtraction to addition and “change” thesign of the number that follows to its opposite.Remember, the sign in front of a term goes with the term.The symbol “–”can have different meanings:1. subtract or “minus” only when it is between 2 terms2. the opposite of3. negativeAnswers: 1. a. –6b. –30c. –8d.3013–e. –2.6f. 50g. –38h. 272. a. –20b. 1.8c. –10d. –1043. a. 4 terms; 4x,–6y, 12, –3yb. 3 terms; 16, –2x, –15c. 5 terms; 15a, –2ab, 3b, –6a, 18d. 4 terms; 5y, –x, 3y, –14xy4. a. 7x+ 7b. –9y– 24c. 3a+ 2d. 5x+ 105. a. 45 degreesb. –214 feet. 10,500 feet; 10, 724 feet

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ML-7Mini Lecture 1.7Multiplication and Division of Real NumbersLearning Objectives:1.Multiply real numbers.2.Multiply more than two real numbers.3.Find multiplicative inverses.4.Use the definition of division.5.Divide real numbers.6.Simplify algebraic expressions involving multiplication.7.Determine whether a number is a solution of an equation.8.Use mathematical models involving multiplication and division.Examples:1.Multiply.a. (3)(–4)b. (–6)(–5)c. (–8)(0)d. (–3.2)(–1.1)e.9243–f. (–5)(2)(–1)g. (–2)(2)(–3)(–3)2.Find the multiplicative inverse of each number.a. –8b.52c. –7d.413.Use the definition of division to find each quotient.a. –49 ÷ 7b.4–24–4.Divide or state that the expression is undefined.a.018–b.252054–c. –32.4 ÷ 8d. 0 ÷ –85. Simplify.a. –3(2x)b. 9x+xc. –12a+ 4ad. –(5x– 3)e. –2(3y+4)f. 2(3x+4) – (4x–6)Teaching Notes:The product of an even number of negative numbers is positive.The product of an odd number of negative numbers is negative.Any product using zero as a factor will equal zero.The quotient of two real numbers with different signs is negative.The quotient of two real numbers with same signs is positive.Division of a non-zero number by zero is undefined.Any non-zero number divided into 0 is 0.Answers: 1. a. –12b. 30c. 0d. 3.52e.61–f. 10g. –362. a.81–b.25c.71–d.143. a. –7b. 64. a. undefinedb. –1c. –4.05d. 05.a. –6xb. 10xc. –8ad. –5x+ 3e. –6y– 8f. 2x+ 14

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ML-8Mini Lecture 1.8Exponents and Order of OperationsLearning Objectives:1.Evaluate exponential expressions.2.Simplify algebraic expressions with exponents.3.Use order of operations agreement.4.Evaluate mathematical models.Examples:1.Identify the base and the exponent, then evaluate.a. 34b. (–4)3c. –82d. (–8)22.Evaluate.a. 132b. 25c. (–3)3d. 523.Simplify if possible.a. 6x2–x2b. 5y3+ 2y– 3y3c. 6a2+ 2a– 4a2– 6ad. 10p3– 8p24.Simplify by using the order of operations.a.30 ÷ 2 · 3 – 52b. 14 – (33 ÷ 11) +4c.(5 + 2)2d. 10 – 7(32 ÷ 8) + 5 · 3e.23141f. 15 – 3[8 – (–12 ÷ 22) – 42]g.)5(––2–84162h. 22 + 5(x+ 7) – 3x– 105.5. Evaluate each expression for the given value.a. –a–a2ifa= –3b. –a–a2ifa= 3c. 4x2–x+ 3xifx= –16.Use the formula for perimeter of a rectangle,P= 2w+ 2lto find the perimeter of arectangle if the length is 28 cm and the width is 15 cm.Teaching Notes:If the negative sign is part of the base, it will be inside the parentheses.NEVERmultiply the base and the exponent together.The exponent tells how many times to write the base as a factor.Always use parentheses when substituting a value for a variable.The Order of Operations must be followed on every problem.Answers: 1. a. 81b. –64c. –64d. 642. a. 169b. 32c. –27d. 253. a. 5x2b. 2y3+ 2yc. 2a2– 4ad. 10p3– 8p24. a. –7b. 15c. 49d. –3e.3613f. 30g. 6h. 2x + 475. a. –6b. –12c. 26. 86 cm

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ML-9Mini Lecture 2.1The Addition Property of EqualityLearning Objectives:1.Identify linear equations in one variable.2.Use the addition property of equality to solve equations.3.Solve applied problems using formulas.Examples:1.Identify the linear equations in one variable.a.107xb.722xc.53xd.6|1|x2.Solve the following equations using the addition property of equality. Be sure tocheck your proposed solution.a.x+ 2 = 17b. –12 =x– 9c.421xd. 3x– 2x= 8e. 5x+ 1 = 4(x– 2)f.x+ 3.5 = 4.8g. 2x+ 5 =x– 2h. 3x+ 5 = 2x+ 53. If Sue is 2 years older than John then we will use S to represent Sue’s age and J torepresent John’s age. Use the equationS=J+ 2 to find John’s age if Sue is 41.Teaching Notes:Solving an equation is the process of finding the number (or numbers) that make theequation a true statement. These numbers are called the solutions, or roots, or theequation.To apply the addition property of equality, one must add the same number or expressionto both sides of the equation.Equivalent equations are equations that have the same solution.Answers: 1. a. linear b. not linear c. not linear d. not linear2. a. 15 b. -3 c.29or214d. 8e. -9f. 1.3 g. -7 h. 0 3. 39

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ML-10Mini Lecture 2.2The Multiplication Property of EqualityLearning Objectives:1.Use multiplication property of equality to solve equations.2.Solve equations in the form.cx3.Use addition and multiplication properties to solve equations.4.Solve applied problems using formulas.Examples:1.Multiply both sides of the equation by the reciprocal of the coefficient of the variable tosolve for the variable.a.63x=b.72x= --c.1015y= -d.83x= -2.Divide both sides of the equation by the coefficient of the variable to solve for thevariable.a. 6x= 18b. –2x= –14c. 15y= –10d. 24 = –3xBoth of the above methods of isolating the variable are effective for solving equations.3.Solve each equation by multiplying or dividing.a. 18y= –108b.1253xc. 124 = 3xd. –7x= –634.Multiply or divide both sides of each equation by –1 to get a positivex.a. –x= –7b. 82 = –xc. –a=73–d. 14 = –x5.Solve each equation using both the addition and multiplication properties of equality.a. 3x– 5 = 13b. 18 – 6x= 14 – 2xc. 23 = 2a– 7d. –6y– 21 = 21e. 33 –x= 3x– 11f.126–32xTeaching Notes:Remind students that reciprocals always have the same sign.When students see –xthey must realize the coefficient is –1.Answers: 1. a.x= 18b.x= 14c.–150y=d.x= –242. a.x= 3b.x= 7c.32–yd.x= –83. a.y= –6b.x= 20c.x= 372d.x= 94. a.x= 7b.x= –82c.73ad.x= –145. a.x= 6b.x= 1c.a= 15d.y= –7e.x= 11f.x= 27

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ML-11Mini Lecture 2.3Solving Linear EquationsLearning Objectives:1.Solve linear equations.2.Solve linear equations containing fractions.3.Solve linear equations containing decimals.4.Identify equations with no solution or infinitely many solutions.5.Solve applied problems using formulas.Examples:1. 3x+ 2x+ 8 = -7 +x+ 112. 6x= 3(x+ 9)3. 5(2x-1) – 15 = 3(4x+ 2) + 44.15732x5x5. 1.2x+ 1.8 = 0.6x6. 1.3x+ 1.7 = –1 – 1.4x7. 2x+ 9 = 2(x+ 4)8. 4(x+ 2) + 5 = 5(x+ 1) + 89. Use the formulaP= 4sto find the length of a side of a square whose perimeter is32 in.Teaching Notes:Simplify the algebraic expression on each side of the equal sign.Collect variable terms on one side of the equal sign and all constant terms on the otherside of the equal sign.Isolate the variable and solve.Check your solution in the original expression.Answers: 1. –12. 93. –154. –15.–36. –17. inconsistent, no solution8. 09. 8 inches

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ML-12Mini Lecture 2.4Formulas and PercentsLearning Objectives:1.Solve a formula for a variable.2.Use the percent formula.3.Solve applied problems involving percent change.Examples:1.Solve the formula for the indicated variable by isolating the variable.a.A=221BBforB1b.P=a+ b +cforcc.A=πr2hforhd. 4p+H=MforpeCByAxforAf.bmxyforb2.Translate each question into an equation using the percent formula,PBA, then solvethe equation.a. What is 15 percent of 60?b. 62% of what number is 31?c. What percent of 132 is 33?d. 60 is what percent of 500?3.The average, or meanAof the three exam grades,x,y,z, is given by formula3xyzA++=.a.Solve the formula forz.b.If your first two exams are 75% and 83% (x= 75,y= 83), what must you get onthe third exam to have an average of 80%?Teaching Notes:Many students have trouble solving formulas for a letter and need to be reminded thesame steps are used when solving for a letter in a formula as are used when solving anyequation for a variable.When changing a decimal to a percent, move the decimal point two places to the rightand use the % symbol.When changing a percent to a decimal, move the decimal point two places to the left anddrop the % symbol.When translating English into a mathematical equation, the word “is” translates to equalsand the word “of” means multiply.Answers:1. a.21–2BABb.baPc––c.2rAhd.4–HMpe.xByCAf.mxyb2. a.)60(15.0x; 9b. 0.62x= 31; 50 c.%25;33132xd.%12;50060x3. a.z= 3A–x–yb. 82%

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ML-13Mini Lecture 2.5An Introduction to Problem SolvingLearning Objectives:1.Translate English phrases into algebraic expressions.2.Solve algebraic word problems using linear equations.Examples:1.Translate each English phrase into an algebraic expression. Let “x” represent theunknown.a. Three times a number decreased by 11.b. The product of seven and a number increased by 2.c. Eight more than a number.2.Translate each sentence into an algebraic equation and then solve the equation.a. Twice a number less five is eleven.b. Five times the sum of a number and eight is 30.3.Identify all unknowns, set up an equation, and then solve.a. Bill earns five dollars more per hour than Joe. Together their pay for one hour totals$21. How much does each man earn per hour?b. Two consecutive even integers equal 42. Find the integers.Teaching Notes for solving algebraic equations:Make sure to familiarize all students with basic mathematical terms and the proper way totranslate to algebraic terms.First, read the problem carefully and assign a variable for one of the unknown quantities.Write expressions if necessary for any other unknown quantities in terms of samevariable.Write an equation for the stated problem.Solve the equation and answer the question.Check the solution in the original stated problem.Answers: 1. a. 3x– 112. a. 2x– 5 = 11b. 7x+ 2x= 8c.x+ 8b. 5(x+ 8) = 30x= –23. a.x= Joeb.x= 1steven integerx+ 5 = Billx+ 2 = 2ndeven integerx+ (x+ 5) = 21x+ (x+ 2) = 42x= $8 (Joe)x= 20x+ 5 = $13 (Bill)x+ 2 = 22

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