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1-4 Solving Absolute Value Equations Evaluate each expression if x = –4 and y = –9. 2. SOLUTION: Substitute –9 for y and solve. 4. SOLUTION: Substitute -4 for x and solve. Solve each equation. Check your solutions. 6. SOLUTION: There appear to be two solutions, 4 and −20. Check: Substitute each value in the original equation. The solution set is . 8. SOLUTION: eSolutions Manual - Powered by Cognero Page 1 1-4 Solving Absolute The solution set is Value Equations . 8. SOLUTION: There appear to be two solutions, 10 and 0. Check: Substitute each value in the original equation. The solution set is . 10. SOLUTION: There appear to be two solutions, 3 and 0. Check: Substitute each value in the original equation. eSolutions Manual - Powered by Cognero Page 2 1-4 Solving Absolute Value Equations The solution set is . 10. SOLUTION: There appear to be two solutions, 3 and 0. Check: Substitute each value in the original equation. The solution set is . 12. SOLUTION: eSolutions Manual - Powered by Cognero Page 3 1-4 Solving Absolute Value Equations The solution set is . 12. SOLUTION: There appear to be two solutions, . Check: Substitute each value in the original equation. Since 3 ≠ –3, a = 1 is an extraneous solution. The solution is . Evaluate each expression if a = –3, b = –5, and c = 4.2. 14. SOLUTION: Substitute 4.2 for c and solve. eSolutions Manual - Powered by Cognero 16. Page 4 1-4 Solving Absolute Equationssolution. The solution is Since 3 ≠ –3, a = 1Value is an extraneous . Evaluate each expression if a = –3, b = –5, and c = 4.2. 14. SOLUTION: Substitute 4.2 for c and solve. 16. SOLUTION: Substitute –3 for a and –5 for b and solve. 18. SOLUTION: Substitute –3 for a and –5 for b and solve. 20. SOLUTION: Substitute –3 for a and 4.2 for c and solve. 22. FOOD To make cocoa powder, cocoa beans are roasted. The ideal temperature for roasting is 300°F, plus or minus 25°. Write and solve an equation describing the maximum and minimum roasting temperatures for cocoa beans. SOLUTION: Solve the equation . eSolutions Manual - Powered by Cognero So, the maximum temperature is 325°F and the minimum temperature is 275°F. Page 5 1-4 Solving Absolute Value Equations 22. FOOD To make cocoa powder, cocoa beans are roasted. The ideal temperature for roasting is 300°F, plus or minus 25°. Write and solve an equation describing the maximum and minimum roasting temperatures for cocoa beans. SOLUTION: Solve the equation . So, the maximum temperature is 325°F and the minimum temperature is 275°F. Solve each equation. Check your solutions. 24. SOLUTION: There appear to be two solutions, 8 and –26. The solution set is . 26. SOLUTION: There appear to be two solutions, 41 and –29. Check: Substitute each value in the original equation. eSolutions Manual - Powered by Cognero Page 6 1-4 Solving Absolute The solution set is Value Equations . 26. SOLUTION: There appear to be two solutions, 41 and –29. Check: Substitute each value in the original equation. The solution set is . 28. SOLUTION: There appear to be two solutions, 3 and –11. Check: Substitute each value in the original equation. The solution set is . 30. eSolutions Manual - Powered by Cognero SOLUTION: Page 7 1-4 Solving Absolute Value Equations The solution set is . 30. SOLUTION: Check: The solution is . 32. SOLUTION: eSolutions Manual - Powered by Cognero Page 8 1-4 Solving Absolute Value Equations There appear to be two solutions, . Check: Substitute each value in the original equation. The solution set is . 34. SOLUTION: eSolutions Manual - Powered by Cognero Page 9 The solution set is . 34. Solving Absolute Value Equations 1-4 SOLUTION: There appear to be two solutions, . Check: Substitute the values in the original equation. z= z= eSolutions Manual - Powered by Cognero Page 10 1-4 Solving Absolute Value Equations z= The solution set is . Evaluate each expression if q = –8, r = –6, and t = 3. 36. SOLUTION: Substitute –6 for r and 3 for t and solve. 38. SOLUTION: Substitute -8 for q, -6 for r, and 3 for t and solve. Solve each equation. Check your solutions. 40. SOLUTION: eSolutions Manual - Powered by Cognero Page 11 Solve each equation. Check your solutions. 40. 1-4 Solving Absolute Value Equations SOLUTION: There appear to be two solutions, . Check: Substitute the values in the original equation. y= . eSolutions Manual - Powered by Cognero y = 8. Page 12 There appear to be two solutions, . Check: Substitute the values in the original equation. 1-4 Solving Absolute Value Equations y= . y = 8. Since −44 ≠ 60, y = 8 is an extraneous solution. The solution set is . 42. SOLUTION: eSolutions Manual - Powered by Cognero Page 13 1-4 Solving Absolute Value Equations There appear to be two solutions, . Check: Substitute the values in the original equation. y= . y= . The solution set is . 44. MULTIPLE REPRESENTATIONS Draw a number line. a. GEOMETRIC Label any 5 integers on the number line points A, B, C, D, and F. b. TABULAR Fill in each blank in the table with either > or < using the points from the number line. eSolutions Manual - Powered by Cognero Page 14 1-4 Solving Absolute The solution set is Value Equations . 44. MULTIPLE REPRESENTATIONS Draw a number line. a. GEOMETRIC Label any 5 integers on the number line points A, B, C, D, and F. b. TABULAR Fill in each blank in the table with either > or < using the points from the number line. c. VERBAL Describe the patterns in the table. d. ALGEBRAIC Describe the patterns algebraically, using the variable x to replace C, D, and F. SOLUTION: a. Draw a number line and label from -3 to 3. Place points A, B, C, D, and F on the number line. Sample answer: b. Fill in the table based on the number line drawn in part a. Since A is -2 and B is -1. A < B. Since C is 0, A + C = 2 + 0 or -2. B + C = -1 + 0 or -1. Therefore, A + C < B + C. Similar reasoning is used to complete the table. c. Sample answer: If A is less than B, then any number added to or subtracted from A will be less than the same number added to or subtracted from B. If B is greater than A, then any number added to or subtracted from B is greater than the same number added to or subtracted from A. d. If A < B, then A + x < B + x. If A < B, then A – x < B – x. If B > A, then B + x > A + x. If B > A, then B – x > A – x. 46. CHALLENGE Solve cases to examine as potential solutions.) List all cases and resulting equations. (Hint: There are four possible SOLUTION: The 4 potential solutions are: 1. (2x − 1) ≥ 0 and (5 − x) ≥ 0 2. (2x − 1) ≥ 0 and (5 − x) < 0 3. (2x − 1) < 0 and (5 − x) ≥ 0 4. (2x − 1) < 0 and (5 − x) < 0 The resulting equations corresponding to these cases are: 1. 2x – 1 + 3 = 5 – x : x = 1 2. 2x – 1 + 3 = x – 5 : x = –7 3. 1 – 2x + 3 = 5 – x : x = –1 4. 1 – 2x + 3 = x – 5 : x = 3 eSolutions Manual - Powered by Cognero The solutions from case 1 and case 3 work. The others are extraneous. The solution set is {–1, 1}. Page 15 REASONING If a, x, and y are real numbers, determine whether each statement is sometimes, always, or number added to or subtracted from B. If B is greater than A, then any number added to or subtracted from B is greater than the same number added to or subtracted from A. B, then AValue + x < Equations B + x. If A < B, then A – x < B – x. d. If A <Absolute 1-4 Solving If B > A, then B + x > A + x. If B > A, then B – x > A – x. 46. CHALLENGE Solve cases to examine as potential solutions.) List all cases and resulting equations. (Hint: There are four possible SOLUTION: The 4 potential solutions are: 1. (2x − 1) ≥ 0 and (5 − x) ≥ 0 2. (2x − 1) ≥ 0 and (5 − x) < 0 3. (2x − 1) < 0 and (5 − x) ≥ 0 4. (2x − 1) < 0 and (5 − x) < 0 The resulting equations corresponding to these cases are: 1. 2x – 1 + 3 = 5 – x : x = 1 2. 2x – 1 + 3 = x – 5 : x = –7 3. 1 – 2x + 3 = 5 – x : x = –1 4. 1 – 2x + 3 = x – 5 : x = 3 The solutions from case 1 and case 3 work. The others are extraneous. The solution set is {–1, 1}. REASONING If a, x, and y are real numbers, determine whether each statement is sometimes, always, or never true. Explain your reasoning. 48. If then SOLUTION: , then x is between –3 or 3. Adding 3 to the absolute value of any of the numbers in this set will Always; if produce a positive number. 50. OPEN ENDED Write an absolute value equation of the form that has no solution. Assume that a, b, c, and SOLUTION: Sample answer: An absolute value expression cannot be negative. Solve the equation and check the solutions. 52. If 4x – y = 3 and 2x + 3y = 19, what is the value of y? A2 B3 C4 D5 SOLUTION: Substitute in the equation 2x + 3y = 19. eSolutions Manual - Powered by Cognero Page 16 SOLUTION: Sample answer: An absolute value expression cannot be negative. Solve the equation and check the solutions. 1-4 Solving Absolute Value Equations 52. If 4x – y = 3 and 2x + 3y = 19, what is the value of y? A2 B3 C4 D5 SOLUTION: Substitute in the equation 2x + 3y = 19. So, the correct choice is D. 54. Which equation is equivalent to 4(9 – 3x) = 7 – 2(6 – 5x)? F 8x = 41 G 22x = 41 H 8x = 24 J 22x = 24 SOLUTION: eSolutions Manual - Powered by Cognero Page 17 1-4 Solving Absolute Value Equations So, the correct choice is D. 54. Which equation is equivalent to 4(9 – 3x) = 7 – 2(6 – 5x)? F 8x = 41 G 22x = 41 H 8x = 24 J 22x = 24 SOLUTION: Therefore, 22x = 41 is equivalent to 4(9 – 3x) = 7 – 2(6 – 5x). So, the correct choice is G. Solve each equation. Check your solution. 56. SOLUTION: Check: Substitute x = 6 in the original equation. The solution is x = 6. 58. SOLUTION: eSolutions Manual - Powered by Cognero Page 18 1-4 Solving Absolute Equations The solution is x = Value 6. 58. SOLUTION: Check: Substitute y = 50 in the original equation. The solution is y = 50. Name the property illustrated by each equation. 60. (1 + 8) + 11 = 11 + (1 + 8) SOLUTION: Commutative Property of Addition; the Commutative Property of Addition states that the order in which terms are added does not affect the sum. Simplify each expression. 62. 7a + 3b – 4a – 5b SOLUTION: eSolutions Manual - Powered by Cognero Page 19 60. (1 + 8) + 11 = 11 + (1 + 8) SOLUTION: Commutative Property of Addition; the Commutative Property of Addition states that the order in which terms are 1-4 Solving Absolute Value added does not affect the Equations sum. Simplify each expression. 62. 7a + 3b – 4a – 5b SOLUTION: 64. 3(15x – 9y) + 5(4y – x) SOLUTION: 66. 8(r + 7t) – 4(13t + 5r) SOLUTION: 68. GEOMETRY The formula for the surface area of a rectangular prism is where represents the length, w represents the width, and h represents the height. Find the surface area of the rectangular prism at the right. SOLUTION: Substitute in the formula . The surface area of the rectangular prism is 358 square inches. eSolutions Manual - Powered by Cognero Solve each equation. 70. 2.4y + 4.6 = 20 Page 20 1-4 Solving Absolute Value Equations 68. GEOMETRY The formula for the surface area of a rectangular prism is where represents the length, w represents the width, and h represents the height. Find the surface area of the rectangular prism at the right. SOLUTION: Substitute in the formula . The surface area of the rectangular prism is 358 square inches. Solve each equation. 70. 2.4y + 4.6 = 20 SOLUTION: 72. 3(w – 1) = 2w – 6 SOLUTION: 74. SOLUTION: eSolutions Manual - Powered by Cognero Page 21 1-4 Solving Absolute Value Equations 74. SOLUTION: eSolutions Manual - Powered by Cognero Page 22