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Solution Manual for Intermediate Algebra, 6th Edition

Solution Manual for Intermediate Algebra, 6th Edition provides the perfect textbook solutions, giving you the help you need to succeed in your studies.

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     R ESOURCE M ANUAL I NTERMEDIATE A LGEBRA S IXTH E DITION Elayn Martin-Gay University of New Orleans

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    Table of Contents Mini-Lectures (M) Chapter 1 ........................................................................................................................................... 1 Chapter 2 ........................................................................................................................................... 9 Chapter 3 ......................................................................................................................................... 17 Chapter 4 ......................................................................................................................................... 25 Chapter 5 ......................................................................................................................................... 31 Chapter 6 ......................................................................................................................................... 39 Chapter 7 ......................................................................................................................................... 45 Chapter 8 ......................................................................................................................................... 53 Chapter 9 ......................................................................................................................................... 59 Chapter 10 ....................................................................................................................................... 67 Answers........................................................................................................................................... 73 Additional Exercises (E) Chapter 1 ........................................................................................................................................... 1 Chapter 2 ......................................................................................................................................... 13 Chapter 3 ......................................................................................................................................... 27 Chapter 4 ......................................................................................................................................... 57 Chapter 5 ......................................................................................................................................... 69 Chapter 6 ......................................................................................................................................... 83 Chapter 7 ......................................................................................................................................... 95 Chapter 8 ....................................................................................................................................... 109 Chapter 9 ....................................................................................................................................... 125 Chapter 10 ..................................................................................................................................... 145 Answers......................................................................................................................................... 161 Group Activities (G) Chapter 1 ........................................................................................................................................... 1 Chapter 2 ........................................................................................................................................... 3 Chapter 3 ........................................................................................................................................... 5 Chapter 4 ........................................................................................................................................... 7 Chapter 5 ........................................................................................................................................... 9 Chapter 6 ......................................................................................................................................... 11 Chapter 7 ......................................................................................................................................... 13 Chapter 8 ......................................................................................................................................... 15 Chapter 9 ......................................................................................................................................... 17 Chapter 10 ....................................................................................................................................... 19 Answers........................................................................................................................................... 23 Tests (T) Chapter 1 ........................................................................................................................................... 1 Chapter 2 ......................................................................................................................................... 21 Chapter 3 ......................................................................................................................................... 41 Chapter 4 ......................................................................................................................................... 73 Chapter 5 ....................................................................................................................................... 105 Chapter 6 ....................................................................................................................................... 121 Chapter 7 ....................................................................................................................................... 141 Chapter 8 ....................................................................................................................................... 161 Chapter 9 ....................................................................................................................................... 177 Chapter 10 ..................................................................................................................................... 197 Final Exam .................................................................................................................................... 217 Test Answers ................................................................................................................................. 239 Final Exam Answers ..................................................................................................................... 269 iii

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    M-1 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-2 Mini-Lecture 1.2 Algebraic Expressions and Sets of Numbers Objectives: A. Identify natural numbers, whole numbers, integers, rational numbers, and irrational real numbers. B. Write phrases as algebraic expressions. Key Vocabulary: variables, algebraic expression (expression), evaluating (an expression), value, number line, origin, unit distance, positive numbers, negative numbers, graphed, ellipsis, set, members (elements), roster, set builder notation, empty (null) set, subset Examples: 1. Define the number sets using set builder notation. a) natural numbers b) whole numbers c) integers d) rational numbers e) irrational numbers f) real numbers 2. Write each set in roster form. a) { x | x is an odd natural number} b) { x | x is an integer less than 2} 3. List elements of the set 1 {5,0, 3, 49, , 112} 9 − that are also elements of the given set. a) natural numbers b) whole numbers c) integers d) rational numbers e) irrational numbers f) real numbers 4. Place ∈ or ∉ in the space provided to make each statement true. a) 9 − { x | x is an integer} b) 2 5 { x | x is a rational number} c) 3 − {1, 3, 5, …} 5. Write each phrase as an algebraic expression. Use x to represent each unknown number. a) three times a number b) a number minus 2 c) the quotient of a number and 5 d) ten and one-tenth plus a number e) five more than twelve times a number f) four less than six times a number Teaching Notes: • Remind students that “rational” starts with “ratio”, and any number that can be expressed as a ratio of integers is a rational number. • Remind students that irrational numbers have non-terminating, non-repeating decimals. • Some have difficulty writing phrases as algebraic expressions. Present several examples. • Refer to Real Numbers and Selected Key Words/Phrases and Their Translations charts. Answers: 1a) {1,2,3,…}, b) {0,1,2,3,…}, c) {…–3, –2, –1,0,1,2,3,…}, d) { } | , a b a and b are integers and b 0 ≠ e) {x|x is a real number and x is not a rational number}, f) {x|x corresponds to a point on the number line}; 2a) {1,3,5,…}, b) {…–3, –2, –1,0,1}, 3a) { , }, 5 49 b) { , , }, 5 0 49 c) { , , , }, 5 0 49 112 − d) , , , , , 1 5 0 49 112 9   −     e) { }, 3 f) , , , , , , 1 5 0 3 49 112 9   −     4a) , ∈ b) , ∈ c) ; ∉ 5a) 3x, b) x–2, c) , x 5 d) , 1 10 x 10 + e) 12x+5, f) 6x–4

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    M-3 Mini-Lecture 1.3 Equations, Inequalities, and Properties of Real Numbers Objectives: A. Write sentences as equations. B. Use inequality symbols. C. Find the opposite, or additive inverse, and the reciprocal, or multiplicative inverse, of a number. D. Identify and use the commutative, associative, and distributive properties. Key Vocabulary: symbols ( ) , , , , , , = ≠ < > ≤ ≥ additive identity, multiplicative identity, opposite (additive inverse), reciprocal (multiplicative inverse), undefined, commutative properties, associative properties, distributive property Examples: 1. Write each sentence as an equation. a) The difference of x and 4 amounts to 15. b) Five more than the product of 3 and b is 7. c) The quotient of 9 and y is 2 more than y. d) 3 added to one-half t is equal to 8 more that t. 2. Insert <, >, or = between each pair of numbers to form a true statement. a) 2 0 − b) 36 36 9 6 c) 2.5 6.7 − d) 3 9 5 15 3. Write the opposite of each number. Then write the reciprocal of each number. a) 8 b) 2 3 − c) 0 d) 45 7 4. Use a commutative property to write an equivalent expression. a) 5 3 x y + b) m n ⋅ c) 3 5 11 x ⋅ 5. Use an associative property to write an equivalent expression. a) 4 (15 ) x ⋅ b) 9 (3 ) p q r + + c) (3.5 ) x y ⋅ 6. Use the distributive property to multiply. a) 3( 5) x + b) 2( 7) m − c) 3(5 ) y − d) 10 ( 4) y z − e) 0.2(2 6 ) x y + f) 1 (9 5 ) 3 x y − Teaching Notes: • In example 2, also discuss how the symbols , , and ≤ ≥ ≠ could be used. • Many students confuse the associative and commutative properties. • Students may have trouble using the distributive property when fractions are involved. Answers: 1a) x–4=15, b) 3b+5=7, c) , 9 y 2 y = + d) ; 1 t 3 t 8 2 + = + 2a) <, b) <, c) >, d) =; 3a) -8, , 1 8 b) , , 2 3 3 2 − c) 0, undefined, d) , ; 45 7 7 45 − 4a) 3y+5x, b) , n m ⋅ c) ; 3 x 11 5 ⋅ , 5a) ( ) , 4 15 x ⋅ b) (9p+3q)+r, c) . ( ), 3 5 x y ⋅ 6a) 3x+15, b) 2m–14, c) 15–3y, d) 10yz-40y, e) 0.4x+1.2y, f ) 5 3x y 3 −

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    M-4 Mini-Lecture 1.4 Operations on Real Numbers Objectives: A. Find the absolute value of a number. B. Add and subtract real numbers. C. Multiply and divide real numbers. D. Simplify expressions containing exponents. E. Find roots of numbers. Key Vocabulary: absolute value, undefined, factor, base, exponent, nth power of a (a raised to the nth power, a n ), square root, positive (principal) square root, cube root Examples: 1. Find each absolute value. a) 6 b) 2 − c) 12 − d) 14 − − 2. Add, subtract, multiply, or divide as indicated. a) 6 ( 3) − + − b) 6 ( 3) + − c) 6 3 − + d) 5 3 12 24 − + e) 2 5 − f) 2 ( 5) − − g) 2 5 − − h) 9.7 ( 4.2) − − − i) 4 5 ⋅ j) 4 5 − ⋅ k) 0( 9) − l) 2 9 3 12   − −     m) 6 ( 2) ÷ − n) 12 4 − − o) 4.6 2.3 − ÷ p) 3 0 − 3. Evaluate each expression. a) 2 5 − b) 2 ( 5) − c) 3 1 2   −     d) 4 1 3   −     e) 25 f) 81 g) 1 100 h) 3 8 4. Mixed practice. Evaluate each expression. a) 6 2 11 − ÷ b) 12 5 7 − − + c) 3(4)( 7) − − d) 4 3 − Teaching Notes: • Some students try to distribute a negative sign through an absolute value symbol. • Students should master the integer examples before trying those with fractions/decimals. • Many students make sign errors when evaluating expressions with exponents. Answers: 1a) 6, b) 2, c) –12, d) –14; 2a) –9, b) 3, c) –3, d) , 7 24 − e) –3, f) 7, g) –7, h) –5.5, i) 20, j) –20, k) 0, l) , 1 2 m) –3, n) 3, 0) –2, p) undefined; 3a) –25, b) 25, c) , 1 8 − d) , 1 81 − e) 5, f) 9, g) , 1 10 h) 2; 4a) , 3 11 − b) –10, c) 84, d) –81

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    M-5 Mini-Lecture 1.5 Order of Operations and Algebraic Expressions Objectives: A. Use the order of operations. B. Identify and evaluate algebraic expressions. C. Identify like terms and simplify algebraic expressions. Key Vocabulary: order of operations, value (of an expression), evaluating an expression, simplified, terms, like terms, combining like terms Examples: 1. Simplify each expression using order of operations. a) 2 6(2 5) − b) 2 3 ( 4) 3 − + c) 5[9 (1 3)] − − d) 3 8 6 4   − − −     e) 2 ( 8 5)( 2 ) 3 3 − + − − − f) 3 ( 8)( 4) ( 16)( 2) − − − g) 25 (5 3.4) 3 − − − − h) | 12 | | 3 8 | 12 − − − − i) 1 28 5 4 1 6 16 4 ⋅ − + ⋅ 2. Evaluate each expression when 5 x = and 3. y = − a) 3 7 x y − b) 2 9 y x − + c) 6 4 4 y x x y + + + 3. Simplify by distributing, if necessary, and combining like terms. a) 4 5 x x + b) 12 y y − c) 3 9 12 x x − − d) 2.5 8.6 3.4 12.3 y y − + − e) 3(6 9) y + f) 2 (5 4) k k − − g) (9 ) (3 6) t t − − + − h) 1 1 (12 4) (30 ) 2 6 x x y − − − i) 7.3 8.1 2(3.2 0.4) b b + − − 4. If C is degrees Celsius, the algebraic expression 1.8 32 C + represents the equivalent temperature in degrees Fahrenheit. Calculate the degrees Fahrenheit for 20, 0, 40. C = − As the temperature Celsius increases, does the temperature Fahrenheit increase/decrease? Teaching Notes: • If students have trouble using order of operations, encourage neatness and organization. • When collecting like terms, some students may assume 2 4 5 9 . x x x + = • It may be helpful to write a “1” in front of the parenthesis in examples 3f) and 3g). Answers: 1a) 54, b) 43, c) 55, d) 0, e) –2, f) 0, g) 2.2, h) , 7 12 − i) ; 1 5 2a) 36, b) –76, c) –2; 3a) 9x, b) 11y, c) –9x–9, d) 5.9y–20.9, e) 18y+27, f) –3k+4, g) 4t–15, h) , 1 x y 2 6 + − i) 0.9b+8.9; 4) –4°F, 32°F, 104°F, increase

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    M-6 Mini-Lecture 1.6 Exponents and Scientific Notation Objectives: A. Use the product rule for exponents. B. Simplify expressions raised to the zero power. C. Use the quotient rule for exponents. D. Simplify expressions raised to negative powers. E. Simplify exponential expressions containing variables in the exponent. F. Convert between scientific notation and standard notation. Key Vocabulary: exponential expression, scientific notation Examples: 1. Use the product rule or the quotient rule to simplify each expression. a) 3 4 2 2 ⋅ b) 9 7 m m m ⋅ ⋅ c) ( 6 )(6 ) x y y − d) 3 2 3 ( 3 )( 5 ) a b a b − − e) 8 3 x x f) 11 7 10 2 y y − g) 6 5 3 15 9 x y xy h) 2 3 12 9 36 4 a b c abc − 2. Evaluate or simplify each expression. Write each answer using positive exponents only. a) 0 2 b) 0 5 − c) 0 ( 10) − d) 0 (2 1) x + e) 4 2 − f) 2 ( 3) − − g) 2 4 x x h) 3 2 a − 3. Simplify and write each answer using positive exponents only. a) 3 6 y y − b) 5 4 2 x x x − − c) 3 3 3 12 4 ab a b − − d) 3 4 6 y y − ⋅ − e) 0 0 2 (2 ) x x − f) 3 2 2 4 − − − g) 8 13 20 2 x yz xyz − − h) 5 2 7 (4 )(3 )( 2 ) a b b a − 4. Simplify. Assume that variables in the exponents represent nonzero integers, x , y , z 0. ≠ a) 2 7 m m x x ⋅ b) 3 2 2 p p y y − c) 5 7 y y x x − ⋅ d) 3 2 x x x y y y − ⋅ 5. Write each number in scientific notation or in standard notation. a) 645,000 b) 0.005621 c) 4 3.6 10 − × d) 5 9.5 10 × Teaching Notes: • Students need repetition and practice in order to master these objectives. • Students often move constants along with a variable that has a negative exponent. For example, in example 2h), a common (incorrect) answer is 2 a -3 = 1/(8 a 3 ). Answers: 1a) 128, b) m 17 , c) –36xy 2 , d) 15a 6 b 3 , e) x 5 , f) –5y 4 , g) , 5 2 5x y 3 h) –9ab 2 c 3 ; 2a) 1, b) –1, c) 1, d) 1, e) , 1 16 f) , 1 9 g) , 2 1 x h) ; 3 2 a 3a) , 9 1 y b) x, c) , 4 6 3a b d) 24y 4 , e) 1, f) , 1 16 g) , 9 14 10 x z h) -24a 12 b 3 ; 4a) x 9m , b) y p-2 , c) , 2 y 1 x d) y 3x-2 ; 5a) . , 5 6 45 10 × b) . , 3 5 621 10 − × c) 0.00036, d) 950,000

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    M-7 Mini-Lecture 1.7 More Work with Exponents and Scientific Notation Objectives: A. Use the power rules for exponents. B. Use exponent rules and definitions to simplify exponential expressions. C. Simplify exponential expressions containing variables in the exponent. D. Use scientific notation to compute. Examples: 1. Simplify using the product rules for exponents. Write answer using positive exponents. a) 3 2 ( ) x b) 2 3 2 ( ) x y c) 2 2 3 x y       d) 3 4 ( ) m − e) 2 3 2 (2 ) x yz f) 5 4 2 3 x y −       g) 5 3 0 3 (4 ) x y z − − h) 3 3 4 ( 2 ) y − − − − 2. Simplify using exponent rules and definitions. Write answer using positive exponents. a) 2 3 4 9 a b c − − − −       b) 2 3 ( 4 ) x − c) 5 6 3 2 n m − −       d) 2 2 10 3 4 7 x y x y − − − e) 0 2 3 ( 2 ) x y − − f) 3 3 3 ( ) x x y − g) 1 3 5 2 2 7 z y y z − − − −             h) 4 2 3 8 3 (3 ) (2 ) x y x y − 3. Simplify. Assume that variables in the exponents represent nonzero integers and that all other variables are not zero. a) 3 5 2 ( ) b x + b) 2 9 y y x x x − + c) 3 6 2 1 ( ) a a y y − d) 4 3 3 2 12 4 a b a b a b a b x y x y − + − + 4. Perform each indicated operation using the properties of exponents. Write each answer in scientific notation. a) 9 7 (4.9 10 )(6 10 ) − × × b) 6 5 (4 10 ) − × c) 8 4 1.2 10 3 10 − × × Teaching Notes: • Some students are confused about when to add exponents or when to multiply exponents. • Encourage students to write exponent rules on a card to view while doing homework. Answers: 1a) x 6 , b) x 4 y 6 , c) , 4 6 x y d) , 12 1 m e) 4x 4 y 2 z 6 , f) 243x 20 y 10 , g) , 15 9 x 64 y h) 4096y 12 ; 2a) , 6 8 18 a b c b) –64x 6 , c) , 15 30 32 m n d) , 14 5 y 49x e) , 6 1 8 y − f) , 6 3 1 x y g) , 4 5 2 y 7 z h) ; 4 3 2 27 x y 3a) x 6b+10 , b) x -y+8 , c) y 16a+1 , d) 3x 3a+2b y 2a-b ; 4a) . , 1 2 94 10 − × b) . , 27 1 024 10 − × c) . 11 4 0 10 ×

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    M-9 Mini-Lecture 2.1 Linear Equations in One Variable Objectives: A. Decide whether a number is a solution of an equation. B. Solve linear equations using properties of equality. C. Solve linear equations that can be simplified by combining like terms. D. Solve linear equations containing fractions or decimals. E. Recognize identities and equations with no solution. Key Vocabulary: equation, solving the equation, linear equations, first-degree equations, linear equation in one variable, solution, solution set, equivalent equations, addition property of equality, multiplication property of equality, contradiction, identity Examples: 1. Determine whether each number is a solution of the given equation. a) 2; 3 5 x + = b) 6; 3 9 x − − = c) 36; 12 6 x − = − 2. Solve each equation. Check the answer. a) 5 7 x − = b) 3 15 x + = c) 6 4 x − = + d) 2 10 x = e) 3 15 x − = f) 3 4 x = g) 4 2 6 3 x x − = + h) 5 4 10 3 y y − = + i) 9.3 4 2.3 x − = − j) 3(2 4) 9 3 x x + = − k) 2(3 1) 5( 4) n n n − − − = − − 3. Solve each equation. Check the answer. a) 1 3 2 4 x x + = b) 2 5 5 3 x x − = c) 2 3 5 10 r r − = d) 28 4 3 x x − = e) 2 6 1 2 5 y y − = − f) 3.4(2 5) 0.2(2 5) x x + = − + 4. Solve each equation. a) 2( 6) 12 2 x x + = + b) 4( 5) 3 5( 2) x x x + + = + − Teaching Notes: • Encourage students to check their solutions. • Some students prefer to end up with the variable on the left, while others prefer to end up with a positive coefficient in front of the variable. • Refer to Addition/Multiplication Property & Solving a Linear Equation in One Variable . Answers: 1a) yes, b) no, c) no; 2a) {12}, b) {12}, c) {–10}, d) {5}, e) {–5}, f) {12}, g) {8], h) {7}, i) {2.9}, j) {5}, k) {–9}; 3a) , 3 10       b) {75}, c) {10}, d) {4}, e) , 11 12       f) {–2.5}; 4a) {x|x is a real number}, b) ∅

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    M-10 Mini-Lecture 2.2 An Introduction to Problem Solving Objectives: A. Write algebraic expressions that can be simplified. B. Apply the steps for problem solving. Key Vocabulary: understand, translate, solve, interpret Examples: 1. Write each of the following as an algebraic expression. Then simplify. a) The sum of three consecutive integers if the first integer is x b) The perimeter of a rectangle with length x and width x – 7 c) The total amount of money (in cents) in x quarters, 5 x dimes, and (3 x – 1) nickels 2. Solve using the General Strategy for Problem Solving. a) Number Problem One number is two times another number. The sum of the numbers is 90. What are the two numbers? b) Number Problem Three times the difference of a number and 5 is the same as 1 increased by five times the number plus twice the number. Find the number. c) Age Problem Today Henry is 7 years older than twice his age of 23 years ago. Find Henry’s age today. d) Car Rental A car rental agency advertised renting a luxury, full-size car for $49.95 per day and $0.69 per mile. If you rent this car for 5 days, how many whole miles can you drive if you only have $500 to spend? e) Carpentry A 7-ft board is cut into 2 pieces so that one piece is 3 feet longer than three times the shorter piece. Find the length of each piece. f) Unknown Sides A triangle has sides measuring 2.5 x cm, 3 x cm, and (2 x + 3) cm. The perimeter measures 60 cm. Find the measures of the sides. g) Unknown Angles Two angles are complementary if their sum is 90 ° . The measure of one angle is x ° , and the measure of the other angle is (3 x – 2) ° . Find the measure of each angle. h) Layoffs A major car manufacturer announced it would lay off 17,000 employees worldwide. This is equivalent to 20% of its work force. Find the size of the work force prior to lay-offs. Teaching Notes: • Many students have difficulty with word problems. • Encourage students to draw and label diagrams when appropriate. • Students may need to see several examples involving consecutive (or odd/even) integers. • Refer to the General Strategy for Problem Solving chart. Answers: 1a) x+x+1+x+2=3x+3, b) 4x–14, c) 90x–5; 2a) 30,60, b) –4, c) 39, d) 362 miles, e) 1 foot, 6 feet, f) 19 cm, 22.8 cm, 18.2 cm, g) 23°, 67°, h) 85,000 employees

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    M-11 Mini-Lecture 2.3 Formulas and Problem Solving Objectives: A. Solve a formula for a specified variable. B. Use formulas to solve problems. Key Vocabulary: formula Examples: 1. Solve each equation for the specified variable. a) M kt = for t b) 2 C r π = for r c) 2 2 2 a b c + = for a 2 d) 4 5 16 x y + = for y e) 2 2 P l w = + for l f) 5 ( 32) 9 C F = − for F 2. Solve. Round all dollar amounts to two decimal places. a) Volume Find the volume of a rectangular crate with dimensions 3 ft by 4 ft by 8 ft. b) Distance Sheranda drives at a constant rate of 65 miles per hour. How far will she travel in 4 hours? c) Compound Interest Emmanuel puts $5010 at 9% compounded semiannually for 12 years. What is the value of his account at the end of the 12 years? d) Circle Crystal is making a cover for a round table that has a diameter of 46 inches. How much fabric will she need if she wants the cover to fit exactly, with no extra material? (Use 3.14 for π and round to two decimal places.) e) Office Rental Rent for a 17 foot by 20 foot office is $2040 per month. What is the rental cost per square foot? f) Temperature Michael’s cousin Luke was visiting from Montreal during the summer. On a news report, Luke heard that the temperature in Montreal that day was 98 ° F. He was used to hearing temperature in degrees Celsius. What is 98 ° F in degrees Celsius? g) Triangle A triangular piece of wood needs to be varnished. The base of the triangle is 3 meters and the height is 13 meters. How many cans of varnish will be needed if each can covers 10 square meters? Teaching Notes: • Students may have difficulty solving for a variable when other variables are present. • Many students benefit from seeing a parallel example with numbers instead of variables. For example, next to (1a) solve: 6 = 3 t • Encourage students to draw and label diagrams when appropriate. • Refer to the Formula and Solving an Equation for a Specified Variable charts. Answers: 1a) , M t k = b) , C r 2 π = c) 2 2 2 , a c b = − d) , 16 4x y 5 − = e) , P 2w l 2 − = f) ; 9 F C 32 5 = + 2a) 96 cu. ft, b) 260 miles, c) $14,408.83, d) 1661.06 sq. in., e) $6.00 per sq. ft, f) 36.67°C, g) 2 cans

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    M-12 Mini-Lecture 2.4 Linear Inequalities and Problem Solving Objectives: A. Use interval notation. B. Solve linear inequalities using the addition property of inequality. C. Solve linear inequalities using the multiplication property of inequality. D. Solve linear inequalities using both properties of inequality E. Solve problems that can be modeled by linear inequalities. Key Vocabulary: linear inequality, solution, solution set, interval notation Examples: 1. Graph the solution set of each inequality on a number line and then write it in interval notation. a) { | 3} x x > b) { | 2} x x < − c) { | 4.2 } x x − ≥ d) { | 3 0} x x − < ≤ 2. Solve. Graph the solution set and write it in interval notation. a) 2 6 x + ≤ b) 10 9 3 x x < + c) 5 5 4 5 x x − ≥ − 3. Solve. Graph the solution set and write it in interval notation. a) 1 2 2 x ≥ b) 2 7.2 x > − c) 3 6 x − ≤ 4. Solve. Show your answer as an inequality. a) 6 9 x − < − b) 7 2.8 x − ≤ c) 5 3 6 4 3 x − > d) 3 ( 2) 2 4 x x + ≥ + e) 0.3(6 1) 1.4( 3) 0.1 x x − < − − f) 4 2 3 1 5 3 10 4 x x + ≥ + Teaching Notes: • Students may have difficulty solving for a variable when other variables are present. • Remind students to reverse the direction of the inequality symbol when multiplying or dividing by a negative number. • Refer to the Addition/Multiplication Property of Inequality and Solving a Linear Inequality in One Variable charts. Answers: 1a)-3c) see graphing answers; 1a) ( , ), 3 ∞ b) ( , ), 2 −∞ − c) ( , . ], 4 2 −∞ − d) ( , ]; 3 0 − 2a) ( , ], 4 −∞ b) ( , ), 3 −∞ c) [ , ); 0 ∞ 3a) [ , ), 4 ∞ b) ( . , ), 3 6 − ∞ c) [ , ); 2 − ∞ 4a) , x 3 < − b) . , x 0 4 ≥ − c) , 1 x 4 < d) , x 2 ≤ − e) , x 10 < − f) 3 x 2 ≥ −

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    M-13 Mini-Lecture 2.5 Sets and Compound Inequalities Objectives: A. Find the intersection of two sets. B. Solve compound inequalities containing “ and. ” C. Find the union of two sets. D. Solve compound inequalities containing “ or. ” Key Vocabulary: compound inequalities, intersection (and), union (or) Examples: 1. If { is an even integer}, A x x = { is an odd integer}, B x x = {1, 2,3, 4}, C = and {3, 4,5,6}, D = list the elements of each set. a) C D ∩ b) C D ∪ c) A D ∪ d) B C ∩ e) A B ∩ 2. Solve each compound inequality by graphing the solution on a number line. a) 1 and 3 x x ≤ ≥ − b) 1 and 4 x x < > c) 3 and 2 x x ≥ − > 3. Solve each compound inequality. Write each solution in interval notation. a) 3 4 and 5 2 8 x x + ≥ − ≥ b) 5 15 and 15 10 x x − < − − < − c) 4 1 2 x − ≤ + ≤ − d) 2 3 1 1 3 x − < − < e) 3 4 1 1 5 x − + − ≤ ≤ 4. Solve each compound inequality by graphing the solution on a number line. a) 3 or 3 x x ≥ − ≤ b) 1 or 1 x x < − < c) 2 or 3 x x ≥ − ≤ − 5. Solve each compound inequality. Write each solution in interval notation. a) 10 20 or 3 4 2 x x − ≤ − ≥ b) 8 1 or 5 15 x x + < − > − c) 6( 2) 12 or 4 10 x x − ≥ − − ≤ 6. Solve each compound inequality. Write each solution in interval notation. a) 2 and 1 3 x x < < b) 2 or 1 3 x x < < c) 1 5 1 9 x < − < d) 2 3 1 and 3 x x − ≥ − > e) 2 1 3 3 2 x − − − < < f) 6 or 4 4 20 x x − < + < − Teaching Notes: • Demonstrate how each inequality can be graphed individually. The graph of the solution is the intersection (or union) of the individual graphs. Answers: 2a)-c), 4a)-c) see graphing answers; 1a) {3,4}, b) {1,2,3,4,5,6}, c) {x|x is an even integer, x=3, x=5}, d) {1,3}, e) ; ∅ 2a) [–3,1], b) no solution, c) ( , ); 2 ∞ 3a) [ , ), 2 ∞ b) (3,5), c) [–5, –3], d) (–3,3), e) , ; 1 3 3   −     4a) all real numbers, b) ( , ), 1 −∞ c) ( , ] [ , ); 3 2 −∞ − ∪ − ∞ 5a) [ , ), 2 − ∞ b) ( , ) ( , ), 9 3 −∞ − ∪ − ∞ c) [ , ); 6 − ∞ 6a) , , 2 3   −∞     b) ( , ), 1 −∞ c) , , 2 2 5       d) , ∅ e) , , 7 5 2 2   −     f) ( , ) ( , ) 6 6 −∞ − ∪ − ∞

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