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Solution Manual for Algebra: A Combined Approach, 6th Edition

Solution Manual for Algebra: A Combined Approach, 6th Edition provides expert solutions to textbook questions, making complex problems easier to solve.

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    R ESOURCE M ANUAL A LGEBRA : A C OMBINED A PPROACH S IXTH E DITION Elayn Martin-Gay University of New Orleans

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    Table of Contents Mini-Lectures (M) Chapter R .......................................................................................................................................... 1 Chapter 1 ........................................................................................................................................... 5 Chapter 2 ......................................................................................................................................... 13 Chapter 3 ......................................................................................................................................... 21 Chapter 4 ......................................................................................................................................... 27 Chapter 5 ......................................................................................................................................... 31 Chapter 6 ......................................................................................................................................... 39 Chapter 7 ......................................................................................................................................... 47 Chapter 8 ......................................................................................................................................... 55 Chapter 9 ......................................................................................................................................... 61 Chapter 10 ....................................................................................................................................... 65 Chapter 11 ....................................................................................................................................... 73 Chapter 12 ....................................................................................................................................... 79 Chapter 13 ....................................................................................................................................... 87 Answers........................................................................................................................................... 93 Additional Exercises (E) Chapter R .......................................................................................................................................... 1 Chapter 1 ......................................................................................................................................... 11 Chapter 2 ......................................................................................................................................... 25 Chapter 3 ......................................................................................................................................... 39 Chapter 4 ......................................................................................................................................... 59 Chapter 5 ......................................................................................................................................... 69 Chapter 6 ......................................................................................................................................... 83 Chapter 7 ......................................................................................................................................... 97 Chapter 8 ....................................................................................................................................... 113 Chapter 9 ....................................................................................................................................... 129 Chapter 10 ..................................................................................................................................... 139 Chapter 11 ..................................................................................................................................... 153 Chapter 12 ..................................................................................................................................... 169 Chapter 13 ..................................................................................................................................... 189 Answers......................................................................................................................................... 205 Group Activities (G) Chapter R .......................................................................................................................................... 1 Chapter 1 ........................................................................................................................................... 3 Chapter 2 ........................................................................................................................................... 5 Chapter 3 ........................................................................................................................................... 7 Chapter 4 ........................................................................................................................................... 9 Chapter 5 ......................................................................................................................................... 11 Chapter 6 ......................................................................................................................................... 13 Chapter 7 ......................................................................................................................................... 16 Chapter 8 ......................................................................................................................................... 18 Chapter 9 ......................................................................................................................................... 20 Chapter 10 ....................................................................................................................................... 22 Chapter 11 ....................................................................................................................................... 24 Chapter 12 ....................................................................................................................................... 26 Chapter 13 ....................................................................................................................................... 28 Answers........................................................................................................................................... 33 iii

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    Mini Lectures Answers

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    M-1 Mini-Lecture R.1 Factors and the Least Common Multiple Objectives: A. Write the factors of a number. B. Write the prime factorization of a number. C. Find the LCM of a list of numbers. Key Vocabulary: factor, factorization, natural numbers, prime number, composite number, prime factorization, divisibility tests (2, 3, 5), multiple, least common multiple (LCM) Examples: 1. List the factors of each number. a) 6 b) 40 c) 72 d) 37 2. Identify each number as prime or composite. a) 17 b) 304 c) 1803 d) 103 3. Write the prime factorization for each number. Do not use exponents. a) 50 b) 80 c) 400 d) 630 4. Find the LCM of each list of numbers. a) 3, 5 b) 6, 10 c) 40, 60 d) 2, 9, 21 Teaching Notes: • Be sure students clearly understand the difference between factors and prime factors. • Have students make a list of the first 15 prime numbers. • Many students prefer to use a factor tree for prime factorization. Encourage students to use the division method for prime factorization. • When finding the LCM, many students will just simply all the factors without regard to repetition. Answers: 1a) 1, 2, 3, 6; b) 1, 2, 4, 5, 8, 10, 20, 40; c) 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; d) 1, 37; 2a) prime, b) composite, c) composite, d) prime; 3a) ; 2 5 5 ⋅ ⋅ b) ; 2 2 2 2 5 ⋅ ⋅ ⋅ ⋅ c) ; 2 2 2 2 5 5 ⋅ ⋅ ⋅ ⋅ ⋅ d) ; 2 3 3 5 7 ⋅ ⋅ ⋅ ⋅ 4a) 15, b) 30, c) 120, d) 126

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    M-2 Mini-Lecture R.2 Fractions and Mixed Numbers Objectives: A. Discover fraction properties having to do with 0 and 1. B. Write equivalent fractions. C. Write fractions in simplest form. D. Multiply and divide fractions. E. Add and subtract fractions. F. Perform operations on mixed numbers. Key Vocabulary: fraction, numerator, denominator, whole numbers, equivalent fractions, simplified (in lowest terms), proper fraction, improper fraction, mixed number, reciprocal, equivalent fractions, least common denominator (LCD) Examples: 1. Simplify by dividing the numerator by the denominator. a) 3 3 b) 40 4 c) 16 1 d) 0 7 e) 12 0 2. Write each fraction as an equivalent fraction with the given denominator. a) 3 10 with a denominator of 40 b) 4 7 with a denominator of 56 3. Simplify the following fractions. a) 5 10 b) 9 15 c) 88 66 d) 300 550 4. Multiply or divide as indicated. a) 1 5 3 7 ⋅ b) 28 8 6 21 ⋅ c) 6 9 19 13 ÷ d) 9 3 14 10 ÷ 5. Add or subtract as indicated. a) 1 3 8 8 + b) 1 7 6 15 + c) 5 1 9 12 − d) 7 5 8 6 − 6. Perform the indicated operations on mixed numbers. a) 2 5 13 6 3 8 + b) 3 5 2 7 − c) 1 3 2 6 3 7 ⋅ d) 4 2 3 1 5 5 ÷ Teaching Notes: • Some students may need a visual review of the meaning of a fraction (i.e. drawing). • Most students have experience with fractions but may have forgotten the procedures. • Stress that all fractions should be written in simplified form. Answers: 1a) 1, b) 10, c) 16, d) 0, e) undefined; 2a) 12/40, b) 32/56; 3a) 1/2, b) 3/5, c) 4/3, d) 6/11; 4a) 5/21, b) 16/9, c) 26/57, d) 15/7; 5a) 1/2, b) 19/30, c) 17/36, d) 1/24; 6a) 20 7/24, b) 2 4/7, c) 15, d) 19/7

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    M-3 Mini-Lecture R.3 Decimals and Percents Objectives: A. Write decimals as fractions. B. Add, subtract, multiply, and divide decimals. C. Round decimals to a given decimal place. D. Write fractions as decimals. E. Write percents as decimals and decimals as percents. Key Vocabulary: decimal notation, place value, round, repeating decimal, percent (per 100) Examples: 1. Write the following decimals as fractions. Do not simplify. a) 0.7 b) 1.22 c) 0.299 d) 47.1 2. Add or subtract as indicated. a) 12.73 + 1.065 + 4.117 b) 44.18 – 13.266 3. Multiply or divide as indicated a) 0.037 6.2 ⋅ b) 0.18 0.0006 ÷ 4. Round each decimal to the given place value. a) 0.539, hundredth b) 2.35, tenth c) 27,003.0637, thousandth 5. Write each fraction as a decimal. If the decimal is a repeating decimal, round to the nearest thousandth. a) 2 5 b) 2 3 c) 3 8 d) 19 6 6. Change the following percents to decimals. a) 16% b) 5.1% c) 300% d) 0.7% 7. Change the following decimals to percents. a) 0.58 b) 2 c) 0.003 d) 3.45 Teaching Notes: • Refer students to the Place Value Chart . • Many students will benefit from a review of the rules for decimals. Answers: 1a) 7/10, b) 122/100, c) 299/1000, d) 471/10; 2a) 17.912, b) 30.914; 3a) 0.2294, b) 300; 4a) 0.54, b) 2.4, c) 27,003.064; 5a) 0.4, b) . . , 0 6 0 667 ≈ c) 0.375, d) . . ; 3 16 3 167 ≈ 6a) 0.16, b) 0.051, c) 3, d) 0.007; 7a) 58%, b) 200%, c) 0.3%, d) 345%

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    M-4 Mini-Lecture R.4 Reading Pictographs and Bar, Line, and Circle Graphs Objectives: A. Read pictographs. B. Read and construct bar graphs. C. Read and construct histograms (or frequency distribution graphs). D. Read line graphs. E. Read circle graphs. Key Vocabulary: pictograph, bar graph, frequency distribution table, histogram, class interval, class frequency, frequency distribution graph and table, line graph, circle graph Examples: Teaching Notes: • Encourage students to explore the internet for use of graphs. Answers: 1a) Jan, b) 1300; 2a) 35, b) 20, c) Lakers: 85, Celtics: 110; 3) 2,4,3,1,1; 4a) June, b) $16,500 Class Intervals Tally Class Frequency a 90–100 b 80–89 c 70–79 d 60–69 e Below 60 a) Which month had the most satisfied customers? b) What was the total number of satisfied customers during the 3-month period? a) How many points did the Celtics score in the 4 th quarter? b) How many points did the Lakers score in the 1 st quarter? c) What was the total score for the game? Month Number of Satisfied Customers Jan Feb Mar = 100 Satisfied Customers 2. The following bar graph shows the points scored per quarter in a basketball game. 1. The following pictograph shows the number of satisfied customers at a restaurant. Use the information to answer the following questions. 3. The following is a list of scores on a recent math exam. Use this list to complete the frequency distribution table: 100, 85, 89, 75, 60, 55, 92, 85, 85, 76, and 77. a) What month had the least sales? b) What is the difference between the highest and lowest sales? 4. The following line graph shows the total sales per month. 0 10 20 30 40 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Lakers Celtics

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    M-5 Mini-Lecture 1.1 Study Skill Tips for Success in Mathematics Objectives: A. Get ready for this course. B. Understand some general tips for success. C. Know how to use this text. D. Know how to use text resources. E. Get help as soon as you need it. F. Learn how to prepare for and take an exam. G. Develop good time management. Examples: 1. Get ready for this course. a) Positive attitude b) Be familiar with course structure c) Avoid schedule conflicts d) Allow adequate time for class arrival e) Bring all required materials 2. Understand some general tips for success. a) Organize materials b) Make contact with other students c) Choose to attend all classes d) Do your homework e) Check your work f) Learn from mistakes g) Ask questions h) Hand in assignments on time 3. Know how to use this text. a) Each example in every section has a Practice exercise associated with it. b) At beginning of each section, a list of icons shows availability of support materials. c) Each chapter ends with Chapter Highlights, Reviews, and Practice Tests. 4. Know how to use video and notebook organizer resources. a) Video resources include interactive lectures, test prep, and student success tips. b) Notebook organizer resources include video and student organizers 5. Get help as soon as you need it. 6. Learn how to prepare for and take an exam. a) Review previous homework assignments, class notes, quizzes, etc. b) Read Chapter Highlights to review concepts and definitions. c) Practice working out exercises in the end-of-the-chapter Review and Test. d) When taking a test, read directions and problems carefully. 7. Develop good time management. a) Make a list of all weekly commitments with estimated time needed. b) Be sure to schedule study time. Don’t forget to eat, sleep, and relax! Teaching Notes: • Many developmental students are hesitant to ask questions and seek extra help. • Be sure to explain your expectations. Keep your expectations clear and concise.

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    M-6 Mini-Lecture 1.2 Symbols and Sets of Numbers Objectives: A. Define the meaning of the symbols = , ≠ , <, >, ≤ , and >. B. Translate sentences into mathematical statements. C. Identify integers, rational numbers, irrational numbers, and real numbers. D. Find the absolute value of a real number. Key Vocabulary: set, member (element), variables, mathematical statements, integers, negative integers, positive integers, rational numbers, irrational numbers, real numbers, absolute value Examples: 1. Insert <, >, or = in the space between the paired numbers to make each statement true. a) 2 ____ 8 b) 41 ____ 14 c) 3 7 ____ 9 21 d) 2.12 ____ 2.12 − Determine whether each statement is true or false. e) 15 20 ≤ f) 3.002 3.202 ≥ g) 14 7 18 9 ≠ h) 6 11 7 14 ≥ 2. Translate each sentence into a mathematical statement. a) Negative eleven is less than or equal to negative four. b) Fourteen is greater than one. 3. Tell which set or sets each number belongs to: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. a) 5 b) 3 − c) 8 3 d) 5 e) 0 4. Find each absolute value. a) 6.2 b) 14 − c) 2 9 − d) 0.03 e) 0 Teaching Notes: • Encourage students to read an inequality statement from left to right. • Remind students that an inequality symbol < or > points toward the smaller value. • Plot values on a number line to visually illustrate less than or greater than. Answers: 1a) <, b) >, c) =, d) >, e) true, f) false, g) false, h) true; 2a) – 11 < – 4, b) 14 > 1, 3a) naturals, whole numbers, integers, rationals, reals; b) integers, rationals, reals; c) rationals, reals; d) irrationals., reals; e) whole numbers, integers, rationals, reals; 4a) 6.2, b) 14, c) 2/9, d) 0.03, e) 0

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    M-7 Mini-Lecture 1.3 Exponents, Order of Operations, and Variable Expressions Objectives: A. Define and use exponents and the order of operations. B. Evaluate algebraic expressions given replacement values for variables. C. Determine whether a number is a solution of a given equation. D. Translate phrases into expressions and sentences into equations. Key Vocabulary: exponential notation, base, exponent, exponential expression, grouping symbols, variables, algebraic expression, evaluate an algebraic expression, equation, solving, solution Examples: 1. Evaluate. a) 3 2 b) 7 1 c) 2 6 7       d) ( ) 3 0.3 2. Using the order of operations, simplify each expression. a) 7 3 · 2 + b) 2 25 3 2 − ⋅ c) ( ) 6 5 6 3 8 − + − +     d) ( ) ( )( ) ( ) 20 1 4 3 2 12 3 3 − − − − − ÷ − −     3. Evaluate each expression when x = 3, y = 2, and z = 6. a) x y z + + b) 3 x z − c) 2 5 z x − d) 2 5 3 z y x z − 4. Determine whether the given number is a solution of the given equation. a) 12 15; 27 x − = b) 12 29 : 7 y + = c) 3 15 ; 5 4 20 x = d) 3 2 : 0 y y = + 5. Write each phrase as an algebraic expression. a) The sum of a number and thirteen b) The quotient of forty-two and a number Write each sentence as an equation. c) The product of one-third and a number is nine. d) A number added to twelve is fourteen. Teaching Notes: • Be sure to identify base and exponent when working with exponential notation. • Some students may find order of operations challenging. • Many students will confuse expressions and equations. Be sure they understand that they will simplify an expression, but solve an equation. • Many students have problems translating sentences into equations. Answers: 1a) 8, b) 1, c) 36/49, d) 0.027; 2a) 13, b) 7, c) 150, d) – 8; 3a) 11, b) 3, c) 3, d) 8; 4a) Solution, b) Not a solution, c) Not a solution, d) Not a solution; 5a) x + 13, b) 42/x, c) 1/3 x = 9, d) 12 + x = 14

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    M-8 Mini-Lecture 1.4 Adding Real Numbers Objectives: A. Add real numbers. B. Find the opposite of a number. C. Evaluate algebraic expressions using real numbers. D. Solve applications that involve addition of real numbers. Key Vocabulary: opposites (additive inverses) Examples: 1. Add the following real numbers. a) 8 11 + b) ( ) ( ) 3 15 − + − c) ( ) ( ) 14 35 − + − d) 3 1 5 2     − + −         e) ( ) 9 5 − + f) ( ) 16 25 + − g) ( ) 15.3 27.03 − + h) 1 5 2 8     + −         i) ( ) 7 23 − + − j) 42 38 − + k) ( ) 53 22 + − l) 5 3 12 8     − +         2. Find the additive inverse (opposite). a) 8 b) 9 − c) 0 d) 17 − 3. a) Evaluate x y + for 2 x = − and 8. y = b) Evaluate 2 3 x y + for 4 x = and 5. y = − 4. Solve each of the following. a) At the beginning of a chemistry experiment, Amy measured the temperature of a liquid to be 5 C. − ° During the experiment, the temperature rose 14°C. What was the liquid’s temperature at the end of the experiment? b) A local restaurant reported net income of $1397, − $2042, and $809 − for the past three months. What was the total net income for the three months? Teaching Notes: • Some students will need to see addition performed on a number line. • Some students will need instruction with inputting negative numbers into a calculator. • Review the definition of absolute value. Answers: 1a) 19, b) – 18, c) – 49, d) – 11/10, e) – 4, f) – 9, g) 11.73, h) – 1/8, i) – 30, j) – 4, k) 31, l) – 1/24; 2a) – 8, b) 9, c) 0, d) – 17; 3a) 6, b) – 7; 4a) 9° C, b) – $164

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    M-9 Mini-Lecture 1.5 Subtracting Real Numbers Objectives: A. Subtract real numbers. B. Evaluate algebraic expressions using real numbers. C. Determine whether a number is a solution of a given equation. D. Solve applications that involve subtraction of real numbers. E. Find complementary and supplementary angles. Key Vocabulary: complementary angles, supplementary angles Examples: 1. Subtract. a) 8 4 − − b) 11 18 − c) ( ) 15 10 − − − d) 12 12 − − e) ( ) 22 13 − − f) ( ) 132 207 − − − g) ( ) 1.3 3.8 − h) 15 9 7 14   − −     2. Evaluate each expression when 3, x = − 7, y = − and 9. z = a) x y − b) 10 2 x y − − c) x y z + − d) 2 x y − 3. Determine whether the given number is a solution of the given equation. a) 8 14; 5 x + = − b) – 8 17; 9 y = − − c) 7 11; 2 x x − + = + − 4. In a game of cards, Alicia won 11 chips, lost 6 chips, won 3 chips, lost 14 chips, and won 1 chip. What was her final count of chips? 5. Find the complementary or supplementary angle. a) b) x ° x ° 42° 53° Teaching Notes: • Remind students to always change subtraction to addition and “add the opposite”. • Some students forget to change the sign of the second value after changing to addition. • Encourage students to take the time to write the steps: ( ) ( ) 3 – 2 3 2 5 − = + + = Answers: 1a) – 12, b) – 7, c) – 5, d) – 24, e) 35, f) 75, g) – 2.5, h) 39/14; 2a) 4, b) – 13/9, c) 1, d) 16; 3a) Not a solution, b) Solution, c) Solution; 4) – 5; 5a) 138°, b) 37°

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    M-10 Mini-Lecture 1.6 Multiplying and Dividing Real Numbers Objectives: A. Multiply real numbers. B. Find the reciprocal of a real number. C. Divide real numbers. D. Evaluate expressions using real numbers. E. Determine whether a number is a solution of a given equation. F. Solve applications that involve multiplication or division of real numbers. Key Vocabulary: reciprocals (multiplicative inverses), undefined Examples: 1. Multiply the real numbers. a) ( ) 6 5 − b) ( ) ( ) 11 3 − − c) ( ) 10 50 − d) ( ) ( ) 3.1 5.01 − − 2. Find the reciprocal of the real number. a) 3 7 b) 5 c) 5 21 − d) 0.3 3. Divide the real numbers. a) 27 3 − b) ( ) 90 5 − ÷ − c) 1 8 2 15   − ÷ −     d) 22 0 − 4. Evaluate each expression. a) 2 9 3 20   − −     b) ( ) ( ) 2 5 4 4 8 4 6 4 + − + ⋅ ⋅ − c) ( ) ( ) ( ) 2 7 5 2 3 9 − − − + − d) ( ) ( ) 9 4 1 7 5 10 − − − − e) Evaluate ( ) 6 4 2 3 10 x y z − − − + − − when 5, x = 1, y = − and 0. z = 5. a) Is –3 a solution of 15 5? x − ÷ = − b) Is 93 − a solution of 3 12 ? 8 x − = − 6. The temperature falls three degrees per hour for eight hours. What is the total change in temperature? Teaching Notes: • Multiplying and dividing real numbers should be relatively easy for most students. • Remind students that 0 5 0 = and 5 0 is undefined. • Many students have difficulty with the fact that ( ) 2 2 5 5 . − ≠ − Answers: 1a) – 30, b) 33, c) – 500, d) 15.531; 2a) 7/3, b) 1/5, c) – 21/5, d) 10/3; 3a) – 9, b) 18, c) 15/16, d) undefined; 4a) 3/10, b) 53/8, c) undefined, d) 3/5, e) – 2; 5a) Not a solution, b) Solutions; 6) –24 degrees

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    M-11 Mini-Lecture 1.7 Properties of Real Numbers Objectives: A. Use the commutative and associative properties. B. Use the distributive property. C. Use the identity and inverse properties. Key Vocabulary: commutative property of addition, commutative property of multiplication, associative property of addition, associative property of multiplication, distributive property of multiplication over addition, identity properties Examples: 1. Use the commutative property of addition or multiplication to complete each statement. a) 3 _____ y + = b) ( ) 9 _____ a + − = c) 10 _____ x − ⋅ = d) _____ s t ⋅ = Use the associative property of addition or multiplication to complete each statement. e) (3 ) _______ x y + + = f) ( ) 2 5 ________ x − ⋅ = Use the commutative and associative properties to simplify each expression. g) ( ) 12 4 x + + h) ( ) 7 5 x − i) 1 5 3 12 x   − + +     j) ( ) 0.13 1.2 y − 2. Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. a) ( ) 8 x y + b) ( ) 3 7 – 9 x − c) ( ) 2 6 – 10 y − − d) ( ) 6 4 – 3 – 9 x y Use the distributive property to write each sum as a product. e) 6 6 x y ⋅ + ⋅ f) 13 13 4 x ⋅ + ⋅ g) ( ) ( ) 2 2 x y − + − h) 1 1 6 3 3 a + ⋅ 3. Name the property that is illustrated by each true statement. a) 0 11 11 + = b) 1 3 1 3 ⋅ = c) ( ) 5 5 0 + − = d) 12 1 12 ⋅ = Teaching Notes: • Many students use the Properties of Real Numbers without realizing that they are using these properties. • Some students, when using the Distributive Property, forget to multiply the second term. Answers: 1a) y+3, b) – 9+a, c) x( – 10), d) t·s, e) 3 + (x + y), f) ( – 2 · 5) x, g) 16 + x, h) – 35x, i) x + 1/12, j) – 0.156y; 2a) 8x + 8y, b) – 21x + 27, c) 12y + 20, d) 24x – 18y – 54, e) 6(x + y), f) 13(x + 4), g) – 2(x + y), h) 1/3(a + 6); 3a) Identity Element for Addition, b) Multiplicative Inverse Property, c) Additive Inverse Property, d) Identity Element for Multiplication

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