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

Solution Manual for Introductory Algebra for College Students, 6th Edition simplifies complex textbook exercises with easy-to-understand solutions and step-by-step guides.

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    ’ S R ESOURCE M ANUAL I NTRODUCTORY A LGEBRA FOR C OLLEGE S TUDENTS E IGHTH E DITION Robert Blitzer Miami Dade College

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    ’s Resource Manual with Tests Introductory Algebra for College Students , Eighth Edition Robert Blitzer TABLE OF CONTENTS MINI-LECTURES (per section) ML-1 Chapter 1 ML-1 Chapter 2 ML-9 Chapter 3 ML-17 Chapter 4 ML-29 Chapter 5 ML-34 Chapter 6 ML-43 Chapter 7 ML-49 Chapter 8 ML-57 Chapter 9 ML-63 Mini-Lectures Answers Included at end of section ADDITIONAL EXERCISES (per section) AE-1 Chapter 1 AE-1 Chapter 2 AE-49 Chapter 3 AE-88 Chapter 4 AE-133 Chapter 5 AE-170 Chapter 6 AE-215 Chapter 7 AE-251 Chapter 8 AE-302 Chapter 9 AE-338 Additional Exercises Answers AE-377 GROUP ACTIVITIES (per chapter) A-1 Chapter 1 A-1 Chapter 2 A-2 Chapter 3 A-3 Chapter 4 A-4 Chapter 5 A-5 Chapter 6 A-6 Chapter 7 A-7 Chapter 8 A-8 Chapter 9 A-9 Group Activities Answers A-10

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    ML-1 Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models Learning 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 for x = 5. a. ) 3 ( 4  x b. x x 3 15 6  2. Evaluate each expression for x = 3 and y = 6. a. ) ( 5 y x  b. y y x 2 3 2  3. Write each English phrase as an algebraic expression. Let x represent the number. a. the difference of a number and six b. eight more than four times a number c. four less than the quotient of a number and twelve 4. Determine whether the given number is a solution of the equation. a. x – 8 = 12; 20 b. 4 x – 7 = 9; 3 c. 3( y – 5) = 6; 7 5. Write each English sentence as an equation. Let x represent 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 see the letters v – a – l – u . This will help them remember that evaluate means to find the value of an expression.  Students often make mistakes with the phrase “less than” so they should be cautioned about the order of the subtraction.  Translating from English to algebra is an important skill that will be used often. Answers: 1a. 8 b. 1 2a. 45 b. 2 3a. x –6 b. 4 x + 8 c. 4 12  x 4a. yes b. not a solution c. yes 5a. 7 x = 21 b. 2 x – 3 = 27 c. 3 x – 6 = x + 12

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    ML-2 Mini Lecture 1.2 Fractions in Algebra Learning 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. 10 7 3 b. 7 3 8 c. 3 2 5 d. 4 1 9 2. Convert each improper fraction to a mixed number. a. 8 13 b. 11 12 c. 3 25 d. 7 37 3. Give the prime factorization of each of the following composite numbers. a. 24 b. 48 c. 90 d. 108 4. What makes a number a prime? 5. Reduce the following fractions to lowest terms by factoring each numerator and denominator and dividing out common factors. a. 12 10 b. 48 32 c. 50 24 d. 98 77 6. Perform the indicated operation. Always reduce answer, if possible. a. 4 3 + 6 1 b. 8 1 8 + 3 1 3 c. 10 7 - 8 3 d. 12 11 10 - 4 1 4 e.             19 18 9 7 f.             4 1 2 3 2 6 g. 8 7 ÷ 4 3 h. 8 3 5 ÷ 4 1 2 Teaching 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 2/7 3. a. 2 · 2 · 2 · 3 b. 2 · 2 · 2 · 2 · 3 c. 2 · 3 · 3 · 5 d. 2 · 2 · 3 · 3 · 3 4. a number whose only factors are 1 and itself 5. a. 5/6 b. 2/3 c. 12/25 d. 11/14 6. a. 11/12 b. 24 275 or 11 11/24 c. 13/40 d. 3 20 or 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-3 Mini Lecture 1.3 The Real Numbers Learning 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. 18 b. –3.5 c. 5 d. 0 e. 4 3  f. π g. –5 h. 0.45 2. Graph each number on the number line. a. 5.5 b. 4 16  c. 4 1 2 d. –3.2 3. Express each rational number as a decimal. a. 8 7 b. 11 9 c. 3 5 d. 4 1 4. Use > or < to compare the numbers. a. 18 □ –20 b. –16 □ –13 c. –4.3 □ –6.2 d. 7 4 □ 11 8 e. 5 3  □ 2 3 5. Give the absolute value. a. 8 b. 5  c. 2 . 3  d. 22 Teaching 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 , real b. rational, real c. irrational, real d. whole, integer, rational, real e. rational, real f. irrational, real g., integer, rational, real h. rational, real 2. See below 3. a. 0.875 b. 0.81 c. 0.6 d. 0.25 4. a. > b. < c. > d. < e. < 5. a. 8 b. 5 c. 3.2 d. 22

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    ML-4 Mini Lecture 1.4 Basic Rules of Algebra Learning 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 terms coefficients like terms a. 6 y – 3 x – 4y +8 ________ __________ ________ b. 5 x 2 + 2 y – 2 x 2 + 9 – 3 y ________ __________ ________ c. 6 x 2 – 9 y + 4 x + 8 – y + 5 ________ __________ ________ 2. Name the property being illustrated and then simplify if possible. a. 6( x + 2) = 6 x + 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. 6 x – x + 2 x = ____________ 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 the same.  Always use parentheses when substituting a value for a variable. Answers: 1 a. 4; 6, –3, –4, 8; 6 y and –4 y b. 5; 5, 2, –2, 9, –3; 5 x 2 and –2 x 2 ; 2 y and –3 y c. 6; 6, –9, 4, 8, –1, 5; 9 y and – y ; 8 and 5 2. a. distributive b. associative of multiplication c. associative of addition d. commutative of multiplication 3. a. 7 x b. 5a +2 c. 6x + 13 d. x – 6 e. 2 y +2

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    ML-5 Mini Lecture 1.5 Addition of Real Numbers Learning 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 + –5 b. –4 + –6 c. –1 + 2 d. 5 + 4 2. Add without using a number line. a. –7 + –11 b. –0.4 + –3.2 c. 10 3 – 5 4 –  d. –15 + 4 e. 7.1 + 8.5 f. –8 + 25 g. –6.4 + 6.1 h. 4 3 – 8 5  3. Simplify the following. a. –30 x + 5 x b. –2 y + 5 x + 8 x + 3 y c. –2(3 x + 5 y ) + 6( x + 2 y ) 4. Write a sum of signed numbers that represents the following situation. Then, add to find the 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 the larger absolute value, and the answer will have the sign of the number with the larger absolute value. Answers: 1. a. –2 b. –10 c. 1 d. 9 2. a. –18 b. –3.6 c. 10 1 1 – or 10 11 – d. –11 e. 15.6 f. 17 g. –0.3 h. 8 1 – 3. a. –25 x b. 13 x + y c. 2 y 4. 2 + (–4) + 1 + (–2) + 1 = –2; fell 2 points

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    ML-6 Mini Lecture 1.6 Subtraction of Real Numbers Learning 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 – 12 b. –15 – 15 c. 13 – 21 d. 6 5 – 5 2 e. 4.2 – 6.8 f. 25 – (–25) g. –51 – (–13) h. 14 – (–13) 2. Simplify. a. –16 – 14 – (–10) b. –20.3 – (–40.1) – 18 c. 15 – (–3) – 10 – 18 d. –11 – 21 – 31 – 41 3. Identify the number of terms in each expression; then name the terms. a. 4 x – 6 y + 12 – 3 y b. 16 – 2 x – 15 c. 15 a – 2 ab + 3 b – 6 a + 18 d. 5 y – x + 3 y – 14 xy 4. Simplify each algebraic expression. a. 8 x + 7 – x b. –11 y – 14 + 2 y – 10 c. 15 a – 10 – 12 a + 12 d. 25 – (–3 x ) – 15 – (–2 x ) 5. Applications. a. The temperature at dawn was –7 degrees but fortunately the sun came out and by 4:00 p.m. the temperature had reached 38 degrees. What was the difference in the temperature at dawn and 4:00 p.m.? b. Express 214 feet below sea level as a negative integer. Express 10,510 above sea level as 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” the sign 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 terms 2. the opposite of 3. negative Answers: 1. a. –6 b. –30 c. –8 d. 30 13 – e. –2.6 f. 50 g. –38 h. 27 2. a. –20 b. 1.8 c. –10 d. –104 3. a. 4 terms; 4 x ,–6 y , 12, –3 y b. 3 terms; 16, –2 x , –15 c. 5 terms; 15 a , –2 ab , 3 b , –6 a , 18 d. 4 terms; 5 y , – x , 3 y , –14 xy 4. a. 7 x + 7 b. –9 y – 24 c. 3 a + 2 d. 5 x + 10 5. a. 45 degrees b. –214 feet. 10,500 feet; 10, 724 feet

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    ML-7 Mini Lecture 1.7 Multiplication and Division of Real Numbers Learning 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.             9 2 4 3 – f. (–5)(2)(–1) g. (–2)(2)(–3)(–3) 2. Find the multiplicative inverse of each number. a. –8 b. 5 2 c. –7 d. 4 1 3. Use the definition of division to find each quotient. a. –49 ÷ 7 b. 4 – 24 – 4. Divide or state that the expression is undefined. a. 0 18 – b. 25 20 5 4 –  c. –32.4 ÷ 8 d. 0 ÷ –8 5. Simplify. a. –3(2 x ) b. 9 x + x c. –12 a + 4 a d. –(5 x – 3) e. –2(3 y +4) f. 2(3 x +4) – (4 x –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. –12 b. 30 c. 0 d. 3.52 e. 6 1 – f. 10 g. –36 2. a. 8 1 – b. 2 5 c. 7 1 – d. 1 4 3. a. –7 b. 6 4. a. undefined b. –1 c. –4.05 d. 0 5.a. –6 x b. 10 x c. –8 a d. –5 x + 3 e. –6 y – 8 f. 2 x + 14

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    ML-8 Mini Lecture 1.8 Exponents and Order of Operations Learning 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. 3 4 b. (–4) 3 c. –8 2 d. (–8) 2 2. Evaluate. a. 13 2 b. 2 5 c. (–3) 3 d. 5 2 3. Simplify if possible. a. 6 x 2 – x 2 b. 5 y 3 + 2 y – 3 y 3 c. 6 a 2 + 2 a – 4 a 2 – 6 a d. 10 p 3 – 8 p 2 4. Simplify by using the order of operations. a. 30 ÷ 2 · 3 – 52 b. 14 – (33 ÷ 11) +4 c. (5 + 2) 2 d. 10 – 7(32 ÷ 8) + 5 · 3 e. 2 3 1 4 1              f. 15 – 3[8 – (–12 ÷ 2 2 ) – 4 2 ] g. ) 5 (– – 2 – 8 4 16 2   h. 22 + 5( x + 7) – 3 x – 10 5. 5. Evaluate each expression for the given value. a. – a – a 2 if a = –3 b. – a – a 2 if a = 3 c. 4 x 2 – x + 3 x if x = –1 6. Use the formula for perimeter of a rectangle, P = 2 w + 2 l to find the perimeter of a rectangle 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.  NEVER multiply 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. 81 b. –64 c. –64 d. 64 2. a. 169 b. 32 c. –27 d. 25 3. a. 5 x 2 b. 2 y 3 + 2 y c. 2 a 2 – 4 a d. 10 p 3 – 8 p 2 4. a. –7 b. 15 c. 49 d. –3 e. 36 13 f. 30 g. 6 h. 2x + 47 5. a. –6 b. –12 c. 2 6. 86 cm

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    ML-9 Mini Lecture 2.1 The Addition Property of Equality Learning 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. 10 7   x b. 7 2 2   x c. 5 3  x d. 6 | 1 |   x 2. Solve the following equations using the addition property of equality. Be sure to check your proposed solution. a. x + 2 = 17 b. –12 = x – 9 c. 4 2 1   x d. 3 x – 2 x = 8 e. 5 x + 1 = 4( x – 2) f. x + 3.5 = 4.8 g. 2 x + 5 = x – 2 h. 3 x + 5 = 2 x + 5 3. If Sue is 2 years older than John then we will use S to represent Sue’s age and J to represent John’s age. Use the equation S = 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 the equation a true statement. These numbers are called the solutions, or roots, or the equation.  To apply the addition property of equality, one must add the same number or expression to 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 linear 2. a. 15 b. -3 c. 2 9 or 2 1 4 d. 8 e. -9 f. 1.3 g. -7 h. 0 3. 39

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    ML-10 Mini Lecture 2.2 The Multiplication Property of Equality Learning Objectives: 1. Use multiplication property of equality to solve equations. 2. Solve equations in the form . c x   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 to solve for the variable. a. 6 3 x = b. 7 2 x = - - c. 10 15 y = - d. 8 3 x = - 2. Divide both sides of the equation by the coefficient of the variable to solve for the variable. a. 6 x = 18 b. –2 x = –14 c. 15 y = –10 d. 24 = –3 x Both of the above methods of isolating the variable are effective for solving equations. 3. Solve each equation by multiplying or dividing. a. 18 y = –108 b. 12 5 3  x c. 124 = 3 x d. –7 x = –63 4. Multiply or divide both sides of each equation by –1 to get a positive x . a. – x = –7 b. 82 = – x c. – a = 7 3 – d. 14 = – x 5. Solve each equation using both the addition and multiplication properties of equality. a. 3 x – 5 = 13 b. 18 – 6 x = 14 – 2 x c. 23 = 2 a – 7 d. –6 y – 21 = 21 e. 33 – x = 3 x – 11 f. 12 6 – 3 2  x Teaching Notes:  Remind students that reciprocals always have the same sign.  When students see – x they must realize the coefficient is –1. Answers: 1. a. x = 18 b. x = 14 c. –150 y = d. x = –24 2. a. x = 3 b. x = 7 c. 3 2 –  y d. x = –8 3. a. y = –6 b. x = 20 c. x = 372 d. x = 9 4. a. x = 7 b. x = –82 c. 7 3  a d. x = –14 5. a. x = 6 b. x = 1 c. a = 15 d. y = –7 e. x = 11 f. x = 27

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    ML-11 Mini Lecture 2.3 Solving Linear Equations Learning 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. 3 x + 2 x + 8 = -7 + x + 11 2. 6 x = 3( x + 9) 3. 5(2 x -1) – 15 = 3(4 x + 2) + 4 4. 15 7 3 2x 5   x 5. 1.2 x + 1.8 = 0.6 x 6. 1.3 x + 1.7 = –1 – 1.4 x 7. 2 x + 9 = 2( x + 4) 8. 4( x + 2) + 5 = 5( x + 1) + 8 9. Use the formula P = 4 s to find the length of a side of a square whose perimeter is 32 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 other side of the equal sign.  Isolate the variable and solve.  Check your solution in the original expression. Answers: 1. –1 2. 9 3. –15 4. –1 5. –3 6. –1 7. inconsistent, no solution 8. 0 9. 8 inches

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    ML-12 Mini Lecture 2.4 Formulas and Percents Learning 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 = 2 2 1 B B  for B 1 b. P = a + b + c for c c. A = π r 2 h for h d. 4 p + H = M for p e C By Ax   for A f. b mx y   for b 2. Translate each question into an equation using the percent formula, PB A  , then solve the 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 mean A of the three exam grades, x , y , z , is given by formula 3 x y z A + + = . a. Solve the formula for z . b. If your first two exams are 75% and 83% ( x = 75, y = 83), what must you get on the third exam to have an average of 80%? Teaching Notes:  Many students have trouble solving formulas for a letter and need to be reminded the same steps are used when solving for a letter in a formula as are used when solving any equation for a variable.  When changing a decimal to a percent, move the decimal point two places to the right and use the % symbol.  When changing a percent to a decimal, move the decimal point two places to the left and drop the % symbol.  When translating English into a mathematical equation, the word “is” translates to equals and the word “of” means multiply. Answers: 1. a. 2 1 – 2 B A B  b. b a P c – –  c. 2 r A h   d. 4 – H M p  e. x By C A   f. mx y b   2. a. ) 60 ( 15 . 0  x ; 9 b. 0.62 x = 31; 50 c. % 25 ; 33 132   x d. % 12 ; 500 60   x 3. a. z = 3 A – x – y b. 82%

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    ML-13 Mini Lecture 2.5 An Introduction to Problem Solving Learning 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 the unknown. 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 to translate 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 same variable.  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. 3 x – 11 2. a. 2 x – 5 = 11 b. 7 x + 2 x = 8 c. x + 8 b. 5( x + 8) = 30 x = –2 3. a. x = Joe b. x = 1 st even integer x + 5 = Bill x + 2 = 2 nd even integer x + ( x + 5) = 21 x + ( x + 2) = 42 x = $8 (Joe) x = 20 x + 5 = $13 (Bill) x + 2 = 22

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