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Multi-Step Equations and Inequalities 3A Solving Multi-Step Equations 3-1 Properties of Rational Numbers 3-2 Simplifying Algebraic Expressions 3-3 Solving Multi-Step Equations LAB Model Equations with Variables on Both Sides 3-4 Solving Equations with Variables On Both Sides 3B Solving Inequalities 3-5 3-6 Inequalities Solving Inequalities by Adding or Subtracting Solving Inequalities by Multiplying or Dividing Solving Two-Step Inequalities 3-7 3-8 KEYWORD: MT8CA Ch3 Equations and inequalities can describe how fast water is flowing in streams and rivers. Merced River, Yosemite National Park 112 Chapter 3 Vocabulary Choose the best term from the list to complete each sentence. 1. A letter that represents a value that can change is called a(n) __?__. algebraic expression equation numerical expression 2. A(n) __?__ has one or more variables. 3. A(n) __?__ is a mathematical sentence that uses an equal sign to show that two expressions have the same value. variable 4. A mathematical phrase that contains operations and numbers is called a(n) __?__. It does not have an equal sign. Complete these exercises to review skills you will need for this chapter. Operations with Fractions Evaluate each expression. 2 1 5. 3 2 5 8 9. 6 1 5 13 19 6. 1 2 8 4 7 6 7. 8 1 1 11 121 10. 1 2 144 1 5 11. 6 8 9 9 8. 1 1 0 3 19 4 12. 2 5 0 Solve One-Step Equations Use mental math to solve each equation. 13. x 7 21 14. p 3 22 15. 14 v 30 16. b 5 6 17. t 33 14 18. w 7 7 Connect Words and Equations Write an equation to represent each situation. 19. The perimeter P of a rectangle is the sum of twice the length and twice the width w. 20. The volume V of a rectangular prism is the product of its three dimensions: length , width w, and height h. 21. The surface area S of a sphere is the product of 4π and the square of the radius r. 22. The cost c of a telegram of 18 words is the cost f of the first 10 words plus the cost a of each additional word. Multi-Step Equations and Inequalities 113 The information below “unpacks” the standards. The academic vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). Academic Vocabulary Chapter Concept variable a symbol, usually a letter, used to show an You rewrite a verbal statement amount that can change using mathematical symbols. Example: x verbal using words Example: a number is greater than –5 n > –5 (Lesson 3-5) AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. property a characteristic of numbers, operations, or equations Example: One property of addition is that you can add numbers in any order without changing the sum. You use mathematical properties to simplify expressions. You give reasons for each step when you simplify expressions. justify give a reason for (Lessons 3-1 and 3-2) AF4.0 Students solve simple linear equations and inequalities over the rational numbers. solve find the value or values of an unknown quantity that make one side of an equation equal to the other side (make the equation true) You find the values of a variable that make an inequality true. (Lessons 3-6 and 3-7) AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solutions in the context from which they arose, and verify the reasonableness of the results. (Lesson 3-8) 114 Chapter 3 interpret to understand and explain the meaning of You understand and can explain the meaning of context in this case, a real-world situation solutions to inequalities. Reading Strategy: Read a Lesson for Understanding Before you begin reading a lesson, find out which standard or standards are the main focus of the lesson. These standards are located at the top of the first page of the lesson. Reading with the standards in mind will help guide you through the lesson material. You can use the following tips to help you follow the math as you read. Lesson Features Reading Tips Identify the standard or standards of the lesson. Then skim through the lesson to get a sense of how the standards are covered. Work through each example. The examples help to demonstrate the standards. Check your understanding of the lesson by answering the Think and Discuss questions. Try This Use Lesson 3-1 in your textbook to answer each question. 1. What is the standard of the lesson? 2. What questions or problems did you have when you read the lesson? 3. Write your own example problem similar to Example 1. 4. What skill is being practiced in the second Think and Discuss question? Multi-Step Equations and Inequalities 115 3-1 California Standards AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Properties of Rational Numbers Why learn this? You can use mental math and properties of rational numbers to calculate costs when shopping. (See Exercises 40 and 41.) In Chapter 2, you performed operations with rational numbers. The following properties are useful when you simplify expressions that contain rational numbers. PROPERTIES OF ADDITION AND MULTIPLICATION Vocabulary Commutative Property Associative Property Distributive Property Words Algebra 3 4 12 4 12 3 a b b a 2 17 17 2 ab ba (6 8) 9 6 (8 9) (a b) c a (b c) Commutative Property You can add numbers in any order. You can multiply numbers in any order. Associative Property When you are only adding or only multiplying, changing the grouping will not affect the sum or product. EXAMPLE Numbers 1 2 18 16 2 18 16 (a b) c a (b c) Identifying Properties of Addition and Multiplication Name the property that is illustrated in each equation. 3 (4 x) (3 4) x 3 (4 x) (3 4) x The factors are grouped differently. Associative Property of Multiplication (9) 2 2 (9) (9) 2 2 (9) The order of the numbers changed. Commutative Property of Addition You can use the properties of rational numbers to rearrange or regroup numbers in a way that helps you do math mentally. 116 Chapter 3 Multi-Step Equations and Inequalities EXAMPLE 2 Using the Commutative and Associative Properties Simplify each expression. Justify each step. 43 29 7 43 29 7 43 7 29 (43 7) 29 50 29 79 Compatible numbers help you do math mentally. Try to make multiples of 5 or 10. They are simpler to use when multiplying. Commutative Property of Addition Associative Property of Addition Add. 1 15 7 5 1 1 15 7 5 15 5 7 Commutative Property of Multiplication 15 5 7 Associative Property of Multiplication 3 7 21 Multiply. 1 The Distributive Property is also helpful when you do math mentally. DISTRIBUTIVE PROPERTY Numbers 7(6 12) 7 6 7 12 a(b c) a b a c Algebra 5(7 3) 5 7 5 3 a(b c) a b a c When you need to find the product of two numbers, write one of the numbers as a sum or difference. Then use the Distributive Property to help you find the product mentally. EXAMPLE 3 Using the Distributive Property Write each product using the Distributive Property. Then simplify. Break the larger factor into a sum or difference that contains a multiple of 10. 5(43) 5(43) 5(40 3) 5 40 5 3 200 15 215 6(28) 6(28) 6(30 2) 6 30 6 2 180 12 168 Rewrite 43 as a sum. Distributive Property Multiply. Then add. Rewrite 28 as a difference. Distributive Property Multiply. Then subtract. Think and Discuss 1. Explain which property you would use to simplify 5.8 (0.2 4). 2. Describe two ways to use the Distributive Property to find 8 45. 3-1 Properties of Rational Numbers 117 3-1 California Standards Practice AF1.3 Exercises KEYWORD: MT8CA 3-1 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Name the property that is illustrated in each equation. 2. (5) 12 12 (5) 1. y 16 16 y See Example 2 See Example 3 Simplify each expression. Justify each step. 3. 17 19 3 4. 51 48 9 5. 4 7 25 1 6. 3 8 9 7. 5 (13 2) 1 3 8. 4 3 4 Write each product using the Distributive Property. Then simplify. 9. 8(21) 10. 5(62) 11. 3(18) 12. 6(49) 13. 4(99) 14. (59)5 INDEPENDENT PRACTICE See Example 1 Name the property that is illustrated in each equation. 15. 4x x 4 See Example 2 See Example 3 16. (7 1.5) 2 7 (1.5 2) Simplify each expression. Justify each step. 17. 4 89 16 18. (0.5 9) 2 19. 2 13 50 20. 69 17 1 21. 8.8 (15 0.2) 1 22. 4 9 12 Write each product using the Distributive Property. Then simplify. 23. 7(19) 24. (53)4 25. 12(11) 26. (98)2 1 27. 2(42) 1 28. 3(87) PRACTICE AND PROBLEM SOLVING Extra Practice See page EP6. Name the property that is illustrated in each equation. 29. 7(9 x) 7 9 7x 30. 16 0 16 31. (5 y) z 5 (y z) 32. 9 1 = 9 33. m 12n 12n m 34. 3(2 t) 3 2 3t Simplify each expression. Justify each step. 35. 13 9 7 11 36. 4 3 25 2 38. Consumer Math Mikiko is buying five DVDs at SaveMart. How can she use the Distributive Property to find the total cost of the DVDs before tax? 39. Consumer Math Jerome is buying a DVD, a pair of jeans, and a t-shirt at SaveMart. Show how he can use properties of rational numbers to find the total cost of the items before tax. 118 Chapter 3 Multi-Step Equations and Inequalities 1 2 1 3 37. 62 75 2 5 SaveMart Price List Item Price DVD $18 Jeans $23 T-Shirt $12 Write an example of each property using rational numbers. 40. Distributive Property 41. Associative Property of Multiplication 42. Commutative Property of Addition Complete each equation. Then name the property that is illustrated in each. (4 7) 8 4 8 7 45. (x y) 12 x ( 44. 4.8 6 6 46. 9(8 z) 9 12) 47. Weather Leann wants to know the total amount of rainfall in Berkeley, California, from 2002 through 2005. Explain how she can use mental math and properties of rational numbers to calculate this amount. 48. Reasoning Make a conjecture: Is division of rational numbers commutative? Explain your thinking. 9z Annual Rainfall, Berkeley, CA 28 Total rainfall (in.) 43. 26 24 22 26 24 M708CS_C03_L01_301_A 21 19 20 18 0 2002 2003 2004 2005 Year 49. What’s the Error? A student writes, “You can use the Associative Property of Addition to change the order of two numbers before you add them.” What is the student’s error? 50. Write About It A case of cat food has 24 cans. Explain how to use mental math and the Distributive Property to find the number of cans in 5 cases. 51. Challenge Simplify the expression 1213 16 14. Justify each step. NS1.2, NS1.5, AF1.3 52. Multiple Choice The equation 3 (5 x) 3 (x 5) is an example of which property? A Associative Property of Addition C Commutative Property of Multiplication B Commutative Property of Addition D Distributive Property 53. Multiple Choice Which is an example of the Associative Property of Multiplication? 6 A (4) y y (4) C 1 1 (18 6) 18 3 3 B 2(9 1) 2 9 2 1 D 3 5 3 2 3(5 2) Write each decimal as a fraction in simplest form. (Lesson 2-1) 54. 0.68 56. 2.01 55. 1.4 57. 0.04 Divide. Write each answer in simplest form. (Lesson 2-5) 1 3 58. 2 8 2 2 59. 43 29 5 60. 8 2 4 1 61. 25 1 0 3-1 Properties of Rational Numbers 119 3-2 California Standards AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Simplifying Algebraic Expressions Who uses this? Consumers can simplify algebraic expressions to find the total cost of tickets. (See Exercise 52.) In the expression below, 7x, 5, 3y, and 2x are terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs. Constant Vocabulary term like terms coefficient constant equivalent expressions Coefficients Like terms , such as 7x and 2x, can be grouped together because they have the same variable raised to the same power. Often, like terms have different coefficients. A coefficient is a number that is multiplied by a variable in an algebraic expression. A constant is a number that does not change. Constants, such as 4, 0.75, and 11, are also like terms. When you combine like terms, you change the way an expression looks but not the value of the expression. Equivalent expressions have the same value for all values of the variables. EXAMPLE 1 Combining Like Terms in One-Variable Expressions Combine like terms. When you rearrange terms, move the operation in front of each term with that term. 7x 2x 7x 2x Identify like terms. 9x Combine coefficients: 7 2 9. 5m 2m 8 3m 6 120 5m 2m 8 3m 6 Identify like terms. 5m 2m 3m 8 6 Commutative Property 0m 14 Combine coefficients. 14 Simplify. Chapter 3 Multi-Step Equations and Inequalities EXAMPLE 2 Combining Like Terms in Two-Variable Expressions Combine like terms. 7a 4a 3b 5 7a 4a 3b 5 11a 3b 5 Identify like terms. Combine coefficients: 7 4 11. k 3n 2n 4k 1k 3n 2n 4k Identify like terms; the coefficient of k is 1 because 1k k. 1k 4k 3n 2n 5k n Commutative Property Combine coefficients: 1 4 5; 3 2 1. 3f – 9g 15 3f 9g 15 No like terms 3f 9g 15 To simplify an expression, perform all possible operations, including combining like terms. You may need to use the Associative, Commutative, or Distributive Properties. EXAMPLE 3 Using the Distributive Property to Simplify Simplify 6(y 8) 5y. 6(y 8) 5y 6(y) 6(8) 5y Distributive Property 6y 48 5y Multiply. 6y 48 5y Identify like terms. 6y 5y 48 Commutative Property 1y 48 Combine coefficients: 6 5 1. y 48 1y y Think and Discuss 1. Describe the first step in simplifying the expression 2 8(3y 5) y. 2. Tell how many sets of like terms are in the expression in Example 1B. What are they? 3-2 Simplifying Algebraic Expressions 121 3-2 California Standards Practice AF1.3 Exercises KEYWORD: MT8CA 3-2 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Combine like terms. 1. 9x 4x 2. 2z 5 3z 3. 6f 3 4f 5 10f 4. 9g 8g 5. 7p 9 p 6. 3x 5 x 3 4x 7. 6x 4y x 4y 8. 4x 5y y 3x 9. 5x 3y 4x 2y 11. 7g 5h 12 12. 3h 4m 7h 4m 13. 4(r 3) 3r 14. 7(3 x) 2x 15. 7(t 8) 5t 16. 3(2 p) 4p 17. 2(5y 4) 9 18. 7(5 2m) m 10. 6p 3p 7z 3z See Example 3 Simplify. INDEPENDENT PRACTICE See Example 1 See Example 2 See Example 3 Combine like terms. 19. 7y 6y 20. 4z 5 2z 21. 3a 6 2a 9 5a 22. 5z z 23. 9x 3 4x 24. 9b 6 3b 3 b 25. 3z 4z b 5 26. 5a a 4z 3z 27. 9x 8y 2x 8 4y 28. 6x 2 3x 6q 29. 7d d 3e 12 30. 16a 7c 5 7a c 31. 5(y 2) y 32. 2(3y 7) 6y 33. 3(x 6) 8x 34. 3(4y 5) 8 35. 6(2x 8) 9x 36. 4(4x 4) 3x Simplify. PRACTICE AND PROBLEM SOLVING Extra Practice See page EP6. 37. Geometry A rectangle has length 5x and width x. Write and simplify an expression for the perimeter of the rectangle. 38. Hobbies Charlie has x state quarters. Ty has 3 more quarters than Charlie has. Vinnie has 2 times as many quarters as Ty has. Write and simplify an expression to show how many state quarters they have in all. 39. Reasoning Determine whether the expression r 17m 8 is equivalent to 3(2r 4m) 5(m 3 r) 7. Use properties to justify your answer. Simplify each expression. Justify each step. 122 40. 6(4 7k) 16 14 41. 5d 7 4d 2d 6 42. 7x 2(y 3x) 43. 3r 6r 2 5r 9r 44. 6y 3 7y 10 3z 45. 2(k 5) 3 k Chapter 3 Multi-Step Equations and Inequalities Write and simplify an expression for each situation. 46. Business A promoter charges $7 for each adult ticket, plus an additional $2 per ticket for tax and handling. What is the total cost of x tickets? 47. Sports Write an expression for the total number of medals won in the 2004 Summer Olympics by the countries shown below. United States Great Britain Brazil Lithuania 35 Gold 39 Silver 29 Bronze 9 Gold 9 Silver 12 Bronze 4 Gold 3 Silver 3 Bronze 1 Gold 2 Silver 0 Bronze Write an algebraic expression for each verbal description. Then simplify the expression. 48. four times the sum of m and p, decreased by six times m 49. y squared minus twice the sum of x and y squared 50. the product of three and r, increased by the sum of nine, 2r, and one 51. What’s the Error? A student said that 3x 4y can be simplified to 7xy by combining like terms. What error did the student make? 52. Write About It Write an expression that can be simplified by combining like terms. Then write an expression that cannot be simplified, and explain why it is already in simplest form. 53. Challenge Simplify the expression 36x 4 9x 5x 1 . 0 1 1 2 AF1.3, AF4.1 54. Multiple Choice Which expression is equivalent to p 3 5t 4p? A 5p 2t B 7p 5t C 5(p t) 3 D 3 5t 4p 55. Gridded Response Simplify 3(2x 7) 10x. What is the coefficient of x? Solve. (Lesson 2-8) x 14 8 56. 3 a 35 57. 5 9 1 58. 4w 7 10.7 Complete each equation. Then name the property that is illustrated in each. (Lesson 3-1) 59. (x 3) 2 x 2 3 60. 4.8 6 6 61. 8(5 9) (8 )9 3-2 Simplifying Algebraic Expressions 123 3-3 Solving Multi-Step Equations Why learn this? You can solve problems about average speed by solving multi-step equations. (See Example 3.) California Standards Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. EXAMPLE A multi-step equation requires more than two steps to solve. To solve a multi-step equation, you may have to simplify the equation first by combining like terms. 1 Solving Equations That Contain Like Terms Solve 3x 5 6x 7 25. 3x 5 6x 7 25 3x 6x 5 7 25 Commutative Property of Addition 9x 2 25 Combine like terms. 2 2 Since 2 is subtracted from 9x, add 2 to 9x 27 both sides. 9x 27 9 9 x 3 Since x is multiplied by 9, divide both sides by 9. If an equation contains fractions, it may help to multiply both sides of the equation by the least common denominator (LCD) to clear the fractions before you isolate the variable. EXAMPLE 2 Solving Equations That Contain Fractions Solve. 3y 5 1 7 7 7 3y 5 1 7 7 7 7 7 73y 75 71 7 7 7 1 5 1 1 3y 1 771 7 71 771 3y 5 1 5 5 3y 6 3y 6 3 3 y 2 124 Chapter 3 Multi-Step Equations and Inequalities Multiply both sides by 7. Distributive Property Simplify. Since 5 is added to 3y, subtract 5 from both sides. Since y is multiplied by 3, divide both sides by 3. Solve. 5p p 1 11 2 6 6 3 5p p 1 11 6 6 3 2 6 6 5p p 1 6 6 6 3 6 1 6 1 2 6 1 11 3 1 2 p 1 5p 6 6 6 3 6 2 6 6 1 1 1 1 The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly. 5p 2p 3 11 7p 3 11 3 3 7p 14 See Skills Bank p. SB9. 7p 14 7 7 p2 EXAMPLE 3 Multiply both sides by 6, the LCD of the fractions. Distributive Property Simplify. Combine like terms. Since 3 is subtracted from 7p, add 3 to both sides. Since p is multiplied by 7, divide both sides by 7. Travel Application On the first day of her vacation, Carly drove m miles in 4 hours. On the second day, she drove twice as far in 7 hours. If her average speed for the two days was 62.8 mi/h, how far did she drive on the first day? Round your answer to the nearest tenth of a mile. Carly’s average speed is her total distance for the two days divided by the total time. total distance average speed total time m 2m 62.8 47 3m 62.8 11 3m 11 11 11(62.8) Substitute m 2m for total distance and 4 7 for total time. Simplify. Multiply both sides by 11. 3m 690.8 3m 690.8 3 3 Divide both sides by 3. m 230.27 Carly drove approximately 230.3 miles on the first day. Think and Discuss 1. List the steps required to solve 3x 4 2x 7. 2. Tell how you would clear the fractions in 34x 23x 58 1. 3-3 Solving Multi-Step Equations 125 3-3 California Standards Practice Extension of AF4.1; AF4.2 Exercises KEYWORD: MT8CA 3-3 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Solve. 1. 7d 12 2d 3 18 2. 3y 4y 6 20 3. 10e 2e 9 39 4. 4c 5 14c 67 5. 5h 6 8h 3h 76 6. 7x 2x 3 32 4x 3 1 1 1 7. 1 3 3 3 8. 2 6 3 2 y 2p 4 6 9. 5 5 5 See Example 3 5y 1 1 15 1 10. 8z 4 4 11. Travel Barry’s family drove 843 mi to see his grandparents. On the first day, they drove 483 mi. On the second day, how long did it take to reach Barry’s grandparents’ house if they averaged 60 mi/h? INDEPENDENT PRACTICE See Example 1 See Example 2 Solve. 12. 5n 3n n 5 26 13. 81 7k 19 3k 14. 36 4c 3c 22 15. 12 5w 4w 15 16. 37 15a 5a 3 17. 30 7y 35 6y p 3 1 18. 8 8 38 4g See Example 3 g 7h 4h 18 19. 1 1 1 2 2 2 20. 1 8 1 1 6 6 6 7 3m m 1 21. 1 6 3 4 2 4 2b 6b 22. 1 13 26 3 3x 21x 1 23. 4 32 18 3 3 24. Recreation Lydia rode 243 miles in a three-day bike trip. On the first day, Lydia rode 67 miles. On the second day, she rode 92 miles. How many miles per hour did she average on the third day if she rode for 7 hours? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP6. Solve and check. 5n 1 3 25. 8 2 4 26. 4n 11 7n 13 27. 7b 2 12b 63 x 2 5 28. 2 3 6 29. 2x 7 3x 10 7 3r 4 30. 4 5 1 0 31. 4y 3 9y 32 32. 7n 10 9n 13 33. Finance Alessia is paid 1.4 times her normal hourly rate for each hour she works over 30 hours in a week. Last week she worked 35 hours and earned $436.60. What is her normal hourly rate? 126 Chapter 3 Multi-Step Equations and Inequalities 34. Geometry The obtuse angle of an isosceles triangle measures 120°. Write and solve an equation to find the measure of the base angles. (Hint: An isosceles triangle has two congruent angles. An obtuse angle measures more than 90° but less than 180°.) Sports 35. Reasoning The sum of two consecutive numbers is 63. What are the two numbers? Explain your solution. You can estimate the weight in pounds of a fish that is L inches long and G inches around at the thickest part by using the formula LG 2 . W 800 36. Sports The average weight of the top 5 fish caught at a fishing tournament was 12.3 pounds. The weights of the second-, third-, fourth-, and fifth-place fish are shown in the table. What was the weight of the heaviest fish? F 32 1.8 37. Science The formula K 273 is used to convert a temperature from degrees Fahrenheit to kelvins. Water boils at 373 kelvins. Use the formula to find the boiling point of water in degrees Fahrenheit. Winning Entries Caught by Weight (lb) Wayne S. Carla P. 12.8 Deb N. 12.6 Virgil W. 11.8 Brian B. 9.7 38. What’s the Error? A student’s work in solving an equation is shown. What error has the student made, and what is the correct answer? 1 x 5x 13 5 x 5x 65 6x 65 65 x 6 39. Write About It Compare the steps you would use to solve the equations 4x 8 16 and 4(x 2) 16. 40. Challenge Solve the following equation. 1 1 4 4 3x 4 3x 1 6 3 AF1.3, AF4.1, Ext. of AF4.1 41. Multiple Choice Solve 4k 7 3 5k 59. A k6 B k 6.6 C k7 D k 11.8 42. Gridded Response Antonio’s first four test grades were 85, 92, 91, and 80. What must he score on the next test to have an 88 test average? Solve. (Lesson 1-9) 43. 5n 6 21 x 44. 17y 31 3 30 45. 41 11 47. 6t 3k 15 48. 5a 3 b 1 Combine like terms. (Lesson 3-2) 46. 9m 8 4m 7 5m 3-3 Solving Multi-Step Equations 127 Model Equations with Variables on Both Sides 3-4 Use with Lesson 3-4 KEYWORD: MT8CA Lab3 KEY REMEMBER Algebra tiles Adding or subtracting zero does not change the value of an expression. + x − + 1 − –x 1 –1 + − 0 + California Standards Extension of AF4.1 Solve two-step linear − 0 To solve an equation with the same variable on both sides of the equal sign, you must first add or subtract to eliminate the variable term from one side of the equation. equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Activity 1 Model and solve the equation x 2 2x 4. − + + + − − − − + + − + + x 2 2x 4 + − + + + + + − − − − Add x to both sides. + + + + + + Remove zero. + + + + + + + + + − − − − + + + − − + + − − + + Add 4 to both sides. + + + − − + + − − + + Remove zero. + + + + + + + + + + + + Divide each side into 3 equal groups. 31 of each side is the solution. 2 x Think and Discuss 1. How would you check the solution to x 2 2x 4 using algebra tiles? 2. Why must you isolate the variable terms by having them on only one side of the equation? Try This Model and solve each equation. 1. x 3 x 3 128 2. 3x 3x 18 3. 6 3x 4x 8 Chapter 3 Multi-Step Equations and Inequalities 4. 3x 3x 2 x 17 3-4 Solving Equations with Variables on Both Sides Who uses this? Consumers can use equations with variables on both sides to compare costs. (See Example 3.) California Standards Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Also covered: AF1.1 The fees for two dogsitting services are shown at right. To find the number of hours for which the costs will be the same for both services, you can write and solve an equation with variables on both sides of the equal sign. To solve an equation like this, first use inverse operations to “collect” variable terms on one side of the equation. EXAMPLE 1 Solving Equations with Variables on Both Sides Solve. You can always check your solution by substituting the value back into the original equation. 3a 2a 3 3a 2a 3 2a 2a a 3 To collect the variable terms on one side, subtract 2a from both sides. Check 3a 2a 3 ? 3(3) 2(3) 3 ? 963 ? 99✔ Substitute 3 for a in the original equation. 3v 8 7 8v 3v 8 7 8v 3v 3v 8 7 5v 7 7 15 5v To collect the variable terms on one side, subtract 3v from both sides. 15 5v 5 5 3 v Since 7 is added to 5v, subtract 7 from both sides. Since v is multiplied by 5, divide both sides by 5. 3-4 Solving Equations with Variables on Both Sides 129 Solve. g7g3 g7 g3 g g 7 3 If the variables in an equation are eliminated and the resulting statement is false, the equation has no solution. To collect the variable terms on one side, subtract g from both sides. There is no solution. There is no number that can be substituted for the variable g to make the equation true. To solve more complicated equations, you may need to first simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable. EXAMPLE 2 Solving Multi-Step Equations with Variables on Both Sides Solve. 2c 4 3c 9 c 5 2c 4 3c 9 c 5 c 4 4 c c c 4 4 2c 4 4 8 2c 8 2c 2 2 Combine like terms. To collect the variable terms on one side, add c to both sides. Since 4 is add to 2c, add 4 to both sides. Since c is multiplied by 2, divide both sides by 2. 4c 2w 5w 1 11 w 3 6 4 9 2w 5w 1 11 w 3 6 4 9 2w 5w 1 11 36 3 6 4 36 w 9 Multiply both sides by 36, the LCD. 12 9 4 2w 6 5w 1 11 3631 3661 3641 36(w) 3691 Distributive Property 24w 30w 9 6w 9 6w 9 44 35 36w 44 36w 44 6w 42w 44 44 42w 35 42w 42 42 5 6 w 130 Chapter 3 Multi-Step Equations and Inequalities Combine like terms. Add 6w to both sides. Subtract 44 from both sides. Divide both sides by 42. A system of equations is a set of two or more equations that contain two or more variables. To solve a system of two equations, you can reduce the system to one equation that has only one variable. EXAMPLE 3 Business Application Happy Paws charges a flat fee of $19.00 plus $1.50 per hour to keep a dog. Woof Watchers charges a flat fee of $15.00 plus $2.75 per hour. Find the number of hours for which you would pay the same amount for both services. What is the cost? Write an equation for each service. Let c represent the total cost and h represent the number of hours. total cost is flat fee plus cost per hour Happy Paws: c 19.00 1.5 h Woof Watchers: c 15.00 2.75 h Now write an equation showing that the costs are equal. 19.00 1.5h 15.00 2.75h 1.5h 1.5h To collect the variable terms on one 19.00 15.00 1.25h side, subtract 1.5h from both sides. 15.00 15.00 Subtract 15.00 from both sides. 4.00 1.25h 4.00 1.25h 1.25 1.25 Divide both sides by 1.25. 3.2 h The two services cost the same when used for 3.2 hours. To find the cost, substitute 3.2 for h in either equation. Happy Paws: Woof Watchers: c 19.00 1.5h c 15.00 2.75h c 19.00 1.5(3.2) c 15.00 2.75(3.2) c 19.00 4.8 c 15.00 8.8 c 23.8 c 23.8 The cost for 3.2 hours at either service is $23.80. Think and Discuss 1. Explain how you would solve the equation 3x 4 2x 6x 2 5x 2. What do you think the solution means? 2. Give a series of steps that you can use to solve any equation with variables on both sides of the equal sign. 3-4 Solving Equations with Variables on Both Sides 131 3-4 California Standards Practice AF1.1, Extension of AF4.1; MG2.1 Exercises KEYWORD: MT8CA 3-4 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Solve. 1. 6x 3 x 8 2. 5a 5 7 2a 3. 2x 7 10x 9 4. 4y 2 6y 6 5. 13x 15 11x 25 6. 5t 5 5t 7 7. 5x 2 3x 17 12x 23 n 3n 6 5 2n 18 8. 4 1 2 5 11d 9. 1 1 3 3d 7 4d 2 2 See Example 3 10. 4(x 5) 2 x 3 11. Business A long-distance phone company charges $0.027 per minute and a $2 monthly fee. Another long-distance phone company charges $0.035 per minute with no monthly fee. Find the number of minutes for which the charges for both companies would be the same. What is the cost? INDEPENDENT PRACTICE See Example 1 See Example 2 See Example 3 Solve. 12. 3n 16 7n 13. 8x 3 11 6x 14. 5n 3 14 6n 15. 3(2x 11) 6x 33 16. 6x 3 x 8 17. 7y 8 5y 4 3p 7p 3 1 p 1 18. 8 16 4 4 1 2 6 19. 4(x 5) 5 6x 7.4 4x 1 20. 2(2n 6) 5n 12 n a 9 4 20a 21. 2 5.5 2a 1 1 1 6 3 3 3 22. Business Al’s Rentals charges $25 per hour to rent a Windsurfer™ and a wet suit. Wendy’s charges $20 per hour plus $15 extra for a wet suit. Find the number of hours for which the total charges for both would be the same. What is the cost? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP6. Solve and check. 23. 3y 1 13 4y 24. 4n 8 9n 7 25. 5n 20n 5(n 20) 26. 3(4x 2) 12x 27. 100(x 3) 450 50x 28. 0.2p 1.2 1.2 0.2p 29. Find two consecutive whole numbers such that 34 of the first number is 5 more than 12 the second number. (Hint: Let n represent the first number. Then n 1 represents the next consecutive whole number.) 132 Chapter 3 Multi-Step Equations and Inequalities Write an equation to represent each relationship. Then solve the equation. 30. Six plus the product of 3 and a number is the same as the product of 9 and the number. 31. A number decreased by 25 is the same as 10 minus 4 times the number. 32. Eight less than 2 times a number is the same as the number increased by 24. Science The figures in each pair have the same perimeter. Find each perimeter. x 15 33. 34. x x6 x x 45 x 40 x x2 x4 x 25 35. Science An atom of chlorine (Cl) has 6 more protons than an atom of sodium (Na). The atomic number of chlorine is 5 less than twice the atomic number of sodium. The atomic number of an element is equal to the number of protons per atom. Sodium and chlorine bond together to form sodium chloride, or salt. The atomic structure of sodium chloride causes it to form cubes. a. How many protons are in an atom of chlorine? b. What is the atomic number of sodium? 36. Choose a Strategy Solve the following equation for t. How can you determine the solution once you have combined like terms? 3(t 24) 7t 4(t 18) 37. Write About It Two cars are traveling in the same direction. The first car is going 45 mi/h, and the second car is going 60 mi/h. The first car left 2 hours before the second car. Explain how you could solve an equation to find how long it will take the second car to catch up to the first car. x2 6 x1 38. Challenge Solve the equation 8 7 2. NS1.1, AF4.1 39. Multiple Choice Find three consecutive integers (x, x 1, and x 2) so that the sum of the first two integers is 10 more than the third integer. A 7, 6, 5 B 4, 5, 6 C 11, 12, 13 D 35, 36, 37 C w 1 D w 5 40. Multiple Choice Solve 6w 15 9w. A w3 B w0 Solve. (Lesson 1-9) 41. 6x 3 15 n 42. 7 2 1 3 43. 72 5g 12 y 44. 4 7 7 Compare. Write , , or . (Lesson 2-2) 5 45. 9 13 21 13 46. 1 1 8 7 1 47. 7 1 8 2 48. 3 14 2 1 3-4 Solving Equations with Variables on Both Sides 133 Quiz for Lessons 3-1 Through 3-4 3-1 Properties of Rational Numbers Name the property that is illustrated in each equation. 1 1 2. m n n m 1. 3 3 7 3 3 7 Simplify each expression. Justify each step. 4. 20 19 5 5. 35.5 12.7 4.5 1 1 1 3. 2 x 6 2 x 2 6 1 6. 4 11 16 Write each product using the Distributive Property. Then simplify. 7. 3(57) 3-2 8. (42)7 9. 5(95) Simplifying Algebraic Expressions Simplify. 10. 5x 3x 11. 6p 6 p 12. 2t 3 t 4 5t 13. 3x 4y x 2y 14. 2(r 1) r 15. 4n 2m 8n 2m 3-3 Solving Multi-Step Equations Solve. 16. 2c 6c 8 32 3x 2 10 17. 7 7 7 t t 7 18. 4 3 1 2 4m m 7 19. 3 6 2 3 1 20. 4b 5b 11 r r 21. 3 7 5 3 22. Marlene drove 540 miles to visit a friend. She drove 3 hours and stopped for gas. She then drove 4 hours and stopped for lunch. How many more hours did she drive if her average speed for the trip was 60 miles per hour? 3-4 Solving Equations with Variables on Both Sides Solve. 23. 4x 11 x 2 24. q 5 2q 7 25. 6n 21 4n 57 26. 2m 6 2m 1 27. 4a 2a 11 6a 7 1 5 y 4 2y 3 28. 1 2 29. The rectangle and the triangle have the same perimeter. Find the perimeter of each figure. x x9 x2 x7 x7 134 Chapter 3 Multi-Step Equations and Inequalities California Standards MR1.1 Analyze problems by identifying relationships, distinguishing relevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Also covered: AF1.1, Extension of Make a Plan AF4.1 • Write an equation Several steps may be needed to solve a problem. It often helps to write an equation that represents the steps. Example: Juan’s first 3 exam scores are 85, 93, and 87. What does he need to score on his next exam to average 90 for the 4 exams? Let x be the score on his next exam. The average of the exam scores is the sum of the 4 scores, divided by 4. This amount must equal 90. Write the equation in words: Exam 1 Exam 2 Exam 3 Exam 4 Number of exams 90 85 93 87 x 90 4 265 x 90 4 265 x 4 4 4(90) 265 x 360 265 265 x 95 Juan needs a 95 on his next exam. Read each problem and write an equation that could be used to solve it. 1 The average of two numbers is 34. The first number is three times the second number. What are the two numbers? 2 Nancy spends 13 of her monthly salary on rent, 0.1 on her car payment, 112 on food, and 20% on other bills. She has $680 left for other expenses. What is Nancy’s monthly salary? 3 A vendor at a concert sells new and used CDs. The new CDs cost 2.5 times as much as the old CDs. If 4 used CDs and 9 new CDs cost $159, what is the price of each item? 4 Amanda and Rick have the same amount to spend on school supplies. Amanda buys 4 notebooks and has $8.60 left. Rick buys 7 notebooks and has $7.55 left. How much does each notebook cost? 135 3-5 Inequalities Why learn this? You can show the maximum capacity of an elevator using an inequality. California Standards AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). An inequality compares two expressions using , , , or . Symbol Vocabulary inequality algebraic inequality solution set EXAMPLE Meaning Word Phrases Is less than Fewer than, below Is greater than More than, above Is less than or equal to At most, no more than Is greater than or equal to At least, no less than An inequality that contains a variable is an algebraic inequality . 1 Translating Word Phrases into Inequalities Write an inequality for each situation. The capacity of an elevator is at most 12 people. Let c the capacity of the elevator. “At most” means less than or equal to. c 12 There are more than 1000 books in the library. Let b the number of books in the library. b 1000 “More than” means greater than. EXAMPLE 2 Writing Inequalities Write an inequality for each statement. A number x plus 14 is greater than or equal to 30. A number x plus 14 is greater than or equal to 30 x 14 30 x 14 30 A number n decreased by 3 is less than 21. A number n decreased by 3 is less than 21 n 3 21 n 3 21 136 Chapter 3 Multi-Step Equations and Inequalities A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set . You can graph the solution set on a number line. The symbols and indicate an open circle. This open circle shows that 5 is not a solution. 0 1 2 3 4 5 6 7 8 9 10 The symbols and indicate a closed circle. This closed circle shows that 3 is a solution. –3 –2 –1 0 EXAMPLE 3 1 2 3 4 5 6 7 Graphing Inequalities Graph each inequality. x 4 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 1 12 m 112 m is the same as m 112. ⫺2 ⫺1 0 1 2 3 4 5 Draw an open circle at 4. The solutions are all values of x greater than 4, so shade to the right of 4. Draw a closed circle at 112. The solutions are 112 and all values of m less than 112, so shade to the left of 112. A compound inequality is the result of combining two inequalities. The words and and or are used to describe how the two parts are related. EXAMPLE 4 Writing Compound Inequalities Write a compound inequality for each statement. The compound inequality in Example 4B can also be written with the variable between the two endpoints. 6 n 9.5 A number t is either less than 2 or greater than or equal to 1. t 2 or t 1 A number n is both greater than or equal to 6 and less than 9.5. n 6 and n 9.5 Think and Discuss 1. Explain how to write “x is no less than 16” as an inequality. 2. Compare the graphs of the inequalities x 3 and x 3. 3-5 Inequalities 137 3-5 California Standards Practice Exercises AF1.1 KEYWORD: MT8CA 3-5 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Write an inequality for each situation. 1. There are no more than 60 people in the theater. 2. The temperature of the water is above 72°F. See Example 2 Write an inequality for each statement. 3. A number m increased by 7 is at least 15. 4. Twice a number x is less than 18. See Example 3 Graph each inequality. 5. x 2 See Example 4 6. w 1 7. 2.5 y 1 8. m 32 Write a compound inequality for each statement. 9. A number s is either less than 5 or greater than or equal to 3. 10. A number t is both greater than 10 and less than 1. INDEPENDENT PRACTICE See Example 1 Write an inequality for each situation. 11. Fewer than 10 students rode their bikes to the game. 12. No more than 18 people may ride the roller coaster at one time. See Example 2 Write an inequality for each statement. 13. A number x decreased by 11 is less than 35. 1 14. Three times a number n is greater than 43. 15. A number y divided by 7 is at most 10. See Example 3 See Example 4 Graph each inequality. 16. m 3 17. s 1.5 18. 2 x 20. b 1 21. x 0 22. n 2 1 19. 4 y 1 23. 22 c Write a compound inequality for each statement. 24. A number x is both less than 1.5 and greater than or equal to 0. 1 25. A number c is either greater than or equal to 2 or less than or equal to 7. PRACTICE AND PROBLEM SOLVING Extra Practice See page EP7. 138 26. Suly earned 87 points on her first test and p points on her second test. She needs a total of at least 140 points on the two tests to pass the class. Write an inequality for this situation. Chapter 3 Multi-Step Equations and Inequalities Write an inequality for each statement. 27. A number w multiplied by 5 is no less than 60. 28. The sum of 10 and a number g is greater than 4.8. 2 1 29. A number m decreased by 25 is at most 35. Write an inequality shown by each graph. 30. 31. 4 2 0 2 4 6 8 4 2 0 2 4 6 8 32. 0 33. 2 4 6 8 10 12 4 2 0 2 4 6 8 34. Business A cafe sells fruit smoothies for $3.50 each. The manager of the cafe wants the total daily revenue from the smoothies to be at least $175. Assume the cafe sells n smoothies per day. Write an inequality that represents the manager’s goal. 35. Astronomy The diameter of Jupiter, the largest planet in the Solar System, is 89,000 miles. Let d be the diameter of any planet in the Solar System. Write an inequality for d. 36. What’s the Error? A student was asked to graph the inequality 2 n. Explain the student’s error in the graph at right. ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 37. Write About It In mathematics, the conventional way to write an inequality is with the variable on the left, such as x 5. Explain how to rewrite the inequality 4 x in the conventional way. 38. Challenge Write an inequality for the statement “14 less than twice a number x is greater than three times the number.” NS1.1, NS1.2, AF1.1 39. Multiple Choice Which inequality represents the statement, “A number z decreased by 9 is no more than 20”? A z 9 20 B z 9 20 9 z 20 C D 9 z 20 D x 4 40. Multiple Choice Which inequality is shown by the graph? ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 A x 4 B x 4 0 C 1 2 x 4 Write each set of integers in order from least to greatest. (Lesson 1-3) 41. 19, 25, 12 42. 4, 0, 4, 3 43. 5, 9, 7, 11 44. 2, 6, 5, 0 Add or subtract. Write each answer in simplest form. (Lesson 2-3) 17 6 45. 121 121 46. 7 7 1 8 29 13 47. 1 1 2 2 29 45 48. 5 5 3-5 Inequalities 139 Solving Inequalities by Adding or Subtracting 3-6 Why learn this? You can solve an inequality to find the amount of a nutrient that you should consume. (See Example 2.) California Standards AF4.0 Students solve simple linear equations and inequalities over the rational numbers. When you add or subtract the same number on both sides of an inequality, the resulting inequality will still be true. 2 5 7 7 5 12 You can use this idea to solve inequalities. You find solution sets of inequalities the same way you find solutions of equations, by isolating the variable. EXAMPLE 1 Solving Inequalities by Adding or Subtracting Solve and graph. x 7 10 x 7 10 7 7 x 17 Since 7 is added to x, subtract 7 from both sides. –21 –20 –19 –18 –17 –16 –15 –14 –13 –12 –11 When checking your solution, choose numbers that are easy to work with. Remember to substitute the numbers into the original inequality. Check According to the graph, 20 should be a solution and 3 should not be a solution. x 7 10 x 7 10 ? ? 20 7 10 Substitute 3 7 10 Substitute ? ? 20 for x. 3 for x. 13 10 ✔ 10 10 ✘ So 20 is a solution. t 11 22 t 11 22 11 11 t 11 Since 11 is subtracted from t, add 11 to both sides. –15 –13 –11 140 So 3 is not a solution. –9 Chapter 3 Multi-Step Equations and Inequalities –7 –5 –3 –1 1 3 5 Solve and graph. z 6 3 z 6 3 6 6 z 9 Since 6 is added to z, subtract 6 from both sides. –10 –9 1 1 1 1 1 24 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 44 n 24 44 n 24 24 1 62 n 1 –1 EXAMPLE 2 1 1 Since 24 is subtracted from n, add 24 to both sides. 0 1 2 3 4 5 6 7 8 9 10 11 Nutrition Application Manganese is a mineral that is found in nuts and grains. It is recommended that men consume at least 2.3 mg of manganese each day. Eric has consumed 0.9 mg today. Write and solve an inequality to find how many additional milligrams he should consume. Let m the number of additional milligrams of manganese. 0.9 milligrams plus additional milligrams is at least 2.3 milligrams 0.9 m 2.3 0.9 m 2.3 0.9 0.9 m 1.4 Since 0.9 is added to m, subtract 0.9 from both sides. Eric should consume at least 1.4 additional milligrams of manganese. Check 0.9 m 2.3 ? 0.9 2 2.3 ? 2.9 2.3 ✔ 2 is greater than 1.4. Substitute 2 for m. 0.9 m 2.3 ? 0.9 1 2.3 ? 1.9 2.3 ✘ 1 is less than 1.4. Substitute 1 for m. Think and Discuss 1. Explain how you know whether to use addition or subtraction to solve an inequality. 2. Describe how to check whether 11 is a solution of 6 t 4. 3-6 Solving Inequalities by Adding or Subtracting 141 3-6 California Standards Practice NS1.2, AF1.1, AF4.0 Exercises KEYWORD: MT8CA 3-6 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Solve and graph. 1. x 3 4 2. 4 b 20 3. 6 f 30 4. z 8 13 5. 2.1 k 7.2 6. x 3 2 1 7. A measuring cup can hold no more than 16 fluid ounces of liquid. Rosa pours 612 fluid ounces of water into the cup. Write and solve an inequality to determine how many additional fluid ounces of water she can add. 8. Paul’s car can go at most 375 miles on one tank of gas. Paul fills the tank and then drives 167 miles. Write and solve an inequality to find out how many more miles Paul can drive before he will have to refill the tank. INDEPENDENT PRACTICE See Example 1 Solve and graph. 9. 7 x 49 12. 0.6 y 0.72 See Example 2 10. 1 t 4 1 11. 3 x 12 2 13. c 53 83 14. 2 a (5) 15. Consumer Math A clothes store gives customers a free gift if they spend at least $50 in the store. Stacey plans to buy a pair of jeans that cost $21.75. Write and solve an inequality to show how much more she must spend in order to get the free gift. 16. Consumer Math Latrell’s cell-phone plan allows him to talk for no more than 500 minutes per month. He has already used 288 minutes this month. Write and solve an inequality to determine how many more minutes he can talk on the phone this month. PRACTICE AND PROBLEM SOLVING Extra Practice See page EP7. Solve and graph. 2 17. z 0.75 0.75 18. 7 x 3 19. 7 y 8.8 20. m (12) 6 4 1 21. 5 k 5 22. 39.5 15.5 g You can use set-builder notation to write the solution of an inequality. For example, {x : x 5} means the set of all real numbers x such that x is less than 5. Solve each inequality and write the solution using set-builder notation. 23. x 12 8 24. z 4 16 25. 3.5 b 7 26. Reasoning When a number is added to 15, the result is greater than 12. What are the possible values of the number? Graph them on a number line. 142 Chapter 3 Multi-Step Equations and Inequalities 27. Business Toshi Business Solutions will make a profit for the current year if their total sales are greater than their operating costs. Their accountants estimate that the company will have operating costs of $201,522 for the entire year. So far this year, the company has sales of $98,200. Califor nia Language Arts a. Write and solve an inequality to find out how much more money Toshi must earn in sales for the remainder of the year to show a profit. b. Check your answer. Then explain why your answer is reasonable. Californian John Steinbeck, author of The Grapes of Wrath and Of Mice and Men, won the Nobel Prize for Literature in 1962. Great Novels 189 494 Title 28. Language Arts Danielle is reading one of the novels in the graph. She has already read 65 pages. Write and solve two different inequalities to find out how many pages she has left to read. (Hint: Write one inequality based on the minimum number of pages and one inequality based on the maximum number of pages.) 359 275 29. Reasoning Substitute the values 1, 2, 3, 4, 5, and 6 for x in 10 x 6. Use the results to make a conjecture about the solution of the inequality. 0 75 150 225 300 375 450 525 Number of pages 30. Write a Problem Write a word problem that can be answered by solving the inequality x 40 75. 31. Write About It Explain how to check the solution of an inequality. 32. Challenge The inequality y 3 is missing a number. The solution of the inequality is shown on the number line. What is the missing number? ⫺2 ⫺1 0 1 2 3 4 5 NS1.2, AF1.1, AF4.0 33. Multiple Choice Solve m 5 8 for m. A m 13 B m 3 C m3 D m 13 34. Multiple Choice In the inequality 200 80 x, x is the length of a movie in minutes. Which phrase most accurately describes the length of the movie? A At least 120 minutes C At most 120 minutes B More than 120 minutes D Less than 120 minutes Add or subtract. (Lesson 2-6) 1 3 35. 4 7 7 5 36. 2 8 0 9 5 37. 1 6 0 4 1 38. 37 25 Write an inequality for each statement. (Lesson 3-5) 39. A number t increased by 2 is less than 8. 40. Twice a number w is no more than 12. 3-6 Solving Inequalities by Adding or Subtracting 143 3-7 Solving Inequalities by Multiplying or Dividing Why learn this? You can solve an inequality to determine how many representatives voted on a bill. (See Exercise 18.) California Standards AF4.0 Students solve simple linear equations and inequalities over the rational numbers. Also covered: AF1.1 When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol to make the statement true. –b –a 0 ab a b Multiply both sides by 1. a b Use the number line to EXAMPLE 1 a b b a b a Multiply both sides by 1. b a Use the number line to determine the direction determine the direction of the inequality symbol. of the inequality symbol. Solving Inequalities by Multiplying or Dividing Solve and graph. h 24 5 h 5 24 5 5 120 h, or h 120 When graphing an inequality on a number line, an open circle means that the point is not part of the solution and a closed circle means that the point is part of the solution. Multiply both sides by 5. 115 116 117 118 119 120 121 122 Check According to the graph, 119 should be a solution and 121 should not be a solution. h h 24 5 24 5 ? 119 24 5 ? 24 23.8 ✔ ? 121 24 5 ? 24 24.2 ✘ Substitute 119 for h. So 119 is a solution. Substitute 121 for h. So 121 is not a solution. 7x 42 42 7x 7 7 Divide both sides by 7; changes to . x 6 12 11 10 144 9 Chapter 3 Multi-Step Equations and Inequalities 8 7 6 5 4 EXAMPLE 2 PROBLEM SOLVING APPLICATION If all the sheets of paper used by personal computer printers each year were laid end to end, they would circle Earth more than 800 times. Earth’s circumference is about 25,120 mi (1,591,603,200 in.), and one letter-size sheet of paper is 11 in. long. About how many sheets of paper are used each year? 1 Understand the Problem The answer is the number of sheets of paper used by personal computer printers in one year. List the important information: • The amount of paper would circle the earth more than 800 times. • Once around Earth is approximately 1.6 billion in. • One sheet of paper is 11 in. long. Show the relationship of the information: the number of the length sheets of paper of one sheet 800 the distance around Earth 2 Make a Plan Use the relationship to write an inequality. Let x represent the number of sheets of paper. x 3 Solve 11x 800 1.6 11x 1280 1280 11x 11 11 11 in. 800 1.6 billion in. Simplify. Divide both sides by 11. x 116.36 More than 116 billion sheets of paper are used by personal computer printers in one year. 4 Look Back 1,600,0 00,000 To circle Earth once takes 145,454,545 11 sheets of paper; to circle it 800 times would take 800 145,454,545 116,363,636,000 sheets. Think and Discuss 1. Give all the symbols that make 5 3 15 true. Explain. 2. Explain how you would solve the inequality 4x 24. 3-7 Solving Inequalities by Multiplying or Dividing 145 3-7 Exercises California Standards Practice AF1.1, AF4.0 KEYWORD: MT8CA 3-7 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Solve and graph. r 1. 3 6 a 5. 10 4 See Example 2 j 2. 4w 12 3. 20 6 4. 6r 30 6. 36 2m r 21 7. 3 8. 20 5x 9. The owner of a sandwich shop is selling the special of the week for $5.90. At this price, he makes a profit of $3.85 on each sandwich sold. To make a total profit of at least $400 from the special, what is the least number of sandwiches he must sell? INDEPENDENT PRACTICE See Example 1 See Example 2 Solve and graph. x p 10. 16 2r 11. 15 5 12. 18w 54 13. 11 7 t 14. 9 4 15. 9h 108 a 14 16. 7 17. 16q 64 18. Social Studies A bill in the U.S. House of Representatives passed because at least 23 of the members present voted in favor of it. If the bill received 284 votes, at least how many members of the House of Representatives were present for the vote? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP7. Solve and graph. x p 19. 18 3r 20. 27 3 21. 17w 51 22. 101 7 t 5 23. 19 24. 3h 108 a 25. 1 12 0 26. 6q 72 Write and solve an algebraic inequality. 27. Nine times a number is less than 99. 28. The quotient of a number and 6 is at least 8. 29. The product of 7 and a number is no more than 63. 30. The quotient of some number and 3 is greater than 18. Write and solve an algebraic inequality. Then explain the solution. 31. A school receives a shipment of books. There are 60 cartons, and each carton weighs 42 pounds. The school’s elevator can hold 2200 pounds. What is the greatest number of cartons that can be carried on the elevator at one time if no people ride with them? 32. Each evening, Marisol spends at least twice as much time reading as she spends doing homework. If Marisol works on her homework for 40 minutes, how much time can she spend reading? 146 Chapter 3 Multi-Step Equations and Inequalities Choose the graph that represents each inequality. 33. 2y 14 A. B. 9 8 7 6 5 4 3 2 1 12 11 10 9 8 7 6 5 C. 5 6 7 8 9 10 11 12 13 h 34. 6 5 A. 28 29 30 31 32 33 34 35 36 B. 25 26 27 28 29 30 31 32 33 C. 25 26 27 28 29 30 31 32 33 35. What’s the Error? A student solved x 3 12 and got an answer of x 36. What error did the student make? 36. Write About It The expressions no more than, at most, and less than or equal to all indicate the same relationship between values. Write a problem that uses this relationship. Write the problem using each of the three expressions. 37. Challenge Angel weighs 5 times as much as his dog. When they stand on a scale together, the scale gives a reading of less than 163 pounds. If both their weights are whole numbers, what is the most each can weigh? NS1.2, AF1.1, AF4.0 38. Multiple Choice Which inequality is shown by the graph? 5 4 3 2 1 A w 3 B w 3 0 1 C 2 3 w 3 3 w D 39. Gridded Response In order to have the $200 he needs for a bike, Kevin plans to put money away each week for the next 15 weeks. What is the minimum amount in dollars that Kevin will need to average each week in order to reach his goal? Multiply. Write each answer in simplest form. (Lesson 2-4) 40. 71 4 3 9 25 11 41. 1 5 121 42. 78 9 1 4 43. 513 2 44. Frank needs to earn at least $350. He earns $15 for each hour h that he babysits. Write an inequality that represents Frank’s goal. (Lesson 3-5) 3-7 Solving Inequalities by Multiplying or Dividing 147 3-8 Solving Two-Step Inequalities Why learn this? Drama club members can use two-step inequalities to determine how many tickets they must sell to a musical to break even. (See Example 3.) California Standards AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. EXAMPLE When you solved two-step equations, you used the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities. 1 Solving Two-Step Inequalities Solve and graph. Math Builders For more on solving two-step inequalities, see the Step-by-Step Solution Builder on page MB4. 7y 4 24 7y 4 24 4 4 7y 28 Since 4 is subtracted from 7y, add 4 to both sides. 7y 28 7 7 Since y is multiplied by 7, divide both sides by 7. y 4 0 1 2 3 4 5 6 7 8 9 10 Check According to the graph, 10 should be a solution and 0 should not be a solution. 7y 4 24 7y 4 24 ? 7(0) 4 24 So 10 is a solution. So 0 is not a solution. Substitute ? 10 for y. 66 24 ✔ If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. 148 ? 7(10) 4 24 2x 4 3 2x 4 3 4 4 2x 1 1 2x 2 2 1 x 2 2 0 Substitute 4 24 ✘ 0 for y. ? Since 4 is added to 2x, subtract 4 from both sides. Since x is multiplied by 2, divide both sides by 2. Change to . 2 4 Chapter 3 Multi-Step Equations and Inequalities 6 EXAMPLE 2 Solving Inequalities That Contain Fractions 7 3x 5 and graph the solution. Solve 8 6 12 3x 5 24 8 6 3x 5 248 246 When an inequality contains fractions, you may want to multiply both sides by the LCD to clear the fractions. 7 241 2 7 24 1 2 Multiply by the LCD, 24. Distributive Property 9x 20 14 20 20 9x 6 Since 20 is added to 9x, subtract 20 from both sides. 6 9 9x 9 Since x is multiplied by 9, divide both sides by 9. Change to . 6 x 9 2 x 3 1 23 1 13 1 EXAMPLE 3 23 Simplify. 13 0 1 3 2 3 1 1 13 1 23 School Application The Drama Club is planning a spring musical. Club members estimate that the entire production will cost $1100.00. If they have $610.75 left from fund-raising, how many tickets must they sell to at least break even? In order to at least break even, ticket sales plus the money in the budget must be greater than or equal to the cost of the production. 4.75t 610.75 1100.00 610.75 610.75 4.75t 489.25 4.75t 489.25 4.75 4.75 Subtract 610.75 from both sides. Divide both sides by 4.75. t 103 The drama club must sell at least 103 tickets in order to break even. Think and Discuss 1. Compare solving a multi-step equation with solving a multi-step inequality. 2. Describe two situations in which you would have to reverse the inequality symbol when solving a multi-step inequality. 3-8 Solving Two-Step Inequalities 149 3-8 California Standards Practice AF4.1 Exercises KEYWORD: MT8CA 3-8 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Solve and graph. 1. 3k 5 11 2. 2z 29.5 10.5 3. 6y 12 36 4. 4x 6 14 5. 2y 2.5 16.5 6. 3k 2 13 x 1 2 7. 15 5 5 b 3 1 8. 1 5 2 0 h 5 9. 3 2 3 c 1 3 10. 8 2 4 See Example 3 1 d 1 11. 2 6 3 2 6m 12. 3 9 13. The chess club is selling caps to raise $425 for a trip. They have $175 already. If the club members sell caps for $12 each, at least how many caps do they need to sell to make enough money for their trip? INDEPENDENT PRACTICE See Example 1 See Example 2 Solve and graph. 14. 8k 6 18 15. 5x 3 23 16. 3p 3 36 17. 13 11q 9 18. 3.6 7.2n 25.2 19. 7x 15 34 a 2 1 21. 9 3 3 22. 3 1 4 2 n 4 3 24. 7 1 7 4 r 1 1 25. 3 1 2 8 p 4 1 20. 1 5 3 5 2 1 5 23. 3 1 k 6 8 See Example 3 1 n 1 26. Josef is on the planning committee for the eighth-grade party. The food, decoration, and entertainment costs a total of $350. The committee has $75 already. If the committee sells the tickets for $5 each, at least how many tickets must be sold to cover the remaining cost of the party? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP7. Solve and graph. 27. 3p 11 11 28. 9n 10 17 29. 3 5w 8 30. 6x 18 6 31. 12a 4 10 32. 4y 3 17 33. 3q 5q 12 3m 5 34. 4 8 35. 4b 3.2 7.6 36. 3k 6 4 90 5 37. 4 6 f 38. 9v 3 5 1 39. Reasoning What is the least whole number that is a solution of 2r 4.4 8.6? 40. Entertainment A speech is being given in a gymnasium that can hold no more than 650 people. A permanent bleacher will seat 136 people. The event organizers are setting up 25 rows with an equal number of chairs. At most, how many chairs can be in each row? 150 Chapter 3 Multi-Step Equations and Inequalities 41. Katie and April are making a string of beads for pi day (March 14). The string already has 70 beads. If there are only 30 more days until pi day, and they want to string 1000 beads by then, at least how many beads do they have to string each day? 3 .1 415 9 26 5 3 5 89 3 79 6 4 3 3 8 3 2 7 4 62 95 8 02 3 2 8 8 4 42. Sports The Astros have won 35 and lost 52 baseball games. They have 75 games remaining. At least how many of the remaining 75 games must the Astros win to have a winning season? (Hint: A winning season means they win more than 50% of their games.) 43. Economics Satellite TV customers can either purchase a dish and receiver for $249 or pay a $50 fee and rent the equipment for $12 a month. a. How much would it cost to rent the equipment for 9 months? b. How many months would it take for the rental charges to be more than the purchase price? 44. Write a Problem Write and solve an inequality using the following shipping rates for orders from a mail-order catalog. 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 .. . Mail-Order Shipping Rates Merchandise Amount $0.01– $25.00 $25.01 –50.00 $50.01 –75.00 $75.01 –125.00 $125.01 and over Shipping Cost $3.95 $5.95 $7.95 $9.95 $11.95 45. Write About It Describe two different ways to solve the inequality 3x 4 x. x x 1 46. Challenge Solve the inequality 5 6 1 . 5 AF1.3, AF4.1 47. Multiple Choice Solve 3g 6 18. A g 21 B g8 C 5x 1 g6 D g4 2 48. Short Response Solve and graph 6 2 3. Name the property that is illustrated in each equation. (Lesson 3-1) 49. 12y y 12 50. a (b c) (a b) c 51. x 13y 13y x Simplify. (Lesson 3-2) 52. 5(x 1) 2x 53. 6(r 10) r 54. 3(8 n) 21 3-8 Solving Two-Step Inequalities 151 Quiz for Lessons 3-5 Through 3-8 3-5 Inequalities Write an inequality for each statement. 1. A number n decreased by 15 is no more than 48. 2. The product of 7 and a number x is above 49. Graph each inequality. 3. r 7 2 4. 4 a 6. h 2 5. c 3 Write a compound inequality for each statement. 7. A number m is both greater than 15 and less than or equal to 4. 8. A number d is either less than 12 or greater than 2 13. 3-6 Solving Inequalities by Adding or Subtracting Solve and graph. 3 4 9. n 15 5 12. 19 t 13 3-7 10. 15 8 y 11. 101 x 89 13. 27 d 22 14. 5.3 n 2.7 Solving Inequalities by Multiplying or Dividing Solve and graph. k y 15. 5x 15 16. 9 3 4 17. 4 18. 24 6m 19. n 10 h 20. 2 42 21. Rachael is serving lemonade from a pitcher that holds 60 ounces. What are the possible numbers of 7-ounce juice glasses she can fill from one pitcher? 3-8 Solving Two-Step Inequalities Solve and graph. 22. 2k 4 10 23. 0.5z 5.5 4.5 9 3x 3 5 24. 5 1 5 t 1 25. 3 9 2 1 3x 5 26. 3 4 6 m 3 2 27. 7 1 7 4 28. Jillian must average at least 90 on two quiz scores before she can move to the next skill level. Jillian got a 92 on her first quiz. What scores could Jillian get on her second quiz in order to move to the next skill level? 152 Chapter 3 Multi-Step Equations and Inequalities Skate Away Ms. Lucinda wants to treat her class of 30 students to a skating party to celebrate the end of the school year. Item Cost Rink rental $50 plus $25 per hour Skate rental (per person) $1.50 plus $0.50 per hour Refreshments (per person) $3.50 1. Ms. Lucinda considers renting the rink at Skate Away. How much would it cost to rent the rink for x hours? 2. Another rink, Skate Palace, charges $100 plus $15 per hour to rent the rink. Write and solve an equation to find the number of hours for which the cost of renting the rink at Skate Palace is the same as the cost of renting the rink at Skate Away. 3. Ms. Lucinda decides to take the class to Skate Away. How much will it cost to rent skates for 30 students for x hours? How much will it cost to buy refreshments for 30 students? 4. Ms. Lucinda has budgeted $400 for the party. Write and solve an inequality to find the maximum number of hours the class can have its party at Skate Away. Be sure to include the cost of the rink, the skates, and the refreshments. 5. The final bill for the party was $380. How long did the party last? Concept Connection 153 Trans-Plants Solve each equation below. Then use the values of the variables to decode the answer to the question. 3a 17 25 24 6n 54 2b 25 5b 7 32 8.4o 6.8 14.2 6.3o 2.7c 4.5 3.6c 9 5 1 1 1 d d d d 6 12 12 6 3 4e 6e 5 15 4p p 8 2p 5 4 3r r 8 420 29f 73 2 5 1 3 s s 3 6 2 2 2(g 6) 20 4 15 4t 17 2h 7 3h 52 45 36u 66 23u 31 96i 245 53 6v 8 4 6v 3j 7 46 1 3 1 k k 2 4 2 30l 240 50l 160 3 67 4m 8 8 4w 3w 6w w 15 2w 3w 16 3q 3q 40 1 x 2x 3x 4x 5 75 2 2y 4y 8 5 11 25 4.5z What happens to plants that live in a math classroom? 7, 9, 10, 11 16, 18, 10, 15 12, 4, 4, 14, 18, 10 18, 10, 10, 7, 12 24 24 Points Points This traditional Chinese game is played using a deck of 52 cards numbered 1–13, with four of each number. The cards are shuffled, and four cards are placed face up in the center. The winner is the first player who comes up with an expression that equals 24, using each of the numbers on the four cards once. Complete rules and a set of game cards are available online. 154 Chapter 3 Multi-Step Equations and Inequalities KEYWORD: MT8CA Games Materials • • • • magazine glue scissors index cards PROJECT Picture Envelopes A Make these picture-perfect envelopes in which to store your notes on the lessons of this chapter. Directions 1 Flip through a magazine and carefully tear out eight pages with full-page pictures that you like. B 2 Lay one of the pages in front of you with the picture face down. Fold the page into thirds as shown, and then unfold the page. Figure A 3 Fold the sides in, about 1 inch, and then unfold. Cut away the four rectangles at the corners of the page. Figure B C 4 Fold in the two middle flaps. Then fold up the bottom and glue it onto the flaps. Figure C 5 Cut the corners of the top section at an angle to make a flap. Figure D D 6 Repeat the steps to make seven more envelopes. Label them so that there is one for each lesson of the chapter. Taking Note of the Math Use index cards to take notes on the lessons of the chapter. Store the cards in the appropriate envelopes. 155 Vocabulary algebraic inequality . . . . . . . . . . . . . . . . . . 136 equivalent expressions . . . . . . . . . . . . . . . 120 Associative Property . . . . . . . . . . . . . . . . . 116 inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 120 like terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Commutative Property . . . . . . . . . . . . . . . 116 solution set . . . . . . . . . . . . . . . . . . . . . . . . . 136 constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Distributive Property . . . . . . . . . . . . . . . . 117 Complete the sentences below with vocabulary words from the list above. 1. A(n) ___?____ is a statement that two quantities are not equal. 2. ___?____ states that two or more numbers can be added in any order or multiplied in any order. 3. ___?____ in an expression are set apart by plus or minus signs. 3-1 Properties of Rational Numbers EXAMPLE ■ AF1.3 (pp. 116–119) EXERCISES Name the property that is illustrated in the equation. 10(x y) 10 x 10 y Distributive Property Name the property that is illustrated in each equation. 1 1 4. 22 5 5 22 5. x (8 y) (x 8) y 6. 5(3 n) 5 3 5n 7. 8 6 7 8 6 7 1 3-2 Simplifying Algebraic Expressions EXAMPLE ■ Simplify. 3(z 6) 2z 3z 3(6) 2z 3z 18 2z 5z 18 (pp. 120–123) EXERCISES Simplify. Distributive Property 3z and 2z are like terms. Combine coefficients. 8. 5(3m 2) 4m 9. 12w 2(w 3) 10. 4x 3y 2x 11. 2t 2 4t 3t 3 156 1 Chapter 3 Multi-Step Equations and Inequalities AF1.3 ■ 3-3 Solving Multi-Step Equations (pp. 124–127) EXAMPLE EXERCISES Ext. of Solve. Solve. x 5x 1 3 6 9 3 2 x 5x 1 3 Multiply both sides 18 9 6 3 18 2 by 18. x 5x 1 3 18 9 18 6 18 3 18 2 Distributive Property 10x 3x 6 27 Simplify. 7x 6 27 Combine like terms. 6 6 Subtract 6 from 7x 21 both sides. 12. 3y 6 4y 7 8 7x 21 7 7 13. 5h 6 h 10 12 2t 1 1 14. 3 3 3 2r 4 2 15. 5 5 5 z 3z 1 1 16. 3 4 2 3 a 3a 7 17. 8 1 2 7 2 Divide both sides by 7. x3 3-4 Solving Equations with Variables on Both Sides Solve. Solve. 3x 5 5x 12 x 2 2x 5 10 x 2x 2x 5 10 3x 10 10 15 3x 18. 12s 8 2(5s 3) 15 3x 3 3 Combine like terms. 19. Add 2x to 20. both sides. Add 10 to both sides. Divide both sides by 3. 5x 3-5 Inequalities ■ AF4.1 c 5c 5c 13 3 8 6 4 5x 3 x 21. 4 2y 4y 22. 2n 8 2n 5 2z 3 3z 17 23. 3 2 2 3 AF1.1 (pp. 136–139) EXERCISES EXAMPLE ■ (pp. 129–133) Ext. of EXERCISES EXAMPLE ■ AF4.1 Write an inequality for the situation. Write an inquality for each situation. The capacity of the elevator was at most 2000 pounds. Let c capacity of elevator c 2000 lb “at most” means less 24. It is no more than a one mile walk from home to the school. Graph x 3. –6 –5 –4 than or equal to 25. The cost of the trip will be at least $1500. 26. Fewer than 45 students are expected to attend the workshop. Graph each inequality. –3 –2 –1 0 27. m 0 28. x 2 1 29. c 4 Study Guide: Review 157 3-6 Solving Inequalities by Adding or Subtracting EXAMPLE Solve and graph. n 5 2 5 5 n 3 30. 13 r 17 0 1 Add 5 to both sides. 2 1 3 2 2 0 5 6 2 Subtract 4 3 from both 2 4 3 x 3 13 4 2 4 3 1 2 31. n 3 6 3 3 32. x 3.8 4 5 3 x 4 3 ■ AF4.0 EXERCISES Solve and graph. ■ (pp. 140–143) 1 3 sides. 2 3 4 3 1 33. 3 2 y 34. Ellory budgets at most $20 each week for lunch. She has spent $17.75 so far this week. Write and solve an inequality to determine how much more Ellory can spend and stay within her lunch budget. 5 3 3-7 Solving Inequalities by Multiplying or Dividing EXAMPLE ■ EXERCISES Solve and graph. Solve and graph. z 10 13 z (13)10 Multiply both sides (13) 13 by 13. Change z 130 to . m 35. 6 3 (pp. 144–147) AF4.0 36. 4n 12 t 37. 8 2 38. 5p 15 b 39. 9 3 40. 6a 48 60 70 80 90 100 110 120 130 140 150 160 3-8 Solving Two-Step Inequalities EXAMPLE ■ EXERCISES Solve and graph 3x 3 9. 3x 3 9 3 3 3x 12 3x 12 3 3 x 4 6 158 4 (pp. 148–151) 2 Add 3 to both sides. Solve and graph. 41. 5z 12 7 42. 2h 7 5 Divide both sides by 3. Change to . a 43. 10 3 2 x 44. 3 8 10 45. 5 3k 4 3 0 2 46. 2y 4 1 Chapter 3 Multi-Step Equations and Inequalities AF4.1 Simplify each expression. Justify each step. 2 1. 12 7 3 2. 39 52 11 3. (25 9) 4 4. 2.1 (6.5 4.9) Simplify. 5. 7x 5x 6. m 3m 3 8. 2y 2z 2 9. 3(s 2) s 7. 6n 1 n 5n 10. 10b 8(b 1) Solve. 11. 10x 2x 16 3y 5y 12. 3 8 13. 4c 6 2c 24 2x 3 11 14. 5 5 5 2 1 15. 5b 4b 3 16. 15 6g 8 19 17. On her last three quizzes, Elise scored 84, 96, and 88. What grade must she get on her next quiz to have an average of 90 for all four quizzes? Solve. 18. 3x 13 x 1 19. q 7 2q 5 20. 8n 24 3n 59 21. m 5 m 3 22. 3a 9 3a 9 3z 17 2z 3 23. 2 3 3 2 24. The square and the equilateral triangle have the same perimeter. Find the perimeter of each figure. x2 x Solve and graph. h 25. 12 4 26. 36 6y 27. 56 7m 29. n 14 3 30. 8 22 p 31. 4 u 20 b 8 28. 4 32. 8 z 6 33. Glenda has a $40 gift certificate to a café that sells her favorite tuna sandwich for $3.75 after tax. What are the possible numbers of tuna sandwiches that Glenda can buy with her gift certificate? Solve and graph. 34. 6m 4 2 35. 8 3p 14 36. 4z 4 8 x 1 2 37. 10 2 5 c 3 1 38. 4 8 2 2 1 d 39. 3 2 6 Chapter 3 Test 159 Gridded Response: Write Gridded Responses When responding to a test item that requires you to place your answer in a grid, examine the grid to be sure you know how to fill it in correctly. Grid formats may vary from test to test. The grid in this book is used often, but it is not used on every test that has gridded-response items. Gridded Response: Divide. 3000 7.5 4 0 0 3000 10 3000 7.5 7.5 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 30,000 75 Divide. 400 Simplify. • Write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. Do not leave a blank box in the middle of an answer. • Shade the bubble for each digit in the column beneath it. 1 1 1 2 1 4 3 x 2 2 3 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2 Gridded Response: Solve x 2 3. 7 / 6 0 1 2 3 4 5 6 7 8 9 10 . 7.5 has 1 decimal place, so multiply by 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 x 6 6 7 1 x 6, 16 , or 1.16 1 Add 2 to both sides of the equation. Find a common denominator. Add. • Mixed numbers and repeating decimals cannot be gridded, so you must grid the answer as 76. • Write your answer in the answer boxes at the top of the grid. • Put only one digit or symbol in each box. On some grids, the fraction bar and the decimal point have a designated box. • Shade the bubble for each digit or symbol in the correct column. 160 Chapter 3 Multi-Step Equations and Inequalities You cannot grid a negative number in a gridded-response item because the grid does not include the negative sign. If you get a negative answer to a test item, recalculate the problem because you probably made a math error. Item C A student found 0.65 as the answer to 5 (0.13). Then the student filled in the grid as shown. Read each statement and then answer the questions that follow. Item A A student correctly evaluated an expression and got 193 as a result. Then the student filled in the grid as shown. 9 / 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2. Explain how to fill in the answer correctly. A student added 0.21 and 0.49 and got an answer of 0.7. This answer is displayed in the grid. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 3 1. What error did the student make when filling in the grid? Item B – 0 . 6 5 5. What error does the grid show? 6. Another student got an answer of 0.65. Explain why the student knew this answer was wrong. Item D A student found that x 512 was the solution to the equation 2x 3 8. Then the student filled in the grid as shown. 5 1 / 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 . 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7. What answer does the grid show? 8. Explain why you cannot fill in a mixed number. 9. Write the answer 512 in two forms that could be entered in the grid correctly. 3. What errors did the student make when filling in the grid? 4. Explain how to fill in the answer correctly. Strategies for Success 161 KEYWORD: MT8CA Practice Cumulative Assessment, Chapters 1–3 Multiple Choice 1. A cell phone company charges $0.21 per minute for phone calls. Which expression represents the cost of a phone call of m minutes? A 0.21m C 0.21 m B 0.21 m D 0.21 m 2. Laurie had $88 in her bank account on Sunday. The table below shows her account activity for the past 5 days. What is the balance in her account on Friday? Day 5. In order to apply for a driver’s permit in Ohio, you have to be at least 16 years old. Which graph correctly represents the possible ages of Ohioans who can apply for a driver’s permit? A 12 13 14 15 16 17 18 19 B 12 13 14 15 16 17 18 19 C 12 13 14 15 16 17 18 19 D 12 13 14 15 16 17 18 19 Deposit Withdraw Monday $25 — Tuesday — $58 A 2(x 3) 2 x 2 3 Wednesday — $45 B 2x 3 3 2x Thursday $32 — C Friday $91 — xyyx D 2(xy) (2x)y A $91 C $133 B $103 D $236 6. Which expression represents the Distributive Property? 7. Which addition equation represents the number line diagram below? 3. Which equation has a solution of x 5? A 2x 8 2 C 1 x 6 10 5 5 4 3 2 1 B 1 x 10 5 5 D 2x 10 5 A B 4. You volunteer to bring 4 gallons of juice for a class party. There are 28 students in the class. You plan to give each student an equal amount of juice. Which equation can you use to determine the amount of juice per student? 162 0 1 2 3 4 5 6 4 (2) 2 C 4 6 10 4 (6) 2 D 4 (6) 10 8. A snack package has 4 ounces of mixed nuts, 112 ounces of wheat crackers, 534 ounces of pretzels, and 218 ounces of popcorn. What is the total weight of the snacks? 3 A 4x 28 C 28 x 4 A 138 ounces B x 4 28 D 28x 4 B 138 ounces Chapter 3 Multi-Step Equations and Inequalities 1 5 C 128 ounces D 98 ounces 3 9. Which inequality is the solution of 2 x 1? 3 2 1 A x 6 B x 6 1 1 C x 16 D x 16 1 10. Which value of x is the solution of the equation 38 x 34 16? 9 A x 22 B x 9 5 5 C x 19 D x 29 4 11. Frank purchased x tickets for a concert. Mark has 1 more ticket than Frank. Karen has twice as many tickets as Mark. Which expression represents how many tickets they have in all? A 4x 2 C 3x 2 B 3x 3 D 4x 3 When finding the solution of an equation on a multiple-choice test, work backward by substituting into the equation the answer choices provided. Gridded Response 12. In 2004, the minimum wage for workers was $5.85 per hour. To find the amount of money someone can make in x hours, use the equation y 5.85x. How much money does a person who works 5 hours earn? 13. Solve the equation 94x 31 for x. 14. The sum of two consecutive integers (x, x 1) is 53. What is the smaller of the two numbers? 15. In a local high school, 15 of the school’s 600 students earned National Merit Scholarships. Write as a decimal the number of scholarships earned per total number of students. Short Response 16. Alfred and Eugene each spent $62 on campsite and gasoline expenses during their camping trip. Each campsite they used had the same per-night charge. Alfred paid for 4 nights of campsites and $30 of gasoline. Eugene paid for 2 nights of campsites and $46 of gasoline. Write an equation that could be used to determine the cost of one night’s stay at a campsite. What was the cost of one night’s stay at a campsite? 17. Omar opened a savings account with a $125 deposit in June. Over the next year, he withdrew $40.50 in September, deposited $35.75 in November, deposited $55 in February, and withdrew $45.25 in May. a. List the withdrawals and deposits in order from least to greatest. b. Over the 1 year period did the value of the account rise or fall? What value represents the total amount of change in the value of the account? Determine the final value of the account. Extended Response 18. Statement 1: Currently there are 8 more students in student council than there are officers. There are 12 students total in student council. Statement 2: In addition, there have to be at least 4 officers in the council. a. Write an equation to represent Statement 1 and an inequality to represent Statement 2. b. Solve the equation, and plot the solution to the equation on a number line. c. Graph the solution set to the inequality. d. Explain what the solution sets have in common, and then explain how they are different. Cumulative Assessment, Chapters 1–3 163