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Solution Manual for Beginning Algebra with Applications and Visualization, 3rd Edition

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    ’ S R ESOURCE M ANUAL WITH T ESTS AND M INI -L ECTURES D EANA R ICHMOND B EGINNING A LGEBRA WITH A PPLICATIONS & V ISUALIZATION T HIRD E DITION Gary Rockswold Minnesota State University, Mankato Terry Krieger Rochester Community and Technical College

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    Contents Mini-Lectures (ML) Chapter 1 .......................................................................................................................................... 1 Chapter 2 ........................................................................................................................................ 11 Chapter 3 ........................................................................................................................................ 20 Chapter 4 ........................................................................................................................................ 29 Chapter 5 ........................................................................................................................................ 36 Chapter 6 ........................................................................................................................................ 43 Chapter 7 ........................................................................................................................................ 50 Chapter 8 ........................................................................................................................................ 59 Chapter 9 ........................................................................................................................................ 67 Chapter 10 ...................................................................................................................................... 70 Chapter 11 ...................................................................................................................................... 80 Chapter 12 ...................................................................................................................................... 89 Chapter 13 ...................................................................................................................................... 97 Chapter 14 .................................................................................................................................... 103 Additional Resources (R) Teaching Tips Correlated to Textbook Sections .............................................................................. 1 Outside Resources .......................................................................................................................... 85 Available Supplements................................................................................................................... 86 Graphing Calculator Resources...................................................................................................... 89 Printable Test Forms (T) Chapter 1 .......................................................................................................................................... 1 Chapter 2 ........................................................................................................................................ 11 Chapter 3 ........................................................................................................................................ 21 Chapter 4 ........................................................................................................................................ 39 Chapter 5 ........................................................................................................................................ 57 Chapter 6 ........................................................................................................................................ 67 Chapter 7 ........................................................................................................................................ 77 Chapter 8 ........................................................................................................................................ 89 Chapter 9 ...................................................................................................................................... 107 Chapter 10 .................................................................................................................................... 123 Chapter 11 .................................................................................................................................... 133 Chapter 12 .................................................................................................................................... 151 Chapter 13 .................................................................................................................................... 169 Chapter 14 .................................................................................................................................... 197 Answers to Chapter Tests............................................................................................................. 213 Cumulative Review Exercises and Answers ................................................................................ 165 Final Exams and Answers ............................................................................................................ 319

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    Mini-Lecture 1.1 BEGINNING ALGEBRA ML-1 Numbers, Variables, and Expressions Section Topics: • Natural Numbers and Whole Numbers • Prime Numbers and Composite Numbers • Variables, Algebraic Expressions, and Equations • Translating Words to Expressions Examples: 1. Classify each number as prime or composite . If the number is composite, write it as a product of prime numbers. a) 37 b) 65 c) 97 d) 216 2. Evaluate the expressions when 2 a = , 3 b = , and 6 c = . a) 2 3 a b + b) 5 ab c) c b d) ( ) c a a − 3. Write each word phrase as an algebraic expression, using x as the variable. a) The quotient of 6 less than a number and 3 b) Fifty more than a number c) The product of 8 and the total of a number and 5 Teaching Notes: • When translating “the difference between a and b,” the order is kept the same: a b − . • Remind students that an expression evaluates to a number, while an equation can be translated into a sentence (with an = symbol). Answers: 1a) prime, b) composite, 5 13 ⋅ , c) prime, d) composite, 2 2 2 3 3 3 ⋅ ⋅ ⋅ ⋅ ⋅ ; 2a) 13, b) 30, c) 2, d) 2; 3a) x 6 3 − , b) x 50 + , c) ( ) 8 x 5 +

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    BEGINNING ALGEBRA Mini-Lecture 1.2 ML-2 Fractions Section Topics: • Basic Concepts • Simplifying Fractions to Lowest Terms • Multiplication and Division of Fractions • Addition and Subtraction of Fractions • An Application Examples: 1. Simplify each fraction to lowest terms. a) 5 10 b) 16 64 c) 42 77 d) 88 90 2. Multiply and simplify to lowest terms. a) 5 3 9 15 i b) 5 9 6 2 i c) 4 18 9 i d) 7 5 2 4 i 3. Divide and simplify to lowest terms. a) 5 3 9 15 ÷ b) 5 9 6 2 ÷ c) 5 36 24 24 ÷ d) 5 15 3 4 ÷ 4. Find each sum or difference and write in lowest terms. a) 3 4 14 14 + b) 2 1 3 3 − c) 2 1 4 8 + d) 5 7 6 8 + e) 7 9 12 30 + f) 2 2 1 3 24 6 + + 5. Rachel is jogging for exercise. This week, she ran 1 2 1 miles on Monday, 1 4 2 miles on Tuesday, and on Thursday she ran 1 2 2 times the distance that she ran on Monday. How many miles did she run this week?

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    Mini-Lecture 1.2 BEGINNING ALGEBRA ML-3 6. In a recent survey, 1000 people were asked about their favorite breed of dog. The table shows the approximate fractions of survey respondents who listed each breed as his or her favorite. Dog Breed Fraction Pug 3/40 Labrador Retriever 91/200 Golden Retriever 27/100 Yorkshire Terrier 1/5 a) What fraction of people chose either Golden Retriever or Labrador Retriever as their favorite breed of dog? b) Of the 1000 people surveyed, how many people chose the Yorkshire Terrier as their favorite breed of dog? c) Of the 1000 people surveyed, how many people did not choose the Pug as their favorite breed of dog? Teaching Notes: • Encourage students to simplify fractions by dividing numerator and denominator by the same number and by factoring into primes and dividing out common factors. • Remind students that “fraction of” phrases generally indicate multiplication. • Some students add/subtract the denominators when adding/subtracting fractions. Answers: 1a) 1 2 , b) 1 4 , c) 6 11 , d) 44 45 ; 2a) 1 9 , b) 15 4 , c) 8, d) 35 8 ; 3a) 25 9 , b) 5 27 , c) 5 36 , d) 4 9 ; 4a) 1 2 , b) 1 3 , c) 5 8 , d) 41 24 , e) 53 60 , f) 11 12 ; 5) 1 2 7 miles ; 6a) 29/40, b) 200, c) 925

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    BEGINNING ALGEBRA Mini-Lecture 1.3 ML-4 Exponents and Order of Operations Section Topics: • Natural Number Exponents • Order of Operations • Translating Words to Expressions Examples: 1. Find the value of each exponential expression. a) 2 6 b) 4 3 c) 4 3 4 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 2. Find the value of each expression. a) 4 8 2 − ÷ b) 2 2 3 3 ⋅ − c) 2 10 3 2 − + d) 3 16 32 2 − ÷ e) 2 16 4 6 2 2 2 − + − + f) 2 3 24 5 9 6 ÷ + − g) ( ) ( ) 2 3 1 3 4 2 4 + + ⎡ − + ⎤ ⎣ ⎦ 3. Translate each phrase into a mathematical expression and then evaluate it. a) Seven squared decreased by fifteen b) Four times twenty increased by eleven c) Five times the quantity of twelve plus nine Teaching Notes: • Some students do not know how to say 2 3 , or 3 2 , or 4 5 , etc., in words and need to see the words written. • Students will often compute 4 2 as 2 4 ⋅ . Illustrate the difference between the expressions with examples. • Students should be reminded to work from inside out when evaluating expressions with grouping symbols. Answers: 1a) 36, b) 81, c) 81/256; 2a) 0, b) 9, c) 3, d) 12, e) 6, f) 13, g) 18; 3a) 7 2 − 15, 34, b) 4 ⋅ 20 + 11, 91, c) 5(12+9), 105

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    Mini-Lecture 1.4 BEGINNING ALGEBRA ML-5 Real Numbers and the Number Line Section Topics: • Signed Numbers • Integers and Rational Numbers • Square Roots • Real and Irrational Numbers • The Number Line • Absolute Value • Inequality Examples: 1. List all numbers from the set 3 10, 5, 2 , 0, 5, 10 4 ⎧ ⎫ − − − ⎨ ⎬ ⎩ ⎭ that are: a) natural numbers b) whole numbers c) integers d) rational numbers e) irrational numbers f) real numbers 2. Evaluate the square root. Approximate your answer to three decimal places when appropriate. a) 121 b) 40 c) 27 3. Plot each of the numbers on a number line: 1 1 2, 5, 4 , 1 3 2 − − . 4. Select the lesser number in each pair. a) –3, 8 b) 4 1 , 5 3 − − c) 10, –12 5. Find the opposite of each number. a) 6 b) –(–(–5)) c) 5 6 − 6. Simplify. a) 3 − b) 6 − − c) 13 16 − 7. Decide whether the statement is true or false. a) 10 2 − > b) 3 5 − ≤ − c) 7 1 − − >

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    BEGINNING ALGEBRA Mini-Lecture 1.4 ML-6 . Teaching Notes: • Some students find it helpful to see each type of number in Example 1 on a number line. • Remind students that integers are rational numbers; any integer can be written as the ratio of itself and 1. • Decimal numbers that terminate or repeat in a fixed block are rational numbers – ask students to give examples of both. • The decimal form of an irrational number neither terminates nor repeats. • The number line is a good way to illustrate that opposite numbers are equidistant from 0 but on opposite sides of 0. • Some students have never seen absolute value before and will need examples. Answers: 1a) 10, b) 0, 10, c) –10, –5, 0, 10, d) –10, –5, 3 2 4 − , 0, 10, e) 5 , f) –10, –5, 3 2 4 − , 0, 5 , 10; 2a) 11, b) 6.325, c) 5.196; 3) ; 4a) –3, b) 4 5 − , c) –12; 5a) –6, b) 5, c) 5 6 ; 6a) 3, b) –6, c) 3; 7a) true, b) true, c) false;

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    Mini-Lecture 1.5 BEGINNING ALGEBRA ML-7 Addition and Subtraction of Real Numbers Section Topics: • Addition of Real Numbers • Subtraction of Real Numbers • Applications Examples: 1. Find each sum or difference. a) 9 12 + b) ( ) 16 10 − + − c) ( ) 14 26 − + − d) 6.3 5.2 − + e) ( ) 12 6 17 − + − + f) 9 – 11 g) ( ) 2 6 1 7 − + ⎡ + − + ⎤ ⎣ ⎦ h) 7 2 10 5 ⎛ ⎞ + − ⎜ ⎟ ⎝ ⎠ i) 1 1 1 3 2 4 − − 2. Write a numerical expression for each phrase, and simplify the expression. a) The sum of –6 and 3 and –1 b) The sum of –10 and –15, increased by 12 c) 0.29 more than the sum of 3.56 and –2.41 Solve each problem. 3. A scuba diver is at a depth of 16 feet below the surface. He descends another 8 feet. What is his new depth? 4. On January 14, in New Market, Indiana, the temperature rose 17º F in three hours. If the starting temperature was –5º F, what was the temperature three hours later? Teaching Notes: • Some students need to see addition problems done on a number line first. • Caution students about the difference between the subtraction key and the change-of-sign key on a calculator. • Refer students to the summary boxes for adding and subtracting real numbers. • Emphasize the uses of the – symbol with students. Answers: 1a) 21, b) –26, c) – 40, d) –1.1, e) –1, f) –2, g) 10, h) 3 10 , i) 5 12 − ; 2a) ( ) ; 6 3 1 4 − + + − − , b) ( ) ; 10 15 12 13 ⎡− + − ⎤ + − ⎣ ⎦ , c) ( ) . . . ; . 3 56 2 41 0 29 1 44 ⎡ + − ⎤ + ⎣ ⎦ ; 3) 24 feet below; 4) 12ºF

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    BEGINNING ALGEBRA Mini-Lecture 1.6 ML-8 Multiplication and Division of Real Numbers Section Topics: • Multiplication of Real Numbers • Division of Real Numbers • Applications Examples: 1. Find each product. a) ( ) 2 0 − b) ( ) 4 15 − c) ( ) ( ) 30 5 − − d) ( ) ( ) 2.2 3.3 − e) 3 8 4 9 ⎛ ⎞ ⎛ ⎞ − − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 2. Find each quotient. a) 16 8 − ÷ b) 9 3 − − c) 4.6 0 − d) 3 15 5 20 − ÷ e) 0 4 − 3. Write each number as a decimal. a) 4 18 b) 43 500 c) 1 12 5 4. Write each decimal as a fraction in lowest terms. a) 0.72 b) 0.375 c) 0.005 5. In a box of 50 premium colored pencils, the fraction of pencils that are shades of blue is 4 25 . Write this fraction as a decimal. Teaching Notes: • Refer students to the rules for signs of products and quotients. • Give examples to show why division by zero is undefined but zero can be divided by any number except zero. • , , x x y y ÷ and y x (where 0 y ≠ ) all indicate division. Answers: 1a) 0, b) –60, c) 150, d) –7.26, e) 2 3 ; 2a) –2, b) 3, c) undefined, d) 4 5 − , e) 0; 3a) 0.2 , b) 0.086, c) 12.2; 4a) 18/25, b) 3/8, c) 1/200; 5) 0.16

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    Mini-Lecture 1.7 BEGINNING ALGEBRA ML-9 Properties of Real Numbers Section Topics: • Commutative Properties • Associative Properties • Distributive Properties • Identity and Inverse Properties • Mental Calculations Examples: 1. Name the property (commutative, associative, identity property, inverse property, or distributive property) illustrated by each statement. a) ( ) 3 7 7 3 + − = − + b) 2 3 1 3 2 ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ c) ( ) ( ) ( ) 3 6 2 3 6 2 − + ⎡ + − ⎤ = − + + − ⎣ ⎦ d) 1 1 0 5 5 − + = e) ( ) 5 5 5 c d c d − + = − − f) ( ) ( ) 5 7 10 10 5 7 − + + = + − + g) 3 3 7 7 7 7 ⎛ ⎞ ⎛ ⎞ − = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ h) 56 1 56 − ⋅ = − 2. Use a distributive property to rewrite each expression. Simplify if possible. a) ( ) 3 6 k + b) ( ) 5 2 h − − c) 3 3 y z ⋅ + ⋅ d) ( ) 2 5 2 a b c − − − e) ( ) 2 7 2 3 − ⋅ + − ⋅ 3. Use properties of real numbers to calculate the expression mentally: 22 + 36 + 58 + 44 Teaching Notes: • Remind students that the commutative property deals with the order of addition (or multiplication), whereas the associative property deals with grouping. • Students often confuse the additive identity (zero) and the multiplicative identity (one). • Have students provide examples to show whether or not the commutative / associative properties hold for subtraction and division. Answers: 1a) commutative (add.), b) inverse (mult.), c) associative (add.), d) inverse (add.), e) distributive, f) commutative (add.), g) commutative (mult.), h) identity (mult.); 2a) 3k 18 + , b) 5h 10 − + , c) ( ) 3 y z + , d) 10a 4b 2c − + + , e) ( ) ; 2 7 3 20 − + − ; 3) 160

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    BEGINNING ALGEBRA Mini-Lecture 1.8 ML-10 Simplifying and Writing Algebraic Expressions Section Topics: • Terms • Combining Like Terms • Simplifying Expressions • Writing Expressions Examples: 1. Simplify each expression. a) 8 3 5 x x − + b) ( ) 3 2 7 x x − + c) ( ) 4 3 5 m − − + 2. Give the numerical coefficient of each term. a) 4 m b) 2 16 x − c) 2 3 p q − d) 4 5 r − 3. Simplify each expression. a) ( ) 2 3 6 4 m m − + + b) 3 6 b b + c) 2 3 b b − d) ( ) 7 5 8 c c − + e) ( ) ( ) 2 2 3 4 2 5 1 e e e − − − 4. Translate each phrase into a mathematical expression. Use x as the variable. Simplify. a) A number increased by the difference between 3 and the number b) Six plus the product of 4 more than a number and 2 c) A number plus 7 added to the difference between 6 and twice the number Teaching Notes: • Students have difficulty distinguishing between terms and factors – terms are separated by a + or – sign, while factors are multiplied. • Like terms not only have the same variables but also the same exponents. • Remind students that expressions can be simplified, whereas equations are solved. Answers: 1a) 13x 3 − , b) 7 x 21 − , c) 3m 9 − − ; 2a) 4, b) –16, c) –1, d) 4 5 − ; 3a) 2m 12 − − , b) 9b , c) 1 3 b − , d) 2c 8 − , e) 2 7e 12e 2 − − + ; 4a) ( ) ; x 3 x 3 + − , b) ( ) ; 6 2 x 4 2x 14 + + + , c) ( ) ( ) ; 6 2x x 7 x 13 − + + − +

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    Mini-Lecture 2.1 BEGINNING ALGEBRA ML-11 Introduction to Equations Section Topics: • Basic Concepts • Equations and Solutions • The Addition Property of Equality • The Multiplication Property of Equality Examples: 1. Which pairs of equations are equivalent equations? a) 5 8 x + = and 3 x = b) 12 10 x − = − and 2 x = − c) 15 5 x − = and 6 x = d) 23 30 x + = and 9 x = − 2. Solve each equation, and check your solution. a) 14 12 x = − b) 15 18 x + = c) 19 16 x − = + d) ( ) 6 18 x − − = e) 15 5 x = − f) 1 5 3 6 x + = 3. By what number is it necessary to multiply each side of each equation in order to isolate x on the left side? Do not solve. a) 1 3 5 x = b) 2 6 x = − c) 3 21 4 x − = d) 41 x − = 4. Tell whether you would use the addition or multiplication property of equality to solve each equation. Do not solve. a) 4 5 x + = − b) 5 25 x = − 5. Solve each equation, and check your solution. a) 1 25 4 x = − b) 5 12 x = c) 7 56 x = − d) 11 2 x − = e) 1.1 88 x = f) 16 x − = − g) 3.2 116.8 x − = − h) 53 8 x = − i) 1 9 6 x = − j) 43 4 x − = k) 9.8 210.994 x − = − l) 8 8 x − =

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    BEGINNING ALGEBRA Mini-Lecture 2.1 ML-12 . Teaching Notes: • Encourage students to write all of the addition property steps and to avoid using shortcuts until they have mastered these types of equations. • Encourage students to write the steps for solving the equations in a neat and organized manner. This habit will help immensely when the equations become more complex. • Refer students to the addition property of equality definition in the text. • Mention that letters besides x can be used for variables in equations. • Some students need to be shown that 1 1 1 12 12 1 12 1 12 x x x x ⋅ = ⋅ = = ⋅ . • Refer to the multiplication property of equality definition in the text. • Ask students what should be done to the right side of an equation to balance it if the left side of the equation is doubled. Answers: 1) choices a and b; 2a) 26, b) 3, c) 35 − , d) 12, e) 20, f) 1/2; 3a) 5, b) 6, c) 4 3 − , d) –1; 4a) addition property of equality, b) multiplication property of equality; 5a) 100 − , b) 60, c) 8 − , d) 11 2 − or 1 5 2 − , e) 80, f) 16, g) 36.5, h) 53 8 − , i) 54 − , j) 43 4 − , k) 21.53, l) 64 −

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    Mini-Lecture 2.2 BEGINNING ALGEBRA ML-13 Linear Equations Section Topics: • Basic Concepts • Solving Linear Equations • Applying the Distributive Property • Clearing Fractions and Decimals • Equations with No Solutions or Infinitely Many Solutions Examples: 1. Which of the following are linear equations in one variable? a) 4 7 3 2 x x − = + b) 2 2 5 x x = + c) 2 8 0 x − = d) 3 8 x x = 2. Determine whether the given solution is correct. a) Is 12 a solution for 5 x + 4 – 2 x = 3 x – 5 + x ? b) Is 9 a solution for 5 x + 4 – 2 x = 3 x – 5 + x ? 3. Solve each equation, and check your solution. a) 10 7 107 x + = b) 7 8 27 x − = c) 33 6 3 x = − d) 164 15 14 x = + e) 1 8 2 2 x − = − f) 2 8 32 3 x − − = − 4. Solve each equation, and check your solution. a) 4 x = –2 x + 60 b) 8 x – 6 = 3 + 9 x c) –6 x – 10 = –7 + 10 x d) 0.6 y – 0.3 = 0.7 – 0.3 y e) x – 12 = 10 – x f) –9 x + 4 + 7 x = –3 x + 9 5. Determine whether the equation has no solutions, one solution, or infinitely many solutions. a) 6(2 1) 12 30 x x − − = b) –1( x + 11) = 20 c) ( ) 6 8 6 48 x x − = − d) ( ) 3 6 9 9 15 x x x + + = − e) 0.4 x – 0.2(3 – x ) = 7.6 f) 7 3( 8) 3 24 x x x − − = +

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