Page 1 of 7 2.6 The Distributive Property How much will you pay for six CDs? Goal Use the distributive property. When you go shopping, you can use estimation or mental math to determine the total cost. In Example 5 you will learn how to use the distributive property to calculate the total cost of six CDs—without using a calculator. Key Words • distributive property The distributive property is an important algebraic property. Example 1 uses geometry to illustrate why the property is true for a single case. On the following page the property is formally defined. EXAMPLE Student Help VOCABULARY TIP To distribute means to give something to each member of a group. In Example 1 you can think of the 3 as being distributed to each term in (x 2). 1 Use an Area Model Find the area of a rectangle whose width is 3 and whose length is x 2. Solution You can find the area in two ways. Remember that the area of a rectangle is the product of the length times the width. Area of One Rectangle Area of Two Rectangles 3 3 x2 Area 3(x 2) ANSWER x 2 Area 3(x) 3(2) Because both expressions represent the same area, the following statement is true. Area 3(x 2) 3(x) 3(2) 3x 6 Use an Area Model 1. Write two expressions for the area of the rectangle. 7 2. Write an algebraic statement that shows that the two expressions from Exercise 1 are equal. 100 Chapter 2 Properties of Real Numbers 3 x Page 2 of 7 Student Help STUDY TIP Although four versions of the distributive property are listed, the last three versions can be derived from the first: a(b c) ab ac THE DISTRIBUTIVE PROPERTY The product of a and (b c): a(b c) ab ac Example: 5(x 2) 5x 10 (b c)a ba ca Example: (x 4)8 8x 32 The product of a and (b c): a(b c) ab ac Example: 4(x 7) 4x 28 (b c)a ba ca Example: (x 5)9 9x 45 2 EXAMPLE Use the Distributive Property with Addition Use the distributive property to rewrite the expression without parentheses. a. 2(x 5) b. (1 2n)8 Solution a. 2(x 5) 2(x) 2(5) Distribute 2 to each term of (x 5). 2x 10 Multiply. b. (1 2n)8 (1)8 (2n)8 8 16n Student Help MORE EXAMPLES Multiply. Use the Distributive Property with Subtraction Use the distributive property to rewrite the expression without parentheses. NE ER T More examples are available at www.mcdougallittell.com INT 3 EXAMPLE Distribute 8 to each term of (1 2n). 1 b. (2x 4) 2 a. 3(1 y) Solution a. 3(1 y) 3(1) 3(y) Distribute 3 to each term of (1 y). 3 3y Multiply. 1 1 1 b. (2x 4) (2x) (4) 2 2 2 x2 1 2 Distribute to each term of (2x 4). Multiply. Use the Distributive Property Use the distributive property to rewrite the expression without parentheses. 3. 5(n 3) 4. (2p 6)3 5. 2(x 5) 2.6 2 6. (3y 9) 3 The Distributive Property 101 Page 3 of 7 Use the Distributive Property 4 EXAMPLE Student Help Use the distributive property to rewrite the expression without parentheses. STUDY TIP Be careful when using the distributive property with negative factors. Forgetting to distribute the negative sign is a common error. a. 3(x 4) b. (y 5)(4) c. (6 3x) d. (1 t)(9) Solution a. 3(x 4) 3(x) (3)(4) Use distributive property. 3x 12 Multiply. b. (y 5)(4) (y)(4) (5)(4) 4y 20 Multiply. c. (6 3x) 1(6) (1)(3x) Use distributive property. 6 3x Multiply. d. (1 t)(9) (1)(9) (t)(9) 9 9t Use distributive property. Use distributive property. Multiply. Use the Distributive Property Use the distributive property to rewrite the expression without parentheses. 7. 5(a 2) 8. (x 7)(3) 9. (4 2x) 10. (4 m)(2) History 5 EXAMPLE Mental Math Calculations You are shopping for compact discs. You want to buy 6 compact discs for $11.95 each including tax. Use the distributive property to mentally calculate the total cost of the compact discs. COMPACT DISCS Solution If you think of $11.95 as $12.00 $.05, the mental math is easier. 6(11.95) 6(12 0.05) COMPACT DISCS CDs were released in the United States in 1984. That year the average cost of a CD was $17.81. ANSWER Write 11.95 as a difference. 6(12) 6(0.05) Use distributive property. 72 0.30 Find products mentally. 71.70 Find difference mentally. The total cost of 6 compact discs at $11.95 each is $71.70. Mental Math Calculations Use the distributive property to mentally calculate the total cost. 11. You are buying birthday cards for 3 of your friends. Each card costs $1.25. What is the total cost of the cards? 102 Chapter 2 Properties of Real Numbers Page 4 of 7 2.6 Exercises Guided Practice Vocabulary Check Explain how you would use the distributive property to simplify the expression. 1. 2(x 3) Skill Check 2. (x 4)5 3. 7(x 3) 4. (x 6)4 Use the area model shown. x 5. Write two expressions for the area of the rectangle. 5 6. Write an algebraic statement that 12 shows that the two expressions from Exercise 5 are equal. Match the expression with its simplified form. 7. 3(x 2) 8. (x 3)(2) 9. 3(x 2) A. 6 2x B. 3x 6 C. 2x 6 10. (3 x)2 D. 3x 6 Use the distributive property and mental math to simplify the expression. ( ) ? ( ? ) ? ( ? ) ? ? ? 11. 4(1.15) 4(1 0.15) 12. 9(1.95) 9 ? ? ? ) ? ( ? ) ( ? ? ? ? Practice and Applications AREA MODEL Use the area model to find two expressions for the area of the rectangle. Then write an algebraic statement that shows the two expressions are equal. 13. 14. 2 3 x HOMEWORK HELP Example 1: Exs. 13–16, 71–73 Example 2: Exs. 17–28 Example 3: Exs. 29–40 Example 4: Exs. 41–54 Example 5: Exs. 55–70 x 4 Student Help 7 15. 16. 2x 11 4 x 5 9 2.6 The Distributive Property 103 Page 5 of 7 DISTRIBUTIVE PROPERTY WITH ADDITION Use the distributive property to rewrite the expression without parentheses. 17. 3(x 4) 18. 5(w 6) 19. 7(1 t) 20. (y 4)5 21. (2 u)6 22. (x 8)7 23. 2(2y 1) 24. (3x 7)4 25. 3(4 6a) 26. (9 3n)2 27. (x 2)1.3 1 28. (10 15r) 5 DISTRIBUTIVE PROPERTY WITH SUBTRACTION Use the distributive property to rewrite the expression without parentheses. 29. 5(y 2) 30. 2(x 3) 31. 9(7 a) 32. (x 2)2 33. (7 m)4 34. (n 7)2 35. 10(1 3t) 36. 7(6w 1) 37. (3x 3)6 38. (9 5a)4 39. (3.1u 0.8)3 1 2 40. 5 x 10 15 DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. 41. 3(r 8) 42. 2(x 6) 43. (1 s) 44. (2 t)(2) 45. (y 9)(1) 46. (x 4)(3) 47. 6(4a 3) 48. (9x 1)(7) 49. (6y 5) 50. (3d 8)(5) 51. (2.3 7w)(6) 3 52. x 24 8 ERROR ANALYSIS In Exercises 53 and 54, find and correct the error. 53. simplify the expression. HOMEWORK HELP Extra help with problem solving in Exs. 55–66 is available at www.mcdougallittell.com INT –2(7 – 8) = –2(7) – 2(8) = –14 – 6 = –30 MENTAL MATH Use the distributive property and mental math to Student Help NE ER T 54. 9(3 – 5) = 9(3) – 5 = 27 – 5 = 22 55. 4(6.11) 56. 10(7.25) 57. 3(9.20) 58. 7(5.98) 59. 2(2.90) 60. 6(8.75) 61. 3(4.10) 62. 9(1.02) 63. 2(11.05) 64. 8(2.80) 65. 5(10.99) 66. 4(5.95) 67. CONTACT LENS SUPPLIES The saline solution that you use to clean your contact lenses is on sale for $4.99 a bottle. You decide to stock up and buy 4 bottles. Use the distributive property to mentally calculate the total cost of the bottles of saline. 104 Chapter 2 Properties of Real Numbers Page 6 of 7 68. SCHOOL SUPPLIES You are shopping for school supplies. You want to buy Careers 7 notebooks for $1.05 each. Use the distributive property to mentally calculate the total cost of the notebooks. 69. DECORATIONS You are volunteering for a charity that is operating a haunted house. You are sent to the store to buy 5 bags of cotton balls that will be used to make spider web decorations. Each bag costs $2.09. Use the distributive property to mentally calculate the total cost of the cotton balls. 70. GROCERIES You see a sign at the grocery store that reads, “Buy 2 half-gallons of frozen yogurt and get one free.” Each half-gallon of frozen yogurt costs $4.95. You decide to get three half-gallons. Use the distributive property to mentally calculate the total cost of the frozen yogurt. AGRONOMISTS use soil and plant science to help farmers increase crop production and maintain soil fertility. FARMING You are trying to determine the size of a cornfield, so you will know how many rows of corn to plant. Let x be the width of the cornfield. Use the diagram of the farming field shown. 71. Use the diagram to find two expressions for the area of the entire field. corn x yd soybeans 225 yd 72. Write an algebraic statement that shows the two expressions from Exercise 71 are equal. 73. You decide to plant the cornfield so that x 75 yards. What is the area of the entire field? Use one expression from Exercise 71 to find the solution and the other to check your solution. 200 yd Standardized Test Practice 74. MULTIPLE CHOICE Which expression is equivalent to (x 7)3? A C B D x 21 3x 10 3x 7 3x 21 75. MULTIPLE CHOICE Which expression is equivalent to 6(x 2)? F H G J 6x 2 6x 2 6x 12 6x 12 76. MULTIPLE CHOICE Which expression is equivalent to (5 x)(17)? A C B D 5 17x 85 17x 5 17x 85 17x 77. MULTIPLE CHOICE You are buying 5 new shirts to wear to school. All of the shirts are on sale for $19.99. What expression would you use to mentally calculate the total cost of the shirts? F 5(20) 0.01 G 5(20 0.01) H 2.6 5(20) 0.01 J 5(20 0.01) The Distributive Property 105 Page 7 of 7 Mixed Review EXPRESSIONS WITH FRACTION BARS Evaluate the expression. Then simplify the answer. (Lesson 1.3) 10 p 8 78. 42 p 4 62 12 79. 32 1 75 52 80. 13 3 p 4 3p79 81. 24 5 11 4 p 2 53 82. 32 2 6 72 83. 3 3 97 NAMING PROPERTIES Name the property shown by the statement. (Lesson 2.3) 84. 10 (25) 25 (10) 85. 19 0 19 86. 32 (32) 0 87. (13 8) 7 13 (8 7) EVALUATING EXPRESSIONS Evaluate the expression. (Lesson 2.4) Maintaining Skills 88. 6 7 89. 9 (3) 90. 4 8 3 91. 6 (8) 11 92. 7.2 9 8.5 1 2 93. 1 3 3 DECIMALS AND FRACTIONS Write the decimal as a fraction in simplest form. (Skills Review p. 767) 94. 0.14 95. 0.25 96. 0.34 97. 0.50 98. 0.75 99. 0.82 100. 0.90 101. 0.96 Quiz 2 Evaluate the function when x 3, 1, 1, and 3. Organize your results in a table. (Lesson 2.4) 1 1. y x 12 2. y 27 x 3. y x 4 Find the terms of the expression. (Lesson 2.4) 4. 2x 9 5. 8 x 6. 10x 4 7. STOCK MARKET The daily closing prices for a company’s stock are $19.63, $19.88, $20.00, $19.88, and $19.75. Find the day-to-day change in the closing price. (Lesson 2.4) Find the product. (Lesson 2.5) 8. 7(9) 11. 1.8(6) 9. 5(6) 1 12. 15 5 10. 35(80) 13. 10(3)(9) Rewrite the expression without parentheses. (Lesson 2.6) 14. (x 2)11 15. 5(12 y) 16. 4(3a 4) 17. SHOPPING You want to buy 2 pairs of jeans for $24.95 each. Use the distributive property to mentally calculate the total cost of the jeans. (Lesson 2.6) 106 Chapter 2 Properties of Real Numbers
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