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Principles of Algebra 1A Expressions and Integers 1-1 Evaluating Algebraic Expressions Writing Algebraic Expressions Integers and Absolute Value Adding Integers Subtracting Integers Multiplying and Dividing Integers 1-2 1-3 1-4 1-5 1-6 1B Solving Equations 1-7 Solving Equations by Adding or Subtracting Solving Equations by Multiplying or Dividing Model Two-Step Equations Solving Two-Step Equations 1-8 LAB 1-9 KEYWORD: MT8CA Ch1 Firefighters can use algebra to find out how fast a fire is spreading and to create a plan to stop it. 2 Chapter 1 Vocabulary addition Choose the best term from the list to complete each sentence. 1. __?__is the __?__ of addition. 2. The expressions 3 4 and 4 3 are equal by the __?__. Associative Property Commutative Property 3. The expressions 1 (2 3) and (1 2) 3 are equal by the __?__. division 4. Multiplication and __?__ are opposite operations. opposite operation 5. __?__ and __?__ are commutative. subtraction multiplication Complete these exercises to review skills you will need for this chapter. Whole Number Operations Simplify each expression. 6. 8 116 43 7. 2431 187 8. 204 38 9. 6447 21 Compare and Order Whole Numbers Order each sequence of numbers from least to greatest. 10. 1050; 11,500; 105; 150 11. 503; 53; 5300; 5030 12. 44,400; 40,040; 40,400; 44,040 Inverse Operations Rewrite each expression using the inverse operation. 13. 72 18 90 14. 12 9 108 15. 100 34 66 16. 56 8 7 Order of Operations Simplify each expression. 17. 2 3 4 18. 50 2 5 19. 6 3 3 3 20. (5 2)(5 2) 21. 5 6 2 22. 16 4 2 3 23. (8 3)(8 3) 24. 12 3 2 5 Evaluate Expressions Determine whether the given expressions are equal. 25. (4 7) 2 and 4 (7 2) 26. (2 4) 2 and 2 (4 2) 27. 2 (3 3) and (2 3) 3 28. 5 (50 44) and 5 50 44 29. 9 (4 2) and (9 4) 2 30. 2 3 2 4 and 2 (3 4) 31. (16 4) 4 32. 5 (2 3) and 16 (4 4) and (5 2) 3 Principles of Algebra 3 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 NS2.5 Understand the meaning of absolute value of a number; interpret the the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. Academic Vocabulary Chapter Concept absolute value the distance a number is from zero on a number line You find the absolute value of a number by determining how far it is from zero on a number line. Example: Since 3 is 3 units from 0 on a number line, the absolute value of 3 is 3. interpret to understand and explain the meaning of (Lessons 1-3 and 1-5) 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). variable a symbol, usually a letter, used to show an amount that can change Example: x verbal using words You rewrite a verbal statement using mathematical symbols. Example: three less than a number n3 (Lesson 1-2) AF1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x 5)2. (Lesson 1-1) AF1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. evaluate find the value Example: x 3 for x 2 x3 23 5 terminology words used to represent certain items or ideas You find the value of expressions by performing basic arithmetic operations in a specific order. You learn the names of different mathematical concepts. (Lessons 1-1 and 1-7) 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. (Lesson 1-9) context the facts or circumstances that surround a situation You perform more than one operation with integers to find reasonableness makes sense given the the value of a variable that makes an equation true. You facts or circumstances also decide if answers make sense for the problems. Standards NS1.1, NS1.2, and AF1.3 are also covered in this chapter. To see these standards unpacked, go to Chapter 2, p. 62 (NS1.2), Chapter 3, p. 114 (AF1.3), and Chapter 4, p. 166 (NS1.1). 4 Chapter 1 Principles of Algebra Reading Strategy: Use Your Book for Success Understanding how your textbook is organized will help you locate and use helpful information. As you read through an example problem, pay attention to the margin notes , such as Helpful Hints, Reading Math notes, and Caution notes. These notes will help you understand concepts and avoid common mistakes. Read (ⴚ4)3 as “ⴚ4 to the 3rd power” or “ⴚ4 cubed.” The glossary is found in the back of your textbook. Use it to find definitions and examples of unfamiliar words or properties. The index is located at the end of your textbook. Use it to find the page where a particular concept is taught. The Skills Bank is found in the back of your textbook. These pages review concepts from previous math courses. Try This Use your textbook for the following problems. 1. Use the glossary to find the definition of absolute value. 2. Where can you review the order of operations? 3. On what page can you find answers to exercises in Chapter 2? 4. Use the index to find the page numbers where algebraic expressions, monomials, and volume of prisms are explained. Principles of Algebra 5 Evaluating Algebraic Expressions 1-1 California Standards AF1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x 5)2. AF1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. Vocabulary expression variable numerical expression algebraic expression evaluate Why learn this? You can evaluate an expression to convert a temperature from degrees Celsius to degrees Fahrenheit. (See Example 3.) An expression is a mathematical phrase that contains operations, numbers, and/or variables. A variable is a letter that represents a value that can change or vary. There are two types of expressions: numerical and algebraic. A numerical expression does not contain variables. An algebraic expression contains one or more variables. Numerical Expressions Algebraic Expressions 32 4(5) x2 4n 27 18 3 4 pr x 4 To evaluate an algebraic expression, substitute a given number for the variable. Then use the order of operations to find the value of the resulting numerical expression. EXAMPLE 1 Evaluating Algebraic Expressions with One Variable Evaluate each expression for the given value of the variable. x 5 for x 11 11 5 Substitute 11 for x. 16 Add. Order of Operations PEMDAS: 1. Parentheses 2. Exponents 3. Multiply and Divide from left to right. 4. Add and Subtract from left to right. See Skills Bank p. SB14. 6 2a 3 for a 4 2(4) 3 Substitute 4 for a. 83 Multiply. 11 Add. 4(3 n) 2 for n 0, 1, 2 n Substitute Parentheses Multiply Subtract 0 4(3 0) 2 4(3) 2 12 2 10 1 4(3 1) 2 4(4) 2 16 2 14 2 4(3 2) 2 4(5) 2 20 2 18 Chapter 1 Principles of Algebra EXAMPLE 2 Evaluating Algebraic Expressions with Two Variables Evaluate each expression for the given values of the variables. 5x 2y for x 13 and y 11 5(13) 2(11) Substitute 13 for x and 11 for y. 65 22 Multiply. 87 Add. p(24 q) for p 8 and q 17 8(24 17) Substitute 8 for p and 17 for q. 8(7) Simplify within parentheses. 56 Multiply. EXAMPLE 3 Science Application If c is a temperature in degrees Celsius, then 1.8c 32 can be used to find the temperature in degrees Fahrenheit. Convert each temperature from degrees Celsius to degrees Fahrenheit. freezing point of water: 0°C 1.8c 32 1.8(0) 32 Substitute 0 for c. 0 32 Multiply. 32 Add. 0°C 32°F Water freezes at 32°F. highest recorded temperature in the United States: 57°C 1.8c 32 1.8(57) 32 Substitute 57 for c. 102.6 32 Multiply. 134.6 Add. 57°C 134.6°F The highest recorded temperature in the United States is 134.6°F. Think and Discuss 1. Give an example of a numerical expression and of an algebraic expression. 2. Tell how to evaluate an algebraic expression for a given value of a variable. 3. Explain why you cannot find a numerical value for the expression 4x 5y for x 3. 1-1 Evaluating Algebraic Expressions 7 1-1 California Standards Practice AF1.2, AF1.4 Exercises KEYWORD: MT8CA 1-1 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Evaluate each expression for the given value of the variable. 1. x 4 for x 11 See Example 2 2. 2a 7 for a 7 Evaluate each expression for the given values of the variables. 4. 3x 2y for x 8 and y 10 See Example 3 3. 2(4 n) 5 for n 0 5. 5r (12 p) for p 15 and r 9 If c is the number of cups of water needed to make papier-mâché paste, then 41c can be used to find the number of cups of flour needed. Find the number of cups of flour needed for each number of cups of water. 6. 12 cups 7. 8 cups 8. 7 cups 9. 10 cups INDEPENDENT PRACTICE See Example 1 Evaluate each expression for the given value of the variable. 10. x 7 for x 23 See Example 2 11. 7t 2 for t 5 Evaluate each expression for the given values of the variables. 13. 4x 7y for x 9 and y 3 See Example 3 12. 4(3 k) 7 for k 0 14. 4m 2n for m 25 and n 2.5 If c is the number of cups, then 12c can be used to find the number of pints. Find the number of pints for each of the following. 15. 26 cups 16. 12 cups 17. 20 cups 18. 34 cups PRACTICE AND PROBLEM SOLVING Extra Practice See page EP2. Evaluate each expression for the given value of the variable. 19. 30 n for n 8 20. x 4.3 for x 6 21. 5t 5 for t 1 22. 11 6m for m 0 23. 3a 4 for a 8 24. 4g 5 for g 12 25. 4y 2 for y 3.5 26. 18 3y for y 6 27. 3(z 9) for z 6 Evaluate each expression for t 0, x 1.5, y 6, and z 23. 28. 3z 3y 29. yz 30. 1.4z y 31. 4.2y 3x 32. 4(y x) 33. 4(3 y) 34. 3(y 6) 8 35. 4(2 z) 5 36. 5(4 t) 6 37. y(3 t) 7 38. x y z 39. 10x z y 40. 4(z 5t) 3 41. 2y 6(x t) 42. 8txz 43. 2z 3xy 44. A rectangular shape has a length-to-width ratio of approximately 5 to 3. A designer can use the expression 13(5w) to find the length of such a rectangle with a given width w. Find the length of such a rectangle with width 6 inches. 8 Chapter 1 Principles of Algebra 45. Finance A bank charges interest on money it loans. Interest is sometimes a fixed amount of the loan. The expression a(1 i) gives the total amount due for a loan of a dollars with interest rate i, where i is written as a decimal. Find the amount due for a loan of $100 with an interest rate of 10%. (Hint: 10% 0.1) Califor nia Entertainment 46. Entertainment There are 24 frames, or still shots, in one second of movie footage. To determine the number of frames in a movie, you can use the expression (24)(60)m, where m is the running time in minutes. Using the running time of E.T. the Extra-Terrestrial shown at right, determine how many frames are in the movie. The photos above were taken in Palo Alto as part of Eadweard Muybridge’s 1878 study on horses in motion. Muybridge’s multi-camera techniques were a landmark in the history of cinematography. E.T. the Extra-Terrestrial (1982) has a running time of 115 minutes, or 6900 seconds. 47. Choose a Strategy A basketball league has 288 players and 24 teams, with an equal number of players per team. If the number of teams is reduced by 6 but the total number of players stays the same, there will be ___?____ players per team. A 6 more B 4 more C 4 fewer D 6 fewer 48. Write About It A student says that the algebraic expression 5 x 7 can also be written as 5 7x. Is the student correct? Explain. 49. Challenge Can the expressions 2x and x 2 ever have the same value? If so, what must the value of x be? AF1.2 50. Multiple Choice What is the value of the expression 3x 4 for x 2? A 4 B 6 C 9 D 10 51. Multiple Choice A bakery charges $7 for a dozen muffins and $2 for a loaf of bread. If a customer bought 2 dozen muffins and 4 loaves of bread, how much did she pay? A $22 B $38 C $80 D $98 52. Gridded Response What is the value of 5(x y) for x 19 and y 6? Identify the odd number(s) in each list of numbers. (Previous course) 53. 15, 18, 22, 34, 21, 61, 71, 100 54. 101, 114, 122, 411, 117, 121 55. 4, 6, 8, 16, 18, 20, 49, 81, 32 56. 9, 15, 31, 47, 65, 93, 1, 3, 43 Find each sum, difference, product, or quotient. (Previous course) 57. 200 2 58. 200 2 59. 200 2 60. 200 2 61. 200 0.2 62. 200 0.2 63. 200 0.2 64. 200 0.2 1-1 Evaluating Algebraic Expressions 9 Writing Algebraic Expressions 1-2 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). Who uses this? Advertisers can write an algebraic expression to represent the cost of airing a commercial a given number of times. (See Example 3.) To use algebra to solve real-world problems, you may need to translate word phrases into mathematical symbols. Sixty-eight different commercials aired during the 2005 Super Bowl. Word Phrases EXAMPLE 1 Expression • add 5 to a number • sum of a number and 5 • 5 more than a number n5 • subtract 11 from a number • difference of a number and 11 • 11 less than a number x 11 • 3 multiplied by a number • product of 3 and a number 3 m or 3m • 7 divided into a number • quotient of a number and 7 a or a 7 7 Translating Word Phrases into Math Expressions Write an algebraic expression for each word phrase. 4 times the difference of a number n and 2 4 times the difference of n and 2 Addition is commutative, so order does not matter in Example 1B. 1 12p and 12p 1 are both correct. Subtraction is not commutative, so “1 less than the product of 12 and p” is written as 12p 1, not 1 12p. 4 4(n 2) 1 more than the product of 12 and p 1 more than the product of 12 and p (12 p) 12p 1 10 (n 2) Chapter 1 Principles of Algebra 1 EXAMPLE 2 Translating Math Expressions into Word Phrases Write a word phrase for the algebraic expression 4 7b. 4 7 b 4 minus the product of 7 and b 4 minus the product of 7 and b To solve a word problem, first interpret the action you need to perform and then choose the correct operation for that action. EXAMPLE 3 A company aired its 30-second commercial during Super Bowl XXXIX at a cost of $2.4 million each time. Write an algebraic expression to determine what the cost would be if the commercial had aired n times. Then evaluate the expression for 2, 3, and 4 times. $2.4 million n Combine n equal amounts of $2.4 million. When a word problem involves groups of equal size, use multiplication or division. Otherwise, use addition or subtraction. EXAMPLE Writing and Evaluating Expressions in Word Problems 2.4n 4 Cost in millions of dollars n 2.4n Cost 2 2.4(2) $4.8 million 3 2.4(3) $7.2 million 4 2.4(4) $9.6 million Evaluate for n 2, 3, and 4. Writing a Word Problem from a Math Expression Write a word problem that can be evaluated by the algebraic expression 14,917 m. Then evaluate the expression for m 633. At the beginning of the month, Benny’s car had 14,917 miles on the odometer. If Benny drove m miles during the month, how many miles were on the odometer at the end of the month? 14,917 m 14,917 633 15,550 Substitute 633 for m. The car had 15,550 miles on the odometer at the end of the month. Think and Discuss 1. Give two words or phrases that can be used to express each operation: addition, subtraction, multiplication, and division. 2. Express 5 7n in words in at least two different ways. 1-2 Writing Algebraic Expressions 11 1-2 Exercises California Standards Practice AF1.1, AF1.2 KEYWORD: MT8CA 1-2 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Write an algebraic expression for each word phrase. 1. 5 less than the product of 3 and p 2. 77 more than the product of 2 and u 3. 16 more than the quotient of d and 7 4. 6 minus the quotient of u and 2 Write a word phrase for each algebraic expression. 5. 18 43s See Example 3 See Example 4 22 6. r 37 y 7. 10 3 1 8. 29b 93 9. Mark is going to work for his father’s pool cleaning business during the summer. Mark’s father will pay him $5 for each pool he helps clean. Write an algebraic expression to determine how much Mark will earn if he cleans n pools. Then evaluate the expression for 15, 25, 35, or 45 pools. 10. Write a word problem that can be evaluated by the algebraic expression x 450. Then evaluate the expression for x 1325. INDEPENDENT PRACTICE See Example 1 Write an algebraic expression for each word phrase. 11. 1 more than the quotient of 5 and n 12. 2 minus the product of 3 and p 13. 45 less than the product of 78 and j 14. 4 plus the quotient of r and 5 15. 14 more than the product of 59 and q See Example 2 Write a word phrase for each algebraic expression. 16. 142 19t 17. 16g 12 5 18. 14 d w 51 19. 182 See Example 3 20. A community center is trying to raise $1680 to purchase exercise equipment. The center is hoping to receive equal contributions from members of the community. Write an algebraic expression to determine how much will be needed from each person if n people contribute. Then evaluate the expression for 10, 12, 14, or 16 people. See Example 4 21. Write a word problem that can be evaluated by the algebraic expression 372 r. Then evaluate the expression for r 137. PRACTICE AND PROBLEM SOLVING Extra Practice See page EP2. 12 Write an algebraic expression for each word phrase. 22. 6 times the sum of 4 and y 23. the product of 6 and y increased by 9 1 24. 3 of the sum of 4 and p 25. 1 divided by the sum of 3 and g 26. half the sum of m and 5 27. 6 less than the product of 13 and y 28. 2 less than m divided by 8 29. twice the quotient of m and 35 3 30. 4 of the difference of p and 7 31. 8 times the sum of 3 and x Chapter 1 Principles of Algebra 2 Translate each algebraic expression into words. 32. 4b 3 16 7 34. 8x 33. 8(m 5) 35. 17 w 36. At age 2, a cat or a dog is considered 24 “human” years old. Each year after age 2 is equivalent to 4 “human” years. Let a represent the age of a cat or dog. Fill in the expression [24 (a 2)] so that it represents the age of a cat or dog in “human” years. Copy the chart and use your expression to complete it. Age 24 (a 2) Age (human years) 2 3 4 5 E RIT W T NO OK DO N BO I 6 37. Reasoning Write two different algebraic expressions for the word phrase “14 the sum of x and 7.” 38. What’s the Error? A student wrote an algebraic expression for “5 less n 5 than the quotient of n and 3” as 3 . What error did the student make? 39. Write About It Paul used addition to solve a word problem about the weekly cost of commuting by toll road for $1.50 each day. Fran solved the same problem by multiplying. They both got the correct answer. How is this possible? 40. Challenge Write an expression for the sum of 1 and twice a number n. If you let n be any number, will the result always be an odd number? Explain. NS1.2, AF1.1, AF1.2 41. Multiple Choice Which expression means “3 times the difference of y and 4”? A 3y4 B 3 (y 4) C 3 (y 4) D 3 (y 4) 42. Multiple Choice Which expression represents the product of a number n and 32? A n 32 B n 32 C n 32 D 32 n 43. Short Response A company prints n books at a cost of $9 per book. Write an expression to represent the total cost of printing n books. What is the total cost if 1050 books are printed? Simplify. (Previous course) 44. 32 8 4 45. 24 2 3 6 1 46. (20 8) 2 2 Evaluate each expression for the given value of the variable. (Lesson 1-1) 47. 2(4 x) 3 for x 1 48. 3(8 x) 2 for x 2 1-2 Writing Algebraic Expressions 13 1-3 California Standards NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. Also covered: NS1.1 Vocabulary integer opposite absolute value Integers and Absolute Value Why learn this? You can use integers to represent scores in disc golf. In disc golf, a player throws a disc toward a target, or “hole,” in as few throws as possible. The expected number of throws to complete an entire course is called “par.” A player’s score can be written as an integer to show how many throws he or she is above or below par. If you complete a course in 5 fewer throws than par your score is 5 under par, or 5. If you complete the same course in 4 more throws than par your score is 4 over par, or 4. Integers are the set of whole numbers and their opposites. Opposites are numbers that are the same distance from 0 on a number line, but on opposite sides of 0. Opposites 5 4 3 2 1 0 Negative integers 1 2 3 4 5 Positive integers 0 is neither negative nor positive. EXAMPLE 1 Sports Application After one round of disc golf, the scores are Fred 5, Trevor 3, Monique 4, and Julie 2. Numbers on a number line increase in value as you move from left to right. Use , , or to compare Trevor’s and Julie’s scores. Trevor’s score is 3, and Julie’s score is 2. 5 4 3 2 1 0 1 2 3 4 5 2 3 Place the scores on a number line. 2 is to the left of 3. Julie’s score is less then Trevor’s. List the golfers in order from the lowest score to the highest. The scores are 5, 3, 4, and 2. Place the scores on a 5 4 3 2 1 0 1 2 3 4 5 number line and read them from left to right. In order from the lowest score to the highest, the golfers are Fred, Julie, Trevor, and Monique. 14 Chapter 1 Principles of Algebra EXAMPLE 2 Ordering Integers Write the integers 7, 4, and 3 in order from least to greatest. Graph the integers on a number line. Then read them from left to right. ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 4 5 6 7 The integers in order from least to greatest are 4, 3, and 7. A number’s absolute value is its distance from 0 on a number line. Absolute value of a nonzero number is always positive because distance is always positive. “The absolute value of 4” is written as ⏐4⏐. Opposites have the same absolute value. 4 units 4 units ⏐4⏐ 4 EXAMPLE 3 ⏐4⏐ 4 5 4 3 2 1 0 1 2 3 4 5 Simplifying Absolute-Value Expressions Simplify each expression. ⏐6⏐ 6 units ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 4 6 is 6 units from 0, so⏐6⏐ 6. ⏐20 20⏐ ⏐20 20⏐⏐0⏐ 0 ⏐9⏐⏐7⏐ ⏐9⏐⏐7⏐ 9 7 16 Subtract first: 20 20 0. Then find the absolute value: 0 is 0 units from 0. Find the absolute values first: 9 is 9 units from 0. 7 is 7 units from 0. Then add. ⏐9 3⏐⏐2 3⏐ ⏐9 3⏐⏐2 3⏐⏐6⏐⏐5⏐ 9 3 6; 2 3 5 65 6 is 6 units from 0. 5 is 5 units from 0. 1 Think and Discuss 1. Explain the steps you would take to simplify ⏐5⏐⏐2⏐. 2. Explain whether an absolute value is ever negative. 1-3 Integers and Absolute Value 15 1-3 California Standards Practice NS1.1, NS2.5 Exercises KEYWORD: MT8CA 1-3 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 1. After the first round of the 2005 Masters golf tournament, scores relative to par were Tiger Woods 2, Vijay Singh 4, Phil Mickelson 2, and Justin Leonard 5. Use , , or to compare Vijay Singh’s and Phil Mickelson’s scores. Then list the golfers in order from the lowest score to the highest. Write the integers in order from least to greatest. 2. 5, 2, 3 See Example 3 3. 17, 6, 8 4. 9, 21, 14 5. 3, 7, 0 8. ⏐3⏐ ⏐11⏐ 9. ⏐10⏐⏐4 1⏐ Simplify each expression. 6. ⏐9⏐ 7. ⏐22 7⏐ 10. ⏐0⏐ 11. ⏐12⏐⏐9⏐ 12. ⏐28 18⏐ 13. ⏐8 7⏐⏐6 5⏐ INDEPENDENT PRACTICE See Example 1 14. During a very cold week, the temperature in Philadelphia was 7°F on Monday, 4°F on Tuesday, 2°F on Wednesday, and 3°F on Thursday. Use , , or to compare the temperatures on Wednesday and Thursday. Then list the days in order from the coldest to the warmest. See Example 2 Write the integers in order from least to greatest. 15. 6, 5, 2 See Example 3 16. 8, 11, 5 17. 25, 30, 27 18. 4, 2, 1 Simplify each expression. 19. ⏐6 3⏐ 20. ⏐3⏐ 21. ⏐6⏐⏐8 3⏐ 22. ⏐7 2⏐⏐9 8⏐ 23. ⏐5⏐ 24. ⏐7⏐⏐14⏐ 25. ⏐8 8⏐ 26. ⏐19⏐⏐13⏐ PRACTICE AND PROBLEM SOLVING Extra Practice See page EP2. Compare. Write , , or . 27. 9 15 31. ⏐7⏐ ⏐6⏐ 17 28. 13 32. ⏐3⏐ ⏐3⏐ 29. 23 23 33. ⏐13⏐ ⏐2⏐ 30. 14 34. ⏐20⏐ 0 ⏐21⏐ Write the integers in order from least to greatest. 35. 24, 16, 12 36. 46, 31, 52 37. 45, 35, 25 Simplify each expression. 38. ⏐17⏐⏐24⏐ 39. ⏐22⏐⏐28⏐ 40. ⏐53 37⏐ 41. ⏐21 20⏐ 42. ⏐7⏐⏐9⏐ 43. ⏐6⏐⏐12⏐ 44. ⏐72⏐⏐8⏐ 45. ⏐3⏐ ⏐3⏐ 46. Reasoning Two integers A and B are graphed on a number line. If A is less than B, is ⏐A⏐ always less than ⏐B⏐? Explain your answer. 16 Chapter 1 Principles of Algebra Simplify each expression. Science 47. ⏐51 23⏐⏐8 11⏐ 48. ⏐61⏐⏐28 9⏐ 49. ⏐37 14⏐⏐25 10⏐ 50. ⏐24⏐⏐13 5⏐ 51. ⏐35 11⏐⏐15 7⏐ 52. ⏐17 26⏐⏐29⏐ 53. Science The table shows the lowest recorded temperatures for each continent. Write the continents in order from the lowest recorded temperature to the highest recorded temperature. The Ice Hotel in Jukkasjärvi, Sweden, is rebuilt every winter from 30,000 tons of snow and 4000 tons of ice. 54. Chemistry The boiling point of nitrogen is 196°C. The boiling point of oxygen is 183°C. Which element has the greater boiling point? Explain your answer. 55. Reasoning Write rules for using absolute value to compare two integers. Be sure to take all of the possible combinations into account. Lowest Recorded Temperatures Continent Temperature Africa 11°F Antarctica 129°F Asia 90°F Australia 9°F Europe 67°F North America 81°F South America 27°F 56. Write About It Explain why there is no number that can replace n to make the equation ⏐n⏐ 1 true. 57. Challenge List the integers that can replace n to make the statement ⏐8⏐ n ⏐5⏐ true. NS1.1, NS2.5, AF1.1, AF1.2 58. Multiple Choice Which set of integers is in order from greatest to least? A 10, 8, 5 B 8, 5, 10 C 5, 8, 10 D 10, 5, 8 D 5 D ⏐21⏐ 59. Multiple Choice Which integer is between 4 and 2? A 0 B 3 C 4 60. Multiple Choice Which expression has the greatest value? A ⏐12⏐ B ⏐15⏐ C ⏐18⏐ Evaluate each expression for a 3, b 2.5, and c 24. (Lesson 1-1) 61. c 15 62. 9a 8 63. 8(a 2b) 64. bc a Write an algebraic expression for each word phrase. (Lesson 1-2) 65. 8 more than the product of 7 and a number t 66. A pizzeria delivered p pizzas on Thursday. On Friday, it delivered 3 more than twice the number of pizzas delivered on Thursday. Write an expression to show the number of pizzas delivered on Friday. 1-3 Integers and Absolute Value 17 1-4 California Standards NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Vocabulary Adding Integers Why learn this? You can track your daily calorie count by adding integers. (See Example 4.) Suppose you eat a piece of grapefruit that contains 35 calories. Then you walk 2 laps around a track and burn 35 calories. Using opposites, you can represent this situation as 35 (35) 0. When you add two opposites, or additive inverses , the sum is zero. You can model integer addition on a number line. additive inverse EXAMPLE 1 Using a Number Line to Add Integers Use a number line to find each sum. 3 (7) To add a positive number, move to the right. To add a negative number, move to the left. 7 3 5 4 3 2 1 0 1 2 3 4 5 Start at 0. Move right 3 units. From 3, move left 7 units. You finish at 4, so 3 (7) 4. 2 (5) 5 2 7 6 5 4 3 2 1 0 1 2 3 Start at 0. Move left 2 units. From 2, move left 5 units. You finish at 7, so 2 (5) 7. Another way to add integers is to use absolute value. ADDING TWO INTEGERS If the signs are the same. . . . . . find the sum of the absolute values. Use the same sign as the integers. 18 Chapter 1 Principles of Algebra If the signs are different. . . . . . find the difference of the absolute values. Use the sign of the integer with the greater absolute value. EXAMPLE 2 Using Absolute Value to Add Integers Add. EXAMPLE 3 4 (6) 4 (6) 10 Think: Find the sum of ⏐4⏐ and ⏐6⏐. 8 (9) 8 (9) 1 Think: Find the difference of ⏐8⏐ and ⏐9⏐. 5 11 5 11 6 Think: Find the difference of ⏐5⏐ and ⏐11⏐. Same sign; use the sign of the integers. 9 8; use the sign of 9. 11 5; use the sign of 11. Evaluating Expressions with Integers Evaluate b 11 for b 6. b 11 6 11 Replace b with 6. Think: Find the difference of ⏐11⏐ and ⏐6⏐. 6 11 5 11 6; use the sign of 11. When finding the sum of three or more integers, you may want to group opposites, or additive inverses. You may also want to group positive integers together and negative integers together. EXAMPLE ing Monday Morn Calories Oatmeal Toast w /jam 8 fl oz juice 145 62 111 Calories burned 110 Walked six laps 40 ps la Swam six 4 Health Application Melanie wants to check her calorie count after breakfast and exercise. Use information from the journal entry to find her total. Use positive for calories eaten and negative for calories burned. 145 62 110 (110) (40) (65) 145 62 (110 110) (40) (65) Group opposites. 145 62 0 (40) (65) (145 62) (40 65) Group integers with same signs. 207 (105) Add integers within each group. 102 207 105; use the sign of 207. Melanie’s calorie count after breakfast and exercise is 102 calories. Think and Discuss 1. Compare the sums of 10 (22) and 10 22. 2. Describe how to add the following expressions on a number line: 9 (13) and 13 9. Then compare the sums. 1-4 Adding Integers 19 1-4 California Standards Practice NS1.2, AF1.3 Exercises KEYWORD: MT8CA 1-4 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Use a number line to find each sum. 1. 5 1 See Example 2 3. 7 9 4. 4 (2) 6. 7 (3) 7. 11 17 8. 6 (8) Add. 5. 12 5 See Example 3 2. 6 (4) Evaluate each expression for the given value of the variable. 9. t 16 for t 5 See Example 4 10. m 7 for m 5 12. Lee opens a checking account. In the first month, he makes two deposits and writes three checks, as shown at right. Find what his balance is at the end of the month. (Hint: Checks count as negative amounts.) 11. p (5) for p 5 Checks Deposits $134 $600 $56 $225 $302 INDEPENDENT PRACTICE See Example 1 Use a number line to find each sum. 13. 5 (7) See Example 2 See Example 3 14. 7 7 15. 4 (9) 16. 4 7 17. 8 14 18. 6 (7) 19. 8 (8) 20. 19 (5) 21. 22 (15) 22. 17 9 23. 20 (12) 24. 18 7 Add. Evaluate each expression for the given value of the variable. 25. q 13 for q 10 See Example 4 26. x 21 for x 7 28. On Monday morning, a mechanic has no cars in her shop. The table at right shows the number of cars dropped off and picked up each day. Find the total number of cars left in her shop on Friday. 27. z (7) for z 16 Cars Dropped Off Cars Picked Up Monday 8 4 Tuesday 11 6 9 12 14 9 7 6 Wednesday Thursday Friday PRACTICE AND PROBLEM SOLVING Extra Practice See page EP2. Write an addition equation for each number line diagram. 29. 8 7 6 5 4 3 2 1 30. 0 1 2 5 4 3 2 1 20 Chapter 1 Principles of Algebra 0 1 2 3 4 5 Find each sum. Economics 31. 9 (3) 32. 16 (22) 33. 34 17 34. 44 39 35. 45 (67) 36. 14 85 37. 52 (9) 38. 31 (31) Evaluate each expression for the given value of the variable. The number one category of imported goods in the United States is industrial supplies, including petroleum and petroleum products. In 2004, this category accounted for over $412 million of imports. 39. c 17 for c 9 40. k (12) for k 4 41. b (6) for b 24 42. 13 r for r 19 43. 9 w for w 6 44. 3 n (8) for n 5 45. Economics Refer to the Exports Imports data at right about U.S. Goods $807,584,000,000 $1,473,768,000,000 international trade for Services $338,553,000,000 $290,095,000,000 the year 2004. Consider Source: U.S. Census Bureau values of exports as positive quantities and values of imports as negative quantities. a. What was the total of U.S. exports in 2004? b. What was the total of U.S. imports in 2004? c. The sum of exports and imports is called the balance of trade. Approximate the 2004 U.S. balance of trade to the nearest billion dollars. 46. What’s the Error? A student evaluated 4 d for d 6 and gave an answer of 2. What might the student have done wrong? Give the correct answer. 47. Write About It Explain the different ways it is possible to add two integers and get a negative answer. 48. Challenge What is the sum of 3 (3) 3 (3) . . . when there are 10 terms? 19 terms? 24 terms? 25 terms? Explain any patterns that you find. NS1.2, NS2.5, AF1.2 49. Multiple Choice Which of the following is the value of 7 3h when h 5? A 22 B 8 C 8 22 D 50. Gridded Response Evaluate the expression 35 y for y 8. Evaluate each expression for the given values of the variables. (Lesson 1-1) 51. 2x 3y for x 8 and y 4 52. 6s t for s 7 and t 12 Simplify each expression. (Lesson 1-3) 53. ⏐3⏐ ⏐9⏐ 54. ⏐22 9⏐ 55. ⏐18⏐ ⏐5⏐ 56. ⏐27⏐ ⏐5⏐ 1-4 Adding Integers 21 1-5 California Standards NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. Subtracting Integers Why learn this? You can represent the differences between the drops and climbs of a roller coaster by subtracting integers. (See Example 3.) In Lesson 1-4 you modeled integer addition on a number line. You can also model integer subtraction on a number line. The model below shows how to find 8 10. 10 8 3 2 1 0 1 2 3 4 5 6 7 8 9 Start at 0. Move right 8 units. From 8, move left 10 units. When you subtract a positive integer, the difference is less than the original number. Therefore, you move to the left on the number line. To subtract a negative integer, you move to the right. You can also subtract an integer by adding its opposite. You can then use the rules for addition of integers. SUBTRACTING TWO INTEGERS Words To subtract an integer, add its opposite. EXAMPLE 1 Numbers Algebra 3 7 3 (7) a b a (b) 5 (8) 5 8 a (b) a b Subtracting Integers Subtract. 7 7 7 7 7 (7) 14 Add the opposite of 7. 2 (4) 2 (4) 2 4 6 Add the opposite of 4. Same sign; add; use the sign of the integers. Same sign; add; use the sign of the integers. 13 (5) 13 (5) 13 5 Add the opposite of 5. 8 Unlike signs; subtract; 13 5; use the sign of 13. 22 Chapter 1 Principles of Algebra EXAMPLE 2 Evaluating Expressions with Integers Evaluate each expression for the given value of the variable. 4 s for s 9 4 s 4 (9) Substitute 9 for s. 4 9 Add the opposite of 9. 5 9 4; use the sign of 9. 3 x for x 5 3 x 3 5 Substitute 5 for x. 3 (5) Add the opposite of 5. 8 Same sign; use the sign of the integers. ⏐6 t⏐ 5 for t 4 ⏐6 t⏐ 5 ⏐6 (4)⏐ 5 Substitute 4 for t. ⏐6 4⏐ 5 Add the opposite of 4. ⏐10⏐ 5 6 4 10 10 5 The absolute value of 10 is 10. 15 Add. EXAMPLE 3 Architecture Application The roller coaster Desperado has a maximum height of 209 feet and a maximum drop of 225 feet. How far underground does the roller coaster go? 209 (225) Subtract the drop Desperado Roller Coaster 209 ft 225 ft from the height. 209 (225) Add the opposite of 225. 16 225 209; use the sign of 225. Ground level, 0 ft ? ft Desperado goes 16 feet underground. Think and Discuss 1. Explain why 10 (10) does not equal 10 10. 2. Describe the answer that you get when you subtract a greater number from a lesser number. Then give an example. 1-5 Subtracting Integers 23 1-5 California Standards Practice NS1.2, NS2.5 Exercises KEYWORD: MT8CA 1-5 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Subtract. 1. 5 9 See Example 2 2. 8 (6) 4. 11 (6) Evaluate each expression for the given value of the variable. 5. 9 h for h 8 See Example 3 3. 8 (4) 6. 7 m for m 5 7. ⏐3 k⏐ ⏐3⏐ for k 12 8. The temperature rose from 4°F to 45°F in Spearfish, South Dakota, on January 22, 1943, in only 2 minutes! By how many degrees did the temperature change? INDEPENDENT PRACTICE See Example 1 Subtract. 9. 3 7 See Example 2 10. 14 (9) 12. 8 (2) Evaluate each expression for the given value of the variable. 13. 14 b for b 3 See Example 3 11. 11 (6) 14. 7 q for q 15 15. ⏐5 f⏐ 7 for f 12 16. A submarine cruising at 27 m below sea level, or 27 m, descends 14 m. What is its new depth? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP3. Write a subtraction equation for each number line diagram. 17. 18. 8 6 4 2 0 2 4 6 5 4 3 2 1 0 1 2 3 4 5 Perform the given operations. 19. 8 (11) 20. 24 (27) 21. 43 13 22. 26 26 23. 13 7 (6) 24. 11 (4) (9) Evaluate each expression for the given value of the variable. 25. x 16 for x 4 26. 8 t for t 5 27. 16 y for y 8 28. s (22) for s 18 29. ⏐r 11⏐ ⏐9⏐for r 7 30. 4 ⏐2 p⏐for p 15 31. Estimation A roller coaster starts with a 160-foot climb and then plunges 228 feet down a canyon wall. It then climbs a gradual 72 feet before a steep climb of 189 feet. Approximately how far is the coaster above or below its starting point? 24 Chapter 1 Principles of Algebra 6 Social Studies Use the timeline to answer the questions. Use negative numbers for years B.C.E. Assume that there was a year 0 (there wasn’t) and that there have been no major changes to the calendar (there have been). 32. How long was the Greco-Roman era, when Greece and Rome ruled Egypt? 33. Which was a longer period of time: from the Great Pyramid to Cleopatra, or from Cleopatra to the present? By how many years? 34. Queen Neferteri ruled Egypt about 2900 years before the Turks ruled. In what year did she rule? 35. There are 1846 years between which two events on this timeline? 36. Write About It What is it about years B.C.E. that make negative numbers a good choice for representing them? 37. Challenge How would your calculations differ if you took into account the fact that there was no year 0? KEYWORD: MT8CA Egypt NS1.2 , NS2.5, AF1.1 38. Multiple Choice Which of the following is equivalent to ⏐7 (3)⏐? A ⏐7⏐ ⏐3⏐ B ⏐7⏐ ⏐3⏐ C 10 D 4 39. Gridded Response Subtract: 4 (12). Write an algebraic expression to evaluate the word problem. (Lesson 1-2) 40. Tate bought a compact disc for $17.99. The sales tax on the disc was t dollars. What was the total cost including sales tax? Evaluate each expression for m 3. (Lesson 1-4) 41. m 6 42. m (5) 43. 9 m 44. m 3 1-5 Subtracting Integers 25 1-6 California Standards NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Multiplying and Dividing Integers Who uses this? Football players can multiply integers to represent the total change in yards during several plays. (See Example 3.) If a football team loses 10 yards in each of 3 plays, the total change in yards can be represented by 3(10). You can write this product as repeated addition. 3(10) 30 3(10) 10 (10) (10) 30 2(10) 20 Notice that a positive integer multiplied by a negative integer gives a negative product. 1(10) 10 You already know that the product of two positive integers is a positive integer. The pattern shown at right can help you understand that the product of two negative integers is also a positive integer. 0(10) 0 1(10) 10 2(10) 20 3(10) 30 10 10 10 10 10 10 MULTIPLYING AND DIVIDING TWO INTEGERS If the signs are the same, the sign of the answer is positive. If the signs are different, the sign of the answer is negative. EXAMPLE 1 Multiplying and Dividing Integers Multiply or divide. The sum of two negative integers is always negative, but the product or quotient of two negative integers is always positive. 26 5(8) Signs are different. 45 9 Signs are the same. 40 Answer is negative. 5 Answer is positive. 4(2)(3) 4(2)(3) 8(3) 8(3) 24 Multiply two integers. 32 8 Signs are different. 4 Answer is negative. Chapter 1 Principles of Algebra Signs are different. Answer is negative. Signs are the same. Answer is positive. EXAMPLE 2 Using the Order of Operations with Integers Simplify. 3(2 8) 3(2 8) 3(2 8) 3(6) 18 Subtract inside the parentheses. Add the opposite of 8 inside the parentheses. Think: The signs are the same. The answer is positive. 4(7) 2 4(7) 2 28 2 30 Multiply first. Think: The signs are different; answer is negative. Same sign; use the sign of the integers. 2(14 6) 2(14 6) 2(20) 40 EXAMPLE 3 Add inside the parentheses. Think: The signs are different. The answer is negative. Sports Application A football team runs 10 plays. On 6 plays, it has a gain of 4 yards each. On 4 plays, it has a loss of 5 yards each. Each gain in yards can be represented by a positive integer, and each loss can be represented by a negative integer. Find the total net change in yards. 6(4) 4(5) 24 (20) 4 Add the losses to the gains. Multiply. Add. The team gained 4 yards. Think and Discuss 1. List all possible multiplication and division statements for the integers with absolute values of 5, 6, and 30. For example, 5 6 30. 2. Compare the sign of the product of two negative integers with the sign of the sum of two negative integers. 3. Suppose the product of two integers is positive. What do you know about the signs of the integers? 1-6 Multiplying and Dividing Integers 27 1-6 California Standards Practice NS1.2, AF1.2 Exercises KEYWORD: MT8CA 1-6 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Multiply or divide. 1. 8(4) See Example 2 3. 7(4)(3) 32 4. 8 6. 4(3) 9 7. 6(5 9) 8. 3 11(7 3) Simplify. 5. 7(5 12) See Example 3 54 2. 9 Business An investor buys shares of stock A and stock B. Stock A loses $8 per share, and stock B gains $5 per share. Given the number of shares, how much does the investor lose or gain? 9. stock A: 20 shares, stock B: 35 shares 10. stock A: 30 shares, stock B: 20 shares INDEPENDENT PRACTICE See Example 1 See Example 2 Multiply or divide. 11. 3(7) 72 12. 6 13. 12(7) 42 14. 6 3(4)(2) 15. 8 16. 13(9) 17. 5(7)(4) 63 18. 3(21) 20. 13(2) 8 21. 13(8 11) 22. 10 4(5 8) Simplify. 19. 12(9 14) See Example 3 Banking A student puts $50 in the bank each time he makes a deposit. He takes $20 each time he makes a withdrawal. Given the number of transactions, what is the net change in the student’s account? 23. deposits: 4, withdrawals: 5 24. deposits: 3, withdrawals: 8 PRACTICE AND PROBLEM SOLVING Extra Practice See page EP3. 25. Science Ocean tides are the result of the gravitational force between the sun, the moon, and the earth. When ocean tides occur, the earth’s crust also moves. This is called an earth tide. The formula for the height of an earth tide is y 3x, where x is the height of the ocean tide. Fill in the table. Ocean Tide Height (x) x 3 Earth Tide Height (y) 15 6 9 15 Perform the given operations. 28 26. 7(6) 144 27. 1 2 28. 7(7) 160 29. 40 30. 2(3)(5) 96 31. 12 32. 12(3)(2) 18(6) 33. 3 Chapter 1 Principles of Algebra Evaluate the expressions for the given value of the variable. Science 34. 4t 5 for t 3 35. x 2 for x 9 36. 6(s 9) for s 1 r 37. 8 for r 64 42 38. t for t 6 for y 35 39. 4 40. Science The ocean floor is extremely uneven. It includes underwater mountains, ridges, and extremely deep areas called trenches. To the nearest foot, find the average depth of the trenches shown. Depths of Ocean Trenches (Sea level) 0 –20,000 Depth (ft) Anoplogaster cornuta, often called a fangtooth or ogrefish, is a predatory fish that reaches a maximum length of 15 cm. It can be found in tropical and temperate waters at 16,000 ft. y 11 –25,000 Yap –27,976 –30,000 Bonin –32,788 –35,000 – 40,000 Kuril –31,988 Mariana –35,840 41. Reasoning A football team runs 11 plays. There are 3 plays that result in a loss of 2 yards each and 8 plays that result in a gain of 4 yards each. To find the total yards gained, Art simplifies the expression 3(2) 8(4). Bella first finds the total yards lost, 6, and the total yards gained, 32. Then she subtracts 6 from 32. Compare these two methods. 42. Choose a Strategy P is the set of positive factors of 30, and Q is the set of negative factors of 18. If x is a member of set P and y is a member of set Q, what is the greatest possible value of x y? A 540 B 180 90 C D 1 43. Write About It If you know that the product of two integers is negative, what can you say about the two integers? Give examples. 44. Challenge A visiting player is positioned on his own 30-yard line. The player loses 3 yards after a gain of 12 yards. What is the player’s position if 4 yards are lost on the next two plays? NS1.2, AF1.1 45. Multiple Choice What is the product of 7 and 10? A 70 B 17 C 3 D 70 46. Short Response Brenda donates part of her salary to the local children’s hospital each month by having $15 taken out of her monthly paycheck. Write an integer to represent the amount taken out of each paycheck. Find an integer to represent the change in the amount of money in Brenda’s paychecks after 1.5 years. Write an algebraic expression for each word phrase. (Lesson 1-2) 47. j decreased by 18 48. twice b less 12 49. 22 less than y Find each sum or difference. (Lessons 1-4 and 1-5) 50. 7 3 51. 5 (4) 52. 3 (6) 53. 513 (259) 1-6 Multiplying and Dividing Integers 29 Quiz for Lessons 1-1 Through 1-6 1-1 Evaluating Algebraic Expressions Evaluate each expression for the given values of the variables. 1. 5x 6y for x 8 and y 4 1-2 2. 6(r 7t) for r 80 and t 8 Writing Algebraic Expressions Write an algebraic expression for each word phrase. 3. one-sixth the sum of r and 7 4. 10 plus the product of 16 and m Write a word phrase for each algebraic expression. 5. 7y 46 1-3 x 10 7. 3 6. 2(18 t) p 8. 15 3 2 Integers and Absolute Value Write the integers in order from least to greatest. 9. 17, 25, 18, 2 10. 0, 8, 9, 1 Simplify each expression. 11. ⏐14 7⏐ 1-4 12. ⏐15⏐ ⏐12⏐ 13. ⏐26⏐ ⏐14⏐ Adding Integers Evaluate each expression for the given value of the variable. 14. p 14 for p 8 15. w (9) for w 4 16. In Loma, Montana, on January 15, 1972, the temperature increased 103 degrees in a 24-hour period. If the lowest temperature on that day was 54°F, what was the highest temperature? 1-5 Subtracting Integers Subtract. 17. 12 (8) 18. 7 (5) 19. 5 (16) 20. 22 5 21. The point of highest elevation in the United States is on Mount McKinley, Alaska, at 20,320 feet. The point of lowest elevation is in Death Valley, California, at 282 feet. What is the difference in the elevations? 1-6 Multiplying and Dividing Integers Multiply or divide. 22. (8)(6) 30 28 23. 7 Chapter 1 Principles of Algebra 39 24. 3 25. (2)(5)(6) California Standards MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Also covered: NS1.2 Solve • Choose an operation: Addition or Subtraction To decide whether to add or subtract to solve a word problem, you need to determine what action is taking place in the problem. If you are combining numbers or putting numbers together, you need to add. If you are taking away or finding out how far apart two numbers are, you need to subtract. Action Combining or putting together Operation Illustration Add Removing or taking away Subtract Finding the difference Subtract Jan has 10 red marbles. Joe gives her 3 more. How many marbles does Jan have now? The action is combining marbles. Add 10 and 3. Determine the action in each problem. Use the actions to restate the problem. Then give the operation that must be used to solve the problem. 1 Lake Superior is the largest of the Great Lakes and contains approximately 3000 mi3 of water. Lake Michigan is the second largest Great Lake by volume and contains approximately 1180 mi3 of water. Approximately how much more water is in Lake Superior? Superior Lake N W Upper Peninsula Lower Peninsula t L. On Michigan Lak E 3 Einar has $18 to spend on his friend’s birthday presents. He buys one present that costs $12. How much does he have left to spend? S uron eH ak Lake Michig an L 2 The average temperature in Homer, Alaska, is approximately 53°F in July and approximately 24°F in December. How much hotter is the average temperature in Homer in July than in December? eE a r io 4 Dinah got 87 points on her first test and 93 points on her second test. What is her point total for the first two tests? ri e Focus on Problem Solving 31 1-7 California Standards Preparation for AF4.0 Students solve simple linear equations and inequalities over the rational numbers. AF1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. Solving Equations by Adding or Subtracting Why learn this? You can use an equation to represent the forces acting on an object. (See Example 3.) equation inverse operations EXAMPLE 11 An equation is a mathematical statement that uses an equal sign to show that two expressions have the same value. All of these are equations. 3 8 11 Vocabulary 38 r 6 14 24 x 7 100 50 2 To solve an equation that contains a variable, find the value of the variable that makes the equation true. This value of the variable is called the solution of the equation. 1 Determining Whether a Number Is a Solution of an Equation Determine which value of x is a solution of the equation. The phrase “subtraction ‘undoes’ addition” can be understood with this example: If you start with 3 and add 4, you can get back to 3 by subtracting 4. 34 4 3 x 7 13; x 12 or 20 Substitute each value for x in the equation. x 7 13 x 7 13 ? ? 20 7 13 Substitute 20 for x. 12 7 13 Substitute 12 for x. ? ? 13 13 ✔ 5 13 ✘ So 20 is a solution. So 12 is not a solution. Adding and subtracting by the same number are inverse operations . Inverse operations “undo” each other. To solve an equation, use inverse operations to isolate the variable. In other words, get the variable alone on one side of the equal sign. The properties of equality allow you to perform inverse operations. These properties show that you can perform the same operation on both sides of an equation. You can use the properties of equality along with the Identity Property of Addition to solve addition and subtraction equations. IDENTITY PROPERTY OF ADDITION Words The sum of any number and zero is that number. 32 Chapter 1 Principles of Algebra Numbers 505 2 0 2 Algebra a0a EXAMPLE 2 Solving Equations Using Addition and Subtraction Properties Solve. 6 t 28 6t 6 0t t 28 6 22 22 Check 6 t 28 ? 6 22 28 ? 28 28 ✔ m 8 14 m 8 14 8 8 m 0 6 m 6 Check m 8 14 ? 6 8 14 ? 14 14 ✔ Since 6 is added to t, subtract 6 from both sides to undo the addition. Identity Property of Addition: 0 t t To check your solution, substitute 22 for t in the original equation. Since 8 is subtracted from m, add 8 to both sides to undo the subtraction. Identity Property of Addition: m 0 m To check your solution, substitute 6 for m in the original equation. 15 w (14) 15 w (14) 15 (14) w (14) (14) 29 w 0 29 w w 29 Since 14 is added to w, subtract 14 from both sides to undo the addition. Identity Property of Addition Definition of Equality PROPERTIES OF EQUALITY Words Addition Property of Equality You can add the same number to both sides of an equation, and the statement will still be true. Numbers Algebra 23 5 4 4 27 9 If x y, then x z y z. 4 7 11 3 3 44 8 If x y, then x z y z. Subtraction Property of Equality You can subtract the same number from both sides of an equation, and the statement will still be true. 1-7 Solving Equations by Adding or Subtracting 33 EXAMPLE 3 PROBLEM SOLVING APPLICATION Net force is the sum of all forces acting on an object. Expressed in newtons (N), it tells you in which direction and how quickly the object will move. If two dogs are playing tug-of-war, and the dog on the right pulls with a force of 12 N, what force is the dog on the left exerting on the rope if the net force is 2 N? 1 Understand the Problem 1. Force is measured in newtons (N). The number of newtons tells the size of the force and the sign tells its direction. Positive is to the right and up, and negative is to the left and down. The answer is the force that the left dog exerts on the rope. List the important information: • The dog on the right pulls with a force of 12 N. • The net force is 2 N. Show the relationship of the information: net force left dog’s force right dog’s force 2.2 Make a Plan Write an equation and solve it. Let f represent the left dog’s force on the rope, and use the equation model. 2 Math Builders For more on writing equations from words, see the Graph and Equation Builder on page MB2. f 12 3.3 Solve 2 f 12 12 12 10 f Subtract 12 from both sides. The left dog is exerting a force of 10 newtons on the rope. 4.4 Look Back The problem states that the net force is 2 N, which means that the dog on the right must be pulling with more force. The absolute value of the left dog’s force is less than the absolute value of the right dog’s force, 10 12, so the answer is reasonable. Think and Discuss 1. Explain what the result would be in the tug-of-war match in Example 3 if the dog on the left pulled with a force of 7 N and the dog on the right pulled with a force of 6 N. 2. Describe the steps to solve y 5 16. 34 Chapter 1 Principles of Algebra 1-7 California Standards Practice NS1.2, AF1.4, Preparation for AF4.0 Exercises KEYWORD: MT8CA 1-7 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Determine which value of x is a solution of each equation. 1. x 6 18; x 10, 12, or 25 See Example 2 See Example 3 2. x 7 14; x 2, 7, or 21 Solve. 3. m 9 23 4. 8 t 13 5. p (13) 10 6. q (25) 81 7. 26 t 13 8. 52 p (41) 9. A team of mountain climbers descended 3600 feet to a camp that was at an altitude of 12,035 feet. At what altitude did they start? INDEPENDENT PRACTICE See Example 1 Determine which value of x is a solution for each equation. 10. x 14 8; x 6, 22, or 32 See Example 2 See Example 3 11. x 23 55; x 15, 28, or 32 Solve. 12. 9 w (8) 13. m 11 33 14. 4 t 16 15. z (22) 96 16. 102 p (130) 17. 27 h (8) 18. Olivia owns 43 CDs. This is 15 more CDs than Angela owns. How many CDs does Angela own? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP3. Solve. Check your answer. 19. 7 t 12 20. h 21 52 21. 15 m (9) 22. m 5 10 23. h 8 11 24. 6 t 14 25. 1785 t (836) 26. m 35 172 27. x 29 81 28. p 8 23 29. n (14) 31 30. 20 8 w 31. 8 t 130 32. 57 c 28 33. 987 w 797 34. Social Studies In 1990, the population of Cheyenne, Wyoming, was 73,142. By 2000, the population had increased to 81,607. Write and solve an equation to find n, the increase in Cheyenne’s population from 1990 to 2000. 35. Astronomy Mercury’s surface temperature has a range of 600°C. This range is the broadest of any planet in the solar system. Given that the lowest temperature on Mercury’s surface is 173°C, write and solve an equation to find the highest temperature. 1-7 Solving Equations by Adding or Subtracting 35 Determine which value of the variable is a solution of the equation. 36. d 4 24; d 6, 20, or 28 37. k (13) 27; k 40, 45, or 50 38. d 17 36; d 19, 17, or 19 39. k 3 4; k 1, 7, or 17 40. 12 14 s; s 20, 26, or 32 41. 32 27 g; g 58, 25, 59 42. Science An ion is a charged particle. Each proton in an ion has a charge of 1 and each electron has a charge of 1. The ion charge is the electron charge plus the proton charge. Write and solve an equation to find the electron charge for each ion. Hydrogen sulfate ion (HSO4) Name of Ion Proton Charge Electron Charge Ion Charge Aluminum ion (Al3) 13 3 Hydroxide ion (OH) 9 1 Oxide ion (O2) 8 2 Sodium ion (Na) 11 1 43. What’s the Error? A student evaluated the expression 7 (3) and came up with the answer 10. What did the student do wrong? 44. Write About It Write a set of rules to use when solving addition and subtraction equations. 45. Challenge Explain how you could solve for h in the equation 14 h 8 using algebra. Then find the value of h. NS1.2, Prep for AF4.0 46. Multiple Choice Which value of x is the solution of the equation x (5) 8? A x3 B x 11 C x 13 D x 15 47. Multiple Choice Len bought a pair of $12 flip-flops and a shirt. He paid $30 in all. Which equation can you use to find the price p he paid for the shirt? A 12 p 30 B 12 p 30 C 30 p 12 D p 12 30 48. Gridded Response What value of x is the solution of the equation x 23 19? Add. (Lesson 1-4) 49. 5 (9) 50. 16 (22) 51. 64 51 52. 82 (75) 38 55. 19 56. 8(13) Multiply or divide. (Lesson 1-6) 53. 7(8) 36 54. 63 (7) Chapter 1 Principles of Algebra 1-8 California Standards Preparation for AF4.0 Students solve simple linear equations and inequalities over the rational numbers. Also covered: AF1.3 Solving Equations by Multiplying or Dividing Why learn this? You can write a multiplication equation to solve problems about person-hours. (See Example 3.) Just as with addition and subtraction equations, you can use an identity property to solve multiplication and division equations. IDENTITY PROPERTY OF MULTIPLICATION Vocabulary Words Identity Property of Multiplication Division Property of Equality Numbers The product of any number and one is that number. 515 Algebra 2 1 2 a1a You can use the properties of equality along with the Identity Property of Multiplication to solve multiplication and division equations. DIVISION PROPERTY OF EQUALITY EXAMPLE 1 Words Numbers You can divide both sides of an equation by the same nonzero number, and the statement will still be true. 4 3 12 4 3 12 2 2 12 6 2 Algebra If x y and z 0, x y then z z . Solving Equations Using Division Solve. 8x 32 8x 32 Since x is multiplied by 8, divide both sides by 8 to undo the multiplication. 8x 32 8 8 1x 4 x4 Identity Property of Multiplication: 1 x x 7y 91 7y 91 7y 91 7 7 1y 13 y 13 Since y is multiplied by 7, divide both sides by 7 to undo the multiplication. Identity Property of Multiplication: 1 yy 1-8 Solving Equations by Multiplying or Dividing 37 MULTIPLICATION PROPERTY OF EQUALITY EXAMPLE 2 Words Numbers Algebra You can multiply both sides of an equation by the same number, and the statement will still be true. 236 42346 8 3 24 If x y, then zx zy. Solving Equations Using Multiplication h 6. Solve 3 h 6 3 h 3 3 3 6 Since h is divided by 3, multiply both sides by 3 to undo the division. 1h 18 h 18 EXAMPLE A person-hour is a unit of work. It represents the amount of work that can be completed by one person in one hour. 3 Identity Property of Multiplication: 1 hh Sports Application It takes 96 person-hours to prepare a baseball stadium for a playoff game. The stadium’s manager assigns 6 people to the job. Each person works the same number of hours. How many hours does each person work? Let x represent the number of hours each person works. number of people 6 number of hours total number of each person works person-hours x 96 6x 96 Write the equation. 6x 96 6 6 Since x is multiplied by 6, divide both sides by 6 to undo the multiplication. 1x 16 Identity Property of Multiplication: 1 x x x 16 Each person works for 16 hours. Think and Discuss 1. Explain the steps you would use to solve 2k.5 6. 2. Tell how the Multiplication and Division Properties of Equality are similar to the Addition and Subtraction Properties of Equality. 38 Chapter 1 Principles of Algebra 1-8 California Standards Practice AF1.1, Preparation for AF4.0 Exercises KEYWORD: MT8CA 1-8 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 See Example 3 Solve. 1. 4x 28 2. 3y 42 3. 12q 24 4. 25m 125 l 4 5. 15 h 6. 1 3 9 m 1 7. 6 t 8. 1 52 3 9. Business It takes 270 person-days to prepare one issue of a magazine for publication. Each week, it takes 5 days to prepare an issue of the magazine. Assuming each person works the same number of days, how many people work on the magazine? INDEPENDENT PRACTICE See Example 1 See Example 2 Solve. 10. 3d 57 11. 7x 105 12. 4g 40 14. 8p 88 15. 17n 34 16. 212b 424 17. 41u 164 n 18. 9 63 h 2 19. 27 a 20. 6 102 21. 8 12 d 23. 7 23 t 24. 5 60 3 25. 84 y 11 22. 9 See Example 3 13. 16y 112 j p 26. A kilowatt-hour is one kilowatt of power that is supplied for one hour. The lighting for a 3-hour concert uses a total of 210 kilowatt-hours of electricity. How many kilowatts of electricity do the lights require? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP3. Solve. 27. 2x 14 28. 4y 80 29. 6y 12 30. 9m 9 k 31. 8 7 x 32. 5 121 b 33. 6 12 n 34. 1 1 5 35. 3x 51 36. 15g 75 37. r 92 115 38. 23 x 36 b 12 39. 4 m 40. 2 24 4 41. a 31 16 42. y 25 5 43. Nutrition One serving of milk contains 8 grams of protein, and one serving of steak contains 32 grams of protein. Write and solve an equation to find the number of servings of milk n needed to get the same amount of protein as there is in one serving of steak. x 11 be greater than 11 or less than 44. Reasoning Will the solution of 5 11? Explain how you know. 45. Multi-Step Joy earns $8 per hour at an after-school job. Each month she earns $128. How many hours does she work each month? After six months, she gets a $2 per hour raise. How much money does she earn per month now? 1-8 Solving Equations by Multiplying or Dividing 39 Translate each sentence into an equation. Then solve the equation. 46. The quotient of a number d and 4 is 3. 47. The sum of 7 and a number n is 15. 48. The product of a number b and 5 is 250. 49. Twelve is the difference of a number q and 8. 50. Zero is the product of 38 and a number t. 51. Recreation While on vacation, Milo drove his car a total of 370 miles. This was 5 times as many miles as he drives in a normal week. How many miles does Milo drive in a normal week? 52. Reasoning In the equation ax 12, a is an integer. Is it possible to choose a value of a so that the solution of the equation is x 0? Why or why not? 53. Music Jason wants to buy the trumpet advertised in the classified ads. He has saved $156. Using the information from the ad, write and solve an equation to find how much more money he needs to buy the trumpet. 54. Write a Problem Write a problem that can be solved by using the equation 15x 75. What is the solution of the problem? m 55. Write About It Explain how to estimate the solution of 97. 8 56. Challenge During a winter storm, the temperature dropped 3°F every hour. The overall change in the temperature was 18°F. Write and solve an equation to determine how long the storm lasted. NS1.2, Prep for AF4.0 57. Multiple Choice Solve the equation 7x 42. A x 49 B x 35 C x 6 D x6 58. Gridded Response On a game show, Paul missed q questions, each worth 100 points. Paul received a total of 900 points. How many questions did he miss? Subtract. (Lesson 1-5) 59. 8 8 60. 3 (7) 61. 10 2 62 11 (9) 65. 17 x 9 66. 19 x 11 Solve each equation. (Lesson 1-7) 63. 4 x 13 40 64. x 4 9 Chapter 1 Principles of Algebra Model Two-Step Equations 1-9 Use with Lesson 1-9 KEYWORD: MT8CA Lab1 KEY REMEMBER = 1 = 1 = variable =0 California Standards AF4.1 Solve two-step • You can perform the same operation with the same numbers on both sides of an equation without changing the value of the equation. 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. You can use algebra tiles to model and solve two-step equations. To solve a two-step equation, you use two different operations. Activity 1 Use algebra tiles to model and solve 3s 4 10. 3s 4 10 Two steps are needed to solve this equation. Step 1: Remove 4 yellow tiles from each side. 3s 6 Step 2: Divide each side into 3 equal groups. s 2 Substitute to check: 3s 4 10 ? 3(2) 4 10 ? 6 4 10 ? 10 10 ✔ Lab continued on next page. 1-9 Hands-On Lab 41 2 Use algebra tiles to model and solve 2r 4 6. 2r Step 1: Since 4 is being added to 2r, add 4 red tiles to both sides and remove the zero pairs on the left side. 4 6 Step 2: Divide each side into 2 equal groups. Add 4 to both sides. r 2r 5 10 Substitute to check: 2r 4 6 ? 2(5) 4 6 ? 10 4 6 ? 6 6 ✔ Think and Discuss 1. Why can you add zero pairs to one side of an equation without having to add them to the other side as well? 2. Show how you could have modeled to check your solution for each equation. Try This Use algebra tiles to model and solve each of the following equations. 1. 2x 3 5 2. 4p 3 9 3. 5r 6 11 4. 3n 5 4 5. 6b 8 2 6. 2a 2 6 7. 4m 4 4 8. 7h 8 41 9. Gerry walked dogs five days a week and got paid the same amount each day. One week his boss added on a $15 bonus. That week Gerry earned $90. What was his daily salary? 42 Chapter 1 Principles of Algebra 1-9 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. Also covered: AF1.1 Solving Two-Step Equations Who uses this? Two-step equations can help pet owners decide how much pet food they should purchase. (See Example 4.) Two-step equations contain two operations. For example, the equation 6x 2 10 contains multiplication and subtraction. Multiplication 6x 2 10 Subtraction EXAMPLE 1 Translating Sentences into Two-Step Equations Translate each sentence into an equation. 3 more than the product of 5 and a number y is 17. 3 more than the product of 5 and a number y is 17. 5 y 3 17 5y 3 17 The quotient of a number n and 4, minus 8, is 42. The quotient of a number n and 4, minus 8, is 42. [n (4)] 8 42 n 8 42 4 Math Builders For more on solving two-step equations, see the Step-by-Step Solution Builder on page MB4. Equations that contain two operations require two steps to solve. Identify the operations and the order in which they are applied to the variable. Then use inverse operations to undo them in the reverse order. Operations in the Equation To Solve 1 First x is multiplied by 6. 1 Add 2 to both sides of the equation. 2 Then 2 is subtracted. 2 Then divide both sides by 6. 1-9 Solving Two-Step Equations 43 EXAMPLE 2 Solving Two-Step Equations Using Division Solve. 2x 1 7 Step 1: 2x 1 7 11 2x 8 Reverse the order of operations when solving equations that have more than one operation. Note that x is multiplied by 2. Then 1 is added. Work backward: Since 1 is added to 2x, subtract 1 from both sides. 2x 8 2 2 Step 2: Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. x 4 Check 2x 1 7 ? 2(4) 1 7 8 1 7 ? 7 7 ✔ To check your solution, substitute 4 for x in the original equation. 4 is a solution. 18 7n 3 18 7n 3 3 3 21 7n Since 3 is subtracted from 7n, add 3 to both sides to undo the subtraction. 21 7n 7 7 Since n is multiplied by 7, divide both sides by 7 to undo the multiplication. 3 n EXAMPLE 3 Solving Two-Step Equations Using Multiplication Solve. k 6 5 13 k 6 5 13 Step 1: 5 5 k 6 k 6 8 (6) (6)8 Step 2: k 48 Note that k is divided by 6. Then 5 is added. k Work backward: Since 5 is added to , 6 subtract 5 from both sides. Since k is divided by 6, multiply both sides by 6 to undo the division. v 1 9 10 v 1 9 10 10 v 11 9 10 v (9)11 (9) 9 99 v 44 Chapter 1 Principles of Algebra v Since 10 is subtracted from , add 10 9 to both sides to undo the subtraction. Since v is divided by 9, multiply both sides by 9 to undo the division. EXAMPLE 4 Consumer Math Application Tyrell is ordering cat food online. Shipping is $12 no matter how many cases he orders. Each case of cat food costs $16. Tyrell has $130 to spend. How many cases of cat food can he order? Let c represent the number of cases that Tyrell can order. That means Tyrell can spend $16c plus the cost of shipping. Reasoning cost of shipping cost of cat food total cost $12 16c 12 16c 130 12 12 16c 118 $130 Subtract 12 from both sides. 16c 118 16 16 Divide both sides by 16. c 7.375 Since you cannot order part of a case, Tyrell can order 7 cases. Think and Discuss 1. Explain how you decide which operation to use first when solving a two-step equation. 2. Describe the steps you would follow to solve 5 7x 16. 1-9 Exercises California Standards Practice AF1.1, AF4.1 KEYWORD: MT8CA 1-9 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Translate each sentence into an equation. 1. 6 more than 9 times a number x is 12. 2. The quotient of a number t and 7, plus 2, is 4. See Example 2 See Example 3 Solve. 3. 4x 7 15 4. 8 5s 53 5. 3n 18 6 6. 12 5p 2 7. 6 2y 4 8. 6w 8 20 Solve. m 3 9. 4 12 x 12. 13 11 3 x 7 10. 7 9 w 11. 2 5 r 13. 6 5 8 14. 8 15 6 z 2 1-9 Solving Two-Step Equations 45 See Example 4 15. Consumer Math A Web site that rents DVDs charges a one-time fee of $7 to become a member of the site. It costs $3 to rent each DVD. Diana joined the site and spent a total of $61. How many DVDs did she rent? INDEPENDENT PRACTICE See Example 1 Translate each sentence into an equation. 16. 11 less than the quotient of a number b and 12 is 5. 17. 18 more than the product of 4 and a number z is 54. 18. The difference of twice a number n and 9 is 17. See Example 2 See Example 3 Solve. 19. 5y 16 21 20. 14 2x 16 21. 6 5t 24 22. 10w 3 53 23. 7q 8 27 24. 12 5m 13 25. 9 6g 3 26. 16y 41 7 27. 3z 11 38 Solve. w 5 x 31. 17 12 8 t 34. 8 5 7 28. 2 9 See Example 4 r 2 y 29. 10 8 30. 4 9 4 w 32. 5 16 33. 3 8 4 g 82 35. 3 n 5 x 36. 2 3 8 37. Bob’s Auto Shop charges $375 to replace the timing belt in a car. The belt costs $50 and the labor to install the belt costs $65 per hour. How many hours does it take to install the timing belt? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP3. Translate each equation into words. Then solve the equation. x 38. 5 12 6 w 2 39. 4y 9 3 40. 7 4 42. 6 4p 2 43. 7j 12 9 45. 5 2z 5 r 46. 3 8 Solve and check. m 3 x 44. 4 1 3 41. 11 4 4 Translate each sentence into an equation. Then solve the equation. 47. Three less than the quotient of a number x and 9 is 8. 48. Thirteen plus the product of 15 and a number a is 58. 49. If 5 is decreased by 3 times a number m, the result is 4. 50. Hobbies Mike has 60 baseball cards in his collection. Each week he buys a pack of cards to add to the collection. Each pack contains the same number of cards. After 12 weeks, Mike has 156 cards. Write and solve an equation to find the number of cards in a pack. 46 Chapter 1 Principles of Algebra Social Studies In 1956, during President Eisenhower’s term, construction began on the United States interstate highway system. The original plan was for 42,000 miles of highways to be completed within 16 years. It actually took 37 years to complete. The last part, Interstate 105 in Los Angeles, was completed in 1993. 51. Write and solve an equation to show how many miles m needed to be completed per year for 42,000 miles of highways to be built in 16 years. 52. Interstate 35 runs north and south from Laredo, Texas, to Duluth, Minnesota, covering 1568 miles. There are 505 miles of I-35 in Texas and 262 miles in Minnesota. Write and solve an equation to find m, the number of miles of I-35 that are not in either state. 53. A portion of I-476 in Pennsylvania, known as the Blue Route, is about 22 miles long. The length of the Blue Route is about one-sixth the total length of I-476. Write and solve an equation to calculate the length of I-476 in miles m. 54. Challenge Interstate 80 extends from California to New Jersey. At right are the number of miles of Interstate 80 in each state the highway passes through. a. __?__ has 134 more miles than __?__. b. __?__ has 174 fewer miles than __?__. Number of I-80 Miles State Miles California Nevada Utah Wyoming Nebraska Iowa Illinois Indiana Ohio Pennsylvania New Jersey 195 410 197 401 455 301 163 167 236 314 68 AF4.1 t 5 55. Multiple Choice Solve the equation 4 8. A t 60 t 40 B C t 20 D t 10 D 12 15 56. Multiple Choice For which equation is m 3 a solution? A 5m 6 21 m 1 1 3 B C 2 3m 11 m 3 Multiply or divide. (Lesson 1-6) 57. 4(11) 12 58. 59. 6(5) 42 60. 62. 16m 48 p 63. 21 64. 12g 108 3 7 Solve. (Lesson 1-8) y 61. 9 8 7 1-9 Solving Two-Step Equations 47 Quiz for Lessons 1-7 Through 1-9 1-7 Solving Equations by Adding or Subtracting Solve. 1. p 12 5 2. w (9) 14 3. t (14) 8 4. 23 k 5 5. 52 p 17 6. y (6) 74 7. The approximate surface temperature of Pluto, the coldest planet, is 391°F. This is approximately 1255 degrees cooler than the approximate surface temperature of Venus, the hottest planet. What is the approximate surface temperature of Venus? 1-8 Solving Equations by Multiplying or Dividing Solve. x 8. 6 48 9. 3x 21 10. 14y 84 y 11. 1 72 2 r d 3 14. 1 10 15. 8y 96 13. 7 2 16. Ahmed’s baseball card collection consists of 228 cards. This is 4 times as many cards as Ming has. How many baseball cards are in Ming’s collection? 12. 5p 75 17. It takes 72 person-hours to clean an office building each night. A crew of 12 people cleans the building and each person works the same number of hours. How many hours does each person work? 1-9 Solving Two-Step Equations Solve. 18. 3x 1 8 k 4 21. 6 14 24. 41 2 13n y 19. 2 5 8 20. 7n 4 24 22. 13 9 24 23. 27 5x 2 y x 25. 3 3 7 k 6 26. 15 5 27. Richard is collecting bottle caps for a school fundraiser. He has collected 335 caps toward his goal of 500 caps. If he collects 15 bottle caps per week, how many more weeks will it take before he reaches his goal? 28. One week Ella worked 36 hours as a prep chef. Her earnings for the week totaled $210, which included $30 in tips that she made as a waitress. How much money does Ella make per hour? 48 Chapter 1 Principles of Algebra Have a Ball A physical education class is playing a variation of basketball. When a team makes a basket from inside the three-point line, the team gets a “Climb” (C), or 2 points. When a team makes a basket from outside the three-point line, the other team gets a “Slide” (S), or 1 point. 1. During the first 20 minutes of the game, a team gets the following: C, C, S, S, C, S, C, C, and S. Simplify the expression 2 2 (1) (1) 2 (1) 2 2 (1) to determine the team’s score. Game Results Team 1 Team 2 C C 2. The points scored by two teams during a game are shown in the table. Which team won the game? What was the difference in the teams’ scores? S C S C S S 3. Diego’s team gets 3 Climbs and 2 Slides, but not necessarily in that order. Find his team’s score by substituting S 1 and C 2 in the expression 3C 2S. S S C S C S C S 4. After four consecutive baskets are made, Leann’s team’s score is 8. After the next basket is made, the team’s score is 6. Write and solve an equation for the last made basket. 5. Daryl’s team finishes the game with a score of 12. If his team scored 9 times, how many Climbs did the team get? 6. Is it possible to finish with a score of 2 after five baskets are made? Explain your reasoning. Concept Connection 49 Math Magic You can guess what your friends are thinking by learning to “operate” your way into their minds! For example, try this math magic trick. Think of a number. Multiply the number by 8, divide by 2, add 5, and then subtract 4 times the original number. No matter what number you choose, the answer will always be 5. Try another number and see. You can use what you know about variables to prove it. Here’s how: What you say: What the person thinks: What the math is: Step 1: Pick any number. 6 (for example) n Step 2: Multiply by 8. 8(6) 48 8n Step 3: Divide by 2. 48 2 24 8n 2 4n Step 4: Add 5. 24 5 29 4n 5 Step 5: Subtract 4 times the original number. 29 4(6) 29 24 5 4n 5 4n 5 Invent your own math magic trick that has at least five steps. Show an example using numbers and variables. Try it on a friend! Crazy Crzzy Cubes This game, called The Great Tantalizer around 1900, was reintroduced in the 1960s as “Instant Insanity™.” Make four cubes with paper and tape, numbering each side as shown. 1 4 4 2 4 2 3 2 1 3 4 2 2 3 3 1 2 2 4 3 4 3 1 1 The goal is to line up the cubes so that 1, 2, 3, and 4 can be seen along the top, bottom, front, and back of the row of cubes. They can be in any order, and the numbers do not have to be right-side up. 50 Chapter 1 Principles of Algebra Materials • sheet of decorative paper (812 by 11 in.) • ruler • pencil • scissors • glue • markers PROJECT Note-Taking Taking Shape A 3 _58 in. 3 _58 in. Make this notebook to help you organize examples of algebraic expressions. Directions 1 Hold the sheet of paper horizontally. Make two vertical lines 358 inches from each end of the sheet. Figure A 2 Fold the sheet in half lengthwise. Then cut it in half by cutting along the fold. Figure A B A 3 On one half of the sheet, cut out rectangles A and B. On the other half, cut out rectangles C and D. Figure B Rectangle A: 34 in. by 358 in. Rectangle B: 112 in. by 358 in. Rectangle C: 214 in. by 358 in. Rectangle D: 3 in. by 358 in. 4 Place the piece with the taller rectangular panels on top of the piece with the shorter rectangular panels. Glue the middle sections of the two pieces together. Figure C 5 Fold the four panels into the center, starting with the tallest panel and working your way down to the shortest. Taking Note of the Math Write “Addition,” “Subtraction,” “Multiplication,” and “Division” on the tabs at the top of each panel. Use the space below the name of each operation to list examples of verbal, numerical, and algebraic expressions. B C C D Vocabulary absolute value . . . . . . . . . . 15 Addition Property of Equality . . . . . . . . . . . . . 33 equation . . . . . . . . . . . . . . . 32 evaluate . . . . . . . . . . . . . . . . . 6 integer . . . . . . . . . . . . . . . . . 14 inverse operations . . . . . . 32 expression . . . . . . . . . . . . . . . 6 additive inverse . . . . . . . . . 8 algebraic expression . . . . . 6 Identity Property of Addition . . . . . . . . . . . . . 32 numerical expression . . . . 6 opposite . . . . . . . . . . . . . . . . 14 Division Property of Equality . . . . . . . . . . . . . 37 Identity Property of Multiplication . . . . . . . 37 Subtraction Property of Equality . . . . . . . . . . . . . 33 variable . . . . . . . . . . . . . . . . . 6 Complete the sentences below with vocabulary words from the list above. 1. A(n) ___?___ is a statement that two expressions have the same value. 2. ___?___ is another word for “additive inverse.” 3. The ___?___ of 3 is 3. 1-1 Evaluating Algebraic Expressions EXAMPLE ■ EXERCISES Evaluate 4x 9y for x 2 and y 5. 4x 9y 4(2) 9(5) 8 45 53 Evaluate each expression. 4. 9a 7b for a 7 and b 12 5. 17m 3n for m 10 and n 6 Substitute 2 for x and 5 for y. Multiply. Add. 6. 1.5r 19s for r 8 and s 14 1-2 Writing Algebraic Expressions Write an algebraic expression for the word phrase “2 less than a number n.” n2 ■ Write as subtraction. Write a word phrase for 25 13t. 25 plus the product of 13 and t Write an algebraic expression for each phrase. 7. twice the sum of k and 4 8. 5 more than the product of 4 and t Write a word phrase for each algebraic expression. 9. 5b 10 10 11. r 12 52 AF1.1 (pp. 10–13) EXERCISES EXAMPLE ■ AF1.2, AF1.4 (pp. 6–9) Chapter 1 Principles of Algebra 10. 32 23s y 12. 16 8 ■ 1-3 Integers and Absolute Value (pp. 14–17) EXAMPLE EXERCISES Simplify each expression. ⏐9⏐ ⏐3⏐ 93 6 13. ⏐7 6⏐ 15. ⏐15⏐ ⏐19⏐ 14. ⏐8⏐ ⏐7⏐ 16. ⏐14 7⏐ 17. ⏐16 2⏐ 18. ⏐7⏐ ⏐8⏐ ⏐9⏐ 9 and ⏐3⏐ 3 Subtract. EXERCISES Add. Add. 8 2 6 Find the difference of⏐8⏐ and⏐2⏐. 8 2; use the sign of the 8. 1-5 Subtracting Integers 20. 3 (9) 21. 4 (7) 22. 4 (3) Evaluate. 24. k 11 for k 3 25. 6 m for m 2 NS1.2, (pp. 22–25) EXAMPLE ■ 19. 6 4 23. 11 (5) (8) Evaluate. 4 a for a 7 Substitute. 4 (7) 11 Same sign ■ NS1.2, AF1.3 (pp. 18–21) EXAMPLE ■ NS2.5 EXERCISES Subtract. Subtract. 3 (5) 3 5 2 26. 7 9 27. 8 (9) 28. 2 (5) 29. 13 (2) 30. 5 17 31. 16 20 Add the opposite of 5. 5 3; use the sign of the 5. Evaluate. 9 d for d 2 Substitute. 9 2 9 (2) Add the opposite of 2. 11 Same sign Evaluate. 32. 9 h for h 7 33. 12 z for z 17 1-6 Multiplying and Dividing Integers EXAMPLE Multiply or divide. 4(9) 36 The signs are different. The answer is negative. ■ 33 11 The signs are the same. 3 The answer is positive. ■ NS2.5 Simplify the expression |9| |3|. 1-4 Adding Integers ■ NS1.1, NS1.2 (pp. 26–29) EXERCISES Multiply or divide. 34. 7(–5) 72 35. 4 36. 4(13) 100 37. 4 38. 8(3)(5) 10(5) 39. 25 Study Guide: Review 53 1-7 Solving Equations by Adding or Subtracting EXAMPLE EXERCISES Solve. ■ ■ (pp. 32–36) x 7 12 7 7 x0 5 x 5 y 3 1.5 3 3 y 0 4.5 y 4.5 40. z 9 14 41. t 3 11 Subtract 7 from both sides. 42. 6 k 21 43. x 2 13 Identity Property of Zero Write an equation and solve. Add 3 to both sides. Identity Property of Zero EXAMPLE 44. A polar bear weighs 715 lb, which is 585 lb less than a sea cow. How much does the sea cow weigh? 45. The Mojave Desert, at 15,000 mi2, is 11,700 mi2 larger than Death Valley. What is the area of Death Valley? (pp. 37–40) Prep for EXERCISES Solve. AF4.0 Solve and check. 4h 24 4h 24 4 4 1h 6 h6 t 16 4 t 4 4 4 16 46. 7g 56 47. 108 12k Divide both sides by 4. 48. 0.1p 8 441 1hh 50. 20 2 49. 4 12 z 51. 2 8 4 ■ 1t 64 t 64 Multiply both sides by 4. 441 1tt y 53. Luz will pay a total of $9360 on her car loan. Her monthly payment is $390. For how many months is the loan? (pp. 43–47) EXAMPLE EXERCISES k 3 k 6 3 5 6 6 k 1 3 k (3) (3)(1) 3 k 3 w 52. The Lewis family drove 235 mi toward their destination. This was 13 of the total distance. What was the total distance? 1-9 Solving Two-Step Equations Solve 6 5. 54 AF4.0 Solve and check. 1-8 Solving Equations by Multiplying or Dividing ■ AF1.4, Prep for AF4.1 Solve. Subtract 6 from both sides. Multiply both sides by 3. Chapter 1 Principles of Algebra y 6 9 25 54. 15 3 3x 55. n 56. 1 8 7 n 58. 17 29 2 57. 8x 10 42 59. 5y 12 67 60. A home computer repair service charges $68 for the first hour and $48 for each additional hour of service. If a repair bill is $212, how many additional hours did the repair take? Evaluate each expression for the given value of the variable. 1. 16 p for p 12 2. t 7 for t 14 3. 13 x (2) for x 4 4. 8y 27 for y 9 Write an algebraic expression for each word phrase. 5. 15 more than the product of 33 and y 6. 18 less than the quotient of x and 7 7. 4 times the sum of 7 and h 8. 18 divided by the difference of t and 9 Write each set of integers in order from least to greatest. 9. 7, 7, 2, 3, 0, 1 10. 12, 45, 13, 100, 20 11. 120, 7, 54, 41, 7 13. ⏐21⏐ ⏐9⏐ 14. ⏐16⏐ ⏐8 3⏐ Simplify each expression. 12. ⏐12 5⏐ ⏐7⏐ Perform the given operations. 15. 9 (12) 16. 11 17 17. 6(22) 18. (20) (4) 19. 42 (5) 20. 18 3 21. 9 (13) 22. 12 (6) (5) 23. 2(21 17) 24. (15 3) (4) 25. (54 6) (1) 26. (16 4) 20 27. The temperature on a winter day increased 37°F. If the beginning temperature was 9°F, what was the temperature after the increase? Solve. 28. y 19 9 t 32. 7 12 29. 4z 32 30. 52 p 3 33. 9p 27 q 18 34. 5 w 31. 9 3 g 35. 4 11 36. Acme Sporting Products manufactures 3216 tennis balls a day. Each container holds 3 balls. How many tennis ball containers are needed daily? 37. It takes 105 person-hours to set up new computers in an office. A crew sets up the computers and each person works the same number of hours. It takes 7 hours to complete the job. How many people are on the crew? Solve. 38. 9x 11 110 y 41. 2 6 13 n 3 39. 8 6x 62 40. 14 4 42. 35 27 4n 43. 40 53 5 y 44. A round-trip ticket to a soccer camp costs $250. If the total cost for the ticket and 5 days at camp is $715, what is the cost to attend the camp per day? 45. Group rates at Putt-Putt Pfun include charges of $4 per person and entry to the game room at $10 per group. How many people are in a group that pays $74 for putt-putt and games? Chapter 1 Test 55 Multiple Choice: Eliminate Answer Choices With some multiple-choice test items, you can use logical reasoning or estimation to eliminate some of the answer choices. Test writers often create the incorrect choices, called distracters, using common student errors. Which expression represents “4 times the sum of x and 8”? A 4 (x 8) C 4x8 B 4 (x 8) D 4 (x 8) Read the question. Then try to eliminate some of the answer choices. Use logical reasoning. Times means “to multiply,” and sum means “to add.” You can eliminate any option without a multiplication symbol and an addition symbol. You can eliminate B and D. The sum of x and 8 is being multiplied by 4, so you need to add before you multiply. Because multiplication comes before addition in the order of operations, x 8 should be in parentheses. The correct answer is A. Which value for k is a solution of the equation k 3.5 12? A k 8.5 C k 42 B k 15.5 D k 47 Read the question. Then try to eliminate some of the answer choices. Use estimation. You can eliminate C and D immediately because they are too large. Estimate by rounding 3.5 to 4. If x 47, then 47 4 43. This is not even close to 12. Similarly, if x 42, then 42 4 38, which is also too large to be correct. Choice A is called a distracter because it was created using a common student error, subtracting 3.5 from 12 instead of adding 3.5 to 12. Therefore, A is also incorrect. The correct answer is B. 56 Chapter 1 Principles of Algebra Even if the answer you calculated is an answer choice, it may not be the correct answer. It could be a distracter. Always check your answers! Read each test item and answer the questions that follow. Item A The table shows average high temperatures for Nome, Alaska. Which answer choice lists the months in order from coolest to warmest? Month Temperature (°C) Jan 11 Feb 10 Mar 8 Apr 3 May 6 Jun 12 Jul 15 Aug 13 Sep 9 Oct 1 Nov 5 Dec 9 Item B Which value for p is a solution of the equation p 5.2 15? A p 30.2 C p 20.2 B p 9.8 D p 78 3. Which choices can you eliminate by using estimation? Explain your reasoning. 4. What common error does choice C represent? Item C Solve x 20 8 for x. 4 A x 112 C x 12 B x 112 D x 48 5. Which two choices can you eliminate by using logic? 6. What common error does choice C represent? Item D Which word phrase can be translated into the algebraic expression 2x 6? Jul, Aug, Jun, Sep, May, Oct, Apr, Nov, Mar, Dec, Feb, Jan A six more than twice a number B the sum of twice a number and six B Jul, Jun, Aug, Jan, Feb, Sep, Dec, Mar, May, Nov, Apr, Oct C twice the difference of a number and six C Jan, Apr, Jun, Jul, Sep, Nov, Feb, Mar, May, Jul, Sep, Nov D six less than twice a number D Jan, Feb, Dec, Mar, Nov, Apr, Oct, May, Sep, Jun, Aug, Jul A 1. Which two choices can you eliminate by using logic? Explain your reasoning. 2. What common error does choice A represent? 7. Can you eliminate any of the choices immediately by using logic? Explain your reasoning. 8. Describe how you can determine the correct answer from the remaining choices. Strategies for Success 57 KEYWORD: MT8CA Practice Cumulative Assessment, Chapter 1 Multiple Choice 1. Which expression has a value of 12 when x 2, y 3, and z 1? A 3xyz C 3xz 2y A ⏐5⏐ 5 B 2x 3y z D 4xyz 2 B 8 0 8 C 6 3 3 (6) D 4 (1 7) (4 1) 7 2. The word phrase “10 less than 4 times a number” can be represented by which expression? A 10 4x C 10 4x B 4x 10 D 10x 4 3. A copy center prints c copies at a cost of $0.10 per copy. What is the total cost of the copies? A 0.10c B 0.10 c 0.10 c c D 0.10 C 4. If both x and y are negative, which expression is always negative? A xy C xy B xy D xy 5. What is the solution of s (12) 16? A s4 C s 28 B s8 D s 192 6. Carlos owes his mother money. His paycheck is $105. If he pays his mother the money he owes her, he will have $63 left. Which equation represents this situation? A x 63 105 B x 63 105 C 105 x 63 D 58 7. Which equation uses the Identity Property of Addition? x 105 63 Chapter 1 Principles of Algebra 8. Which addition equation represents the number line diagram below? 5 4 3 2 1 0 1 A 4 (2) 2 B 4 (6) 2 C 4 6 10 D 4 (6) 10 2 3 4 5 6 9. Which equation has the solution x 16? A x 16 4 C 2x 32 B x 32 2 D x 2 16 10. Solve 6x 6 12 for x. A x1 C x 3 B x3 D x 1 11. A scuba diver swimming at a depth of 35 ft below sea level, or 35 ft, dives another 15 ft deeper to get a closer look at a fish. Which integer represents the diver’s new depth? A 50 ft C 20 ft B 20 ft D 50 ft The incorrect answer choices in a multiple-choice test item are called distracters. They are the results of common mistakes. Be sure to check your work! 12. Which set of numbers is in order from least to greatest? A 15, 13, 10 C 10, 15, 13 B 13, 10, 15 D 15, 10, 13 13. Which expression is equivalent to ⏐9 (5)⏐? A ⏐9⏐ ⏐5⏐ C 14 B ⏐9⏐ ⏐5⏐ D 4 14. Dennis spent a total of $340 on flowers and bushes for his yard. He spent $116 on flowers, and the bushes cost $28 each. To find how many bushes b Dennis bought, solve the equation 28b 116 340. A 3 bushes C 10 bushes B 8 bushes D 16 bushes Gridded Response 15. What is the value of the expression 2xy y when x 3 and y 5? 16. What is the solution of the equation x 27 16? 17. Evaluate m 11 (3) for m 5. 18. Nora collects 15 magazines every week for 6 weeks to use for an art project. After 6 weeks, she still does not have enough magazines to complete the project. If Nora needs 20 more magazines, how many total magazines does she need? 19. Patricia works twice as many days as Laura works each month. Laura works 3 more days than Jaime. If Jaime works 10 days each month, how many days does Patricia work? Short Response 20. The Hun family plans to visit the Sea Center. Admission tickets cost $7 each. a. Write an expression to represent the cost of t tickets. b. How much will it cost the Hun family if they buy 6 tickets? Explain your answer. c. Mrs. Hun pays with three $20 bills. How much change should she get back? Explain your answer. 21. It costs $0.15 per word to place an advertisement in the school newspaper. Let w represent the number of words in an advertisement and C represent the cost of the advertisement. a. Write an equation that relates the number of words to the cost of the advertisement. b. If Bernard has $12.00, how many words can he use in his advertisement? Explain your answer. Extended Response 22. Student Council members at a school are either elected members or work-on members. The number of elected student council members at a school is ten times the number of officers. The 55-member student council includes 5 work-on members. a. Write an equation to represent the number of student council officers x. Explain the meaning of each term in the equation. b. Solve the equation in part a. Check your answer. c. At the beginning of the school year, the school’s enrollment is 1106 students. By the end of the year, 6 students have transferred out of the school. Write and solve an equation to find the number of students each student council member represented at the end of the year. Cumulative Assessment, Chapter 1 59