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Using Variables

Learn how to write algebraic expressions and equations for various operations and amounts in this lesson on using variables in algebra.

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Using Variables

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  1. Using Variables ALGEBRA 1 LESSON 1-1 Write the operation (+, –, , ÷) that corresponds to each phrase. 1. divided by 2. difference 3. more than 4. product 5. minus 6. sum 7. multiplied by 8. quotient Find each amount. 9. 12 more than 9 10. 8 less than 13 11. 16 divided by 4 12. twice 25 1-1

  2. Using Variables ALGEBRA 1 LESSON 1-1 1. divided by: ÷ 2. difference: – 3. more than: + 4. product:  5. minus: – 6. sum: + 7. multiplied by: 8. quotient: ÷ 9. 12 more than 9: 9 + 12 = 21 10. 8 less than 13: 13 – 8 = 5 11. 16 divided by 4: 16 ÷ 4 = 4 12. twice 25: 2(25) = 50 Solutions 1-1

  3. Using Variables ALGEBRA 1 LESSON 1-1 Write an algebraic expression for “the sum of n and 8.” “Sum” indicates addition. Add the first number, n, and the second number, 8. n + 8 1-1

  4. Relate: ten more than twice a number Define: Let y = the number. Write: 2 • y + 10 Using Variables ALGEBRA 1 LESSON 1-1 Define a variable and write an algebraic expression for “ten more than twice a number.” 2y + 10 1-1

  5. Relate: The total income is 5 times the number of tickets sold. Define: Let t = the number of tickets sold. Let i = the total income Write: i = 5 • t Using Variables ALGEBRA 1 LESSON 1-1 Write an equation to show the total income from selling tickets to a school play for $5 each. i = 5t 1-1

  6. Gallons 4 6 8 10 Miles 80 120 160 200 Relate: Miles traveled equals 20 times the number of gallons. Define: Let m = the number of miles traveled. Let g = the number of gallons. Write:        m = 20 • g Using Variables ALGEBRA 1 LESSON 1-1 Write an equation for the data in the table. m = 20g 1-1

  7. n 82 Using Variables ALGEBRA 1 LESSON 1-1 pages 6–8  Exercises 19. = total length in feet, n = number of tents, = 60n 20. = number of slices left, e = number of slices eaten, = 8 – e 11. 9 – n 12. 13. 5n 14. 13 + 2n 15. 16. 17.c = total cost, n = number of cans, c = 0.70n 18. p = perimeter, s = length of a side, p = 4s 1.p + 4 2. y – 12 3. 12 – m 4. 15c 5. 6. 7.x – 23 8. v + 3 9–16.Choice of variable may vary. 9. 2n + 2 10. n – 11 n 6 n 8 11 n 17 k 1-1

  8. 25. 9 + k – 17 26. 5n + 6.7 27. 37t – 9.85 28. 29. 15 + 30. 7 – 3v 31. 5m – 32. + 33. 8 – 9r 3b 4.5 60 w t 7 p 14 q 3 Using Variables ALGEBRA 1 LESSON 1-1 21–24.Choices of variables may vary. Samples are given. 21.w = number of workers, r = number of radios, r = 13w 22.n = number of tapes, c = cost, c = 8.5n 23.n = number of sales, t = total earnings, t = 0.4n 24.n = number of hours, p = pay, p = 8n 34–38.Answers may vary. Samples are given. 34. 5 more than q 35. the difference of 3 and t 36. one more than the product of 9 and n 37. the quotient of y and 5 38. the product of 7 times h and b 1-1

  9. 39–40.Choices of variables may vary. Samples are given. 39.n = number of days, c = change in height (m), c = 0.165n 40.t = time in months, = length in inches, = 4.1t 41. a. i. yes; 6 = 3 • 2 ii. yes; 6 = 3 + 3 b. Answers may vary. Sample: i; it makes sense that an equation relating lawns mowed and hours worked would be a multiple of the number of lawns mowed. 42–44.  Choices of variables may vary. Samples are given. 42.a. Let a = the amount in dollars and n = the number of quarters, a = 0.25q. b. $3.25 43.a.d = drop height (ft), f = height of first bounce (ft), f = d b. 10 ft 44.s = height of second bounce (ft), s = d 1 2 1 4 Using Variables ALGEBRA 1 LESSON 1-1 1-1

  10. Using Variables ALGEBRA 1 LESSON 1-1 45–47.Answers may vary. Samples are given. 45. You jog at a rate of 5 miles per hour. How far do you jog in 2 hours? Let d = distance in miles and t = time in hours. 46. Anabel is three years older than her brother Barry. How old will Anabel be when Barry is 12? Let a = Anabel’s age in years and b = Barry’s age in years. 47. The Merkurs have budgeted $40 for a baby sitter. What hourly rate can they afford to pay if they need the sitter for 5 hours? Let h = the number of hours and c the cost per hour. 48. D 49. H 50. A 51. G 52. D 53. F 54. 0.9 55. 1.04 56. 1.35 57. 1.46 58. 0.63 59. 0.09 60. 0.14 61. 0.93 62. 1 63. 0.23 64. 3 65. 0.11 66. any four of 23, 29, 31, 37, 41, 43, and 47 1-1

  11. Using Variables ALGEBRA 1 LESSON 1-1 Write an algebraic expression for each phrase. 1. 7 less than 9 2. the product of 8 and p 3. 4 more than twice c Define variables and write an equation to model each situation. 4. The total cost is the number of sandwiches times $3.50. 5. The perimeter of a regular hexagon is 6 times the length of one side. 9 – 7 8p 2c + 4 Let c = the total cost and s = the number of sandwiches; c = 3.5s Let p = the perimeter and s = the length of a side; p = 6s 1-1

  12. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Find each product. 1. 4 • 4 2. 7 • 7 3. 5 • 5 4. 9 • 9 Perform the indicated operations. 5. 3 + 12 – 7 6. 6 • 1 ÷ 2 7. 4 – 2 + 9 8. 10 – 5 – 4 9. 5 • 5 + 7 10. 30 ÷ 6 • 2 1-2

  13. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Solutions 1. 4 • 4 = 16 2. 7 • 7 = 49 3. 5 • 5 = 25 4. 9 • 9 = 81 5. 3 + 12 – 7 = (3 + 12) – 7 = 15 – 7 = 8 6. 6 • 1 ÷ 2 = (6 • 1) ÷ 2 = 6 ÷ 2 = 3 7. 4 – 2 + 9 = (4 – 2) + 9 = 2 + 9 = 11 8. 10 – 5 – 4 = (10 – 5) – 4 = 5 – 4 = 1 9. 5 • 5 + 7 = (5 • 5) + 7 = 25 + 7 = 32 10. 30 ÷ 6 • 2 = (30 ÷ 6) • 2 = 5 • 2 = 10 1-2

  14. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify 32 + 62 – 14 • 3. 32 + 62 – 14 • 3 = 32 + 36 – 14 • 3 Simplify the power: 62 = 6 • 6 = 36. = 32 + 36 – 42Multiply 14 and 3. = 68 – 42 Add and subtract in order from left to right. = 26 Subtract. 1-2

  15. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Evaluate 5x = 32 ÷ p for x = 2 and p = 3. 5x + 32 ÷ p = 5 • 2 + 32 ÷ 3Substitute 2 for x and 3 for p. = 5 • 2 + 9 ÷ 3 Simplify the power. = 10 + 3Multiply and divide from left to right. = 13 Add. 1-2

  16. total cost      original price      sales tax C = p + r • p sales tax rate Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Find the total cost of a pair of jeans that cost $32 and have an 8% sales tax. C = p + r • p = 32 + 0.08 • 32Substitute 32 for p. Change 8% to 0.08 and substitute 0.08 for r. = 32 + 2.56 Multiply first. = 34.56 Then add. The total cost of the jeans is $34.56. 1-2

  17. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify 3(8 + 6) ÷ (42 – 10). 3(8 + 6) ÷ (42 – 10) = 3(8 + 6) ÷ (16 – 10) Simplify the power. = 3(14) ÷ 6 Simplify within parentheses. = 42 ÷ 6 Multiply and divide from left to right. = 7 Divide. 1-2

  18. (xz)2 = (11 • 16)2 xz2 = 11 • 162 Substitute 11 for x and 16 for z. = 11 • 256 = (176)2 Simplify within parentheses. Multiply. = 30,976 Simplify. = 2816 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Evaluate each expression for x = 11 and z = 16. a. (xz)2 b.xz2 1-2

  19. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify 4[(2 • 9) + (15 ÷ 3)2]. 4[(2 • 9) + (15 ÷ 3)2] = 4[18 + (5)2] First simplify (2 • 9) and (15 ÷ 3). = 4[18 + 25] Simplify the power. = 4[43] Add within brackets. = 172 Multiply. 1-2

  20. pa 2 Use the formula A = 3 () to find the total area of all three decks. Substitute 60 for p and 8.7 for a. pa 2 A = 3 () 60 • 8.7 2 Simplify the numerator. = 3 () = 3(261) Simplify the fraction. 522 2 = 3 () = 783 Multiply. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 A carpenter wants to build three decks in the shape of regular hexagons. The perimeter p of each deck will be 60 ft. The perpendicular distance a from the center of each deck to one of the sides will be 8.7 ft. The total area of all three decks is 783 ft2. 1-2

  21. 7 15 5 8 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 pages 12–15  Exercises 1. 59 2. 22 3. 7 4. 113 5. 32 6. 29 7. 21 8. 27 9. 124 10. 15 11. 180 23. 185 24. 361 25. 57 26. 9 27. 1936 28. 3872 29. 18 30. 186 31. 0 32. 7 33. 81 34. 36 35. 8 cm3 36. 91 in.3 37. 21 ft3 38. 28 cm3 39. 15,000 ft3 40. 2596 m3 41. 15 42. 7 43. 111 44. 1 45. 51 46. 50.4 47. 242 48. 61 49. 71.6 50. 58 51. 52. 7 53. 1 54. 7 12. 169 13. $37.09 14. $347.10 15. 22 16. 3 17. 44 18. 5 19. 16 20. 4 21. 704 22. 7744 1 3 5 6 1-2

  22. 3 4 1 3 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 55.a. left side = 1 right side = 1 b. left side = 4 right side = 2 c. Answers may vary. Sample: For a = 2 and b = 3, left side = 25, right side = 13 d. No; as seen in part (b), (a + b)2 = a2 + b2 is not true for all values of a and b. 56. 17 57. 9 58. 143 59. 135 60. 143 61. 27 62. 14 63. 308 64.a. 135 in.2 b. The frame is 2 in. wide, so it adds 2 in. on each side. 65. $.16 66. 407.72 cm3 67.a. 523.60 cm3 b. 381.70 cm3 c. about 73% 68. 38 69. 127 70. 9 71. 10 72. 10 73. a. 23.89 in.3 b. 2.0 in.3 c. 47.38 in.2 1-2

  23. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 74. Yes; the rules for simplifying are designed to produce exactly one result. 75. (10 + 6) ÷ 2 – 3 = 5 76. 14 – (2 + 5) – 3 = 4 77. (32 + 9) ÷ 9 = 2 78. (6 – 4) ÷ 2 = 1 79.a. 22; 22 b. No; part (a) shows the value is unaffected for the given numbers. 80.a. 16, 2 b. Yes; part (a) shows that the placement of parentheses can affect the value of the expression, when both add. and subtr. are involved. 5 + 4 – 1 • 2 = 7 25 ÷ (4 • 1) = 8 25 ÷ 4 + 1 = 9 42 – (1 + 5) = 10 (42 – 5) ÷ 1 = 11 1 + 2 + 4 + 5 = 12 2(4 – 1) + 5 = 13 2 • 5 + 1 • 4 = 14 5(4 – 12) = 15 2(1 + 5) + 4 = 16 5 + 4(2 + 1) = 17 4(5 – 1) + 2 = 18 4 • 5 – 12 = 19 42 + 5 – 1 = 20 81. Answers may vary. Samples: 2(4 – 1) – 5 = 1 5 + 2 – (1 + 4) = 2 (24 – 1) ÷ 5 = 3 1 + 2 + 5 – 4 = 4 2 • 5 – (1 + 4) = 5 (52 – 1) ÷ 4 = 6 1-2

  24. / / 2(b1 + b2) 2 2(b1 + b2) 2 2b1 + b2 2 b1 + b2 2 b1 + b2 2 b1 + b2 2 b1 + b2 2 y 5 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 82.a. 38 ft2 b. Yes; 2h = 2 h . No; h = 2h since b2 = 0. Yes; h = h = 2 h . 95. 95% 96. 145% 97. 6% 98. 43 99. 18.9 100. 1.25 101. 60.3 105. 60, 150, 240 106. 26, 39, 52 102–106.Answers may vary. Samples are given. 102. 8, 24, 56 103. 55, 100, 250 104. 44, 66, 121 83. D 84. F 85. B 86. H 87. B 88. F 89.c + 2 90. 36m 91.t – 21 92. 93. 50% 94. 34% 1-2

  25. Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify each expression. 1. 50 – 4 • 3 + 6 2. 3(6 + 22) – 5 3. 2[(1 + 5)2 – (18 ÷ 3)] Evaluate each expression. 4. 4x + 3y for x = 2 and y = 4 5. 2 • p2 + 3s for p = 3 and s = 11 6.xy2 + z for x = 3, y = 6 and z = 4 44 25 60 20 51 112 1-2

  26. 2 5 3 8 2 3 5 9 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 (For help, go to skills handbook page 725.) Write each decimal as a fraction and each fraction as a decimal. 1. 0.5 2. 0.05 3. 3.25 4. 0.325 5.6.7.8. 3 1-3

  27. 1. 0.5 = = = 2. 0.05 = = = 3. 3.25 = 3 = 3 = 3 or 4. 0.325 = = = 5. = 2 ÷ 5 = 0.4 6. = 3 ÷ 8 = 0.375 7. = 2 ÷ 3 = 0.6 8. 3 = 3 + (5 ÷ 9) = 3.5 Solutions 5 10 1 2 5 • 1 5 • 2 5 • 1 5 • 20 5 100 1 20 25 • 1 25 • 4 1 4 25 100 13 4 325 1000 13 40 25 • 13 25 • 40 2 5 3 8 2 3 5 9 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 1-3

  28. Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Name the set(s) of numbers to which each number belongs. a. –13 b. 3.28 integers rational numbers rational numbers 1-3

  29. Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Which set of numbers is most reasonable for displaying outdoor temperatures? integers 1-3

  30. 2 3 A negative number can be a fraction, such as – . This is not an integer. Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Determine whether the statement is true or false. If it is false, give a counterexample. All negative numbers are integers. The statement is false. 1-3

  31. 3 4 7 12 5 8 – = –0.75 Write each fraction as a decimal. – = –0.583 – = –0.625 3 4 7 12 5 8 –0.75 < –0.625 < –0.583 Order the decimals from least to greatest. 3 4 5 8 7 12 From least to greatest, the fractions are – , – , and – . Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Write – , – , and – , in order from least to greatest. 1-3

  32. Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Find each absolute value. a. |–2.5| b. |7| –2.5 is 2.5 units from 0 on a number line. 7 is 7 units from 0 on a number line. |–2.5| = 2.5 |7| = 7 1-3

  33. 2 3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 pages 20–23  Exercises 1. integers, rational numbers 2. rational numbers 3. rational numbers 4. natural numbers, whole numbers, integers, rational numbers 5. rational numbers 6. integers, rational numbers 7. whole numbers, integers, rational numbers 8. rational numbers 9. rational numbers 10. irrational numbers 11. Answers may vary. Sample: –17 12. Answers may vary. Sample: 53 13. Answers may vary. Sample: 0.3 14. rational numbers 15. whole numbers 16. integers 17. whole numbers 18. rational numbers 19. true 20. False; answers may vary. Sample: – 21. False; answers may vary. Sample: 6 22. true 23. False; answers may vary. Sample: –6 < | –6 | 24. > 25. < 26. > 1-3

  34. 27. = 28. 2.001, 2.01, 2.1 29. –9 , –9 , –9 30. – , – , 31. –1.01, –1.001, –1.0009 32. 0.63, 0.636, 33. , 0.8888, 34. 4 35. 9 36. 37. 0.5 38. 39. 0 40. 1295 41. 42. Answers may vary. Sample: 43. Answers may vary. Sample: 44. Answers may vary. Sample: 45. Answers may vary. Sample: 46. Answers may vary. Sample: 47. natural numbers, whole numbers, integers, rational numbers 48. natural numbers, whole numbers, integers, rational numbers 49. rational numbers 50. rational numbers – 4 1 3 4 2 3 7 12 4 5 5 6 1 2 2 3 1 5 7 11 5 1 22 25 8 9 213 10 9 14 1034 1000 3 5 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 1-3

  35. 63.a. irrational b. No; has no final digit. 64. 0 65. Neg.; 0 is the point between R and S since the coordinates of Q and T are opposites, so if R is to the left of 0, then R is neg. 66.Q; 0 is the point to the right of S because the coordinates of R and T are opposites, therefore the point Q is the farthest from 0, so it has the greatest absolute value. Exploring Real Numbers ALGEBRA 1 LESSON 1-3 51. = 52. > 53. < 54. > 55. = 56. > 57. 6 58. 9 59.a 60. 28 61. 48 62. 5 67. Answers may vary. Sample: 25, |25| + | –25| = 50 68. sometimes 69. sometimes 70. sometimes 71. always 72. Yes; all can be expressed as ratios of themselves to 1. 73. –6 74. 17 75. 1 76. 10 1-3

  36. g cm3 g cm3 g cm3 g cm3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 92. 25 93. 4 94. 195 95–98.Choices of variables may vary. 95.n = number of tickets, 6.25n 96.i = cost of item, i + 3.98 97.n = number of hours, d = distance traveled, d = 7n 98.c = total cost, n = number of books, c = 3.5n 77.a. 2.75 , 19.3 , 10.5 , 3.5 b. aluminum, diamond, silver, gold 78. Answers may vary. Samples are given. a. –2.2 b. –2.81 c. –2 d. Yes; find the average of the two given numbers. 79. A 80. G 81. D 82. G 83. C 84. I 85. C 86. 33 87. 48 88. 315 89. 7 90. 0 91. 7 1 8 1 2 1-3

  37. 3 4 5 8 7 12 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Name the set(s) of numbers to which each given number belongs. 1. –2.7 2. 11 3. 16 Use <, =, or > to compare. 4.5. 6. Find |– |. rational numbers irrational numbers natural numbers, whole numbers integers, rational numbers 3 4 5 8 > < – – 7 12 1-3

  38. Find each sum. 1. 4 + 2 2. 10 + 7 3. 9 + 5 4. 27 + 32 5. 0.4 + 0.9 6. 5.2 + 0 7. 4.1 + 6.8 8. 7.6 + 9.5 9. + 10. + 11. + 12. + 1 5 3 5 4 9 7 9 1 2 3 4 3 8 1 4 Adding Real Numbers ALGEBRA 1 LESSON 1-4 (For help, go to skills handbook page 726.) 1-4

  39. 1. 4 + 2 = 6 2. 10 + 7 = 17 3. 9 + 5 = 14 4. 27 + 32 = 59 5. 0.4 + 0.9 = 1.3 6. 5.2 + 0 = 5.2 7. 4.1 + 6.8 = 10.9 8. 7.6 + 9.5 = 17.1 9. + = = 10. + = = = 1 11. + = + = = = 1 12. + = + = = Solutions 1 5 3 5 1 + 3 5 4 5 2 9 4 9 7 9 4 + 7 9 11 9 1 2 3 4 2 4 3 4 2 + 3 4 5 4 1 4 3 8 1 4 3 8 2 8 3 + 2 8 5 8 Adding Real Numbers ALGEBRA 1 LESSON 1-4 1-4

  40. Start at –3. Start at 3. Move right 5 units. Move left 5 units. Start at –3. Move left 5 units. Adding Real Numbers ALGEBRA 1 LESSON 1-4 Use a number line to simplify each expression. Use a number line to simplify each expression. a. 3 + (–5) b. –3 + 5 3 + (–5) = –2 –3 + 5 = 2 c. –3 + (–5) –3 + (–5) = –8 1-4

  41. Adding Real Numbers ALGEBRA 1 LESSON 1-4 Simplify each expression. a. 12 + (–23) = –11 b. –6.4 + (–8.6) = –15.0 The difference of the absolute values is 11. Since both addends are negative, add their absolute values. The negative addend has the greater absolute value, so the sum is negative. The sum is negative. 1-4

  42. Adding Real Numbers ALGEBRA 1 LESSON 1-4 The water level in a lake rose 6 inches and then fell 11 inches. Write an addition statement to find the total change in water level. 6 + (–11) = –5   The water level fell 5 inches. 1-4

  43. Adding Real Numbers ALGEBRA 1 LESSON 1-4 Evaluate 3.6 + (–t) for t = –1.7. 3.6 + (–t) = 3.6 + [–(–1.7)] Substitute –1.7 for t. = 3.6 + [1.7] –(–1.7) means the opposite of –1.7, which is 1.7. = 5.3 Simplify. 1-4

  44. Relate: 88 ft below sea level plus  feet diver rises   Define: Let r = the number of feet the diver rises. Write: –88 + r –88 + r = –88 + 37Substitute 37 for r. = – 51 Simplify. Adding Real Numbers ALGEBRA 1 LESSON 1-4 A scuba diver who is 88 ft below sea level begins to ascend to the surface. a.  Write an expression to represent the diver’s depth below sea level after rising any number of feet. –88 + r b.  Find the new depth of the scuba diver after rising 37 ft. The scuba diver is 51 ft below sea level. 1-4

  45. –6 8.6 11 2.3 5 –3 7 –5.4 –2 11.1 3 –1 Add + –68.611 2.3 5 –3 7–5.4–2 11.1 3 –1 + –6 + 78.6 + (–5.4)11 + (–2) 2.3 + 11.1 5 + 3 –3 + (–1) Add corresponding elements. = 1 3.2 9 13.4 8 –4 Simplify. = Adding Real Numbers ALGEBRA 1 LESSON 1-4 1-4

  46. Adding Real Numbers ALGEBRA 1 LESSON 1-4 pages 27–30  Exercises 1. 6 + (–3); 3 2. –1 + (–2); –3 3. –5 + 7; 2 4. 3 + (–4); –1 5. 15 6. –11 7. –19 8. 12.14 9. –4 10. 5 11. –8 1 6 12. –42 13. 2.2 14. –0.65 15. –7.49 16. 1.33 17. 18. – 19. 6 20. –6 21. 0 22. – 23. 5 24. – 25. –47 + 12 = –35, 35 ft below the surface 26. 8 + (–5) = 3, 3 yd gain 27. –6 + 13 = 7, 7°F 28. 8.7 29. –1.7 30. 1.7 31. –8.7 32. 12.6 13 14 14 15 8 9 3 16 1 8 13 18 1-4

  47. 17 60 Adding Real Numbers ALGEBRA 1 LESSON 1-4 38. 39. 40. 41. 42. –2.7 43. –13 44. 6.6 33. –5.6 34. 5.6 35. –12.6 36–37. Choices of variable may vary. 36.c = change in temp., –8 + c a. –1°F b. –11°F c. 11°F 37. c = change in amount of money, 74 + c a. $92 b. $45 c. $27 19 24 –1 1.4 –21 23.2 45. 11 46. 4 47. –3 48. –18.53 49. –20.83 50. –1.72 51. – 52. –5 53. 0.8 54. 4 55. –8.8 22 35 –18.2 11.6 19.1 0 25 –12 1.8 22 – 7 11 120 1 2 1 3 1-4

  48. 65. 5 66. –1 67. 1 68. The sum of –227 and 319; the sum of –227 and 319 is positive, while the sum of 227 and –319 is negative. 69. Answers may vary. Sample: Although 20 and –20 are opposite numbers, there is no such thing as opposite temperatures. 70. –0.3 71. –13.7 72. –0.6 73. 8.7 74. 0.1 75. –1.9 76. +2 56. 13.8 million people 57. 6.3 million people 58. Weaving; add the numbers in each column. 59. a. = b. 0.23 c. about 23% 60. 0 61. –2 62. 1 63. –5 64. 7 100 442 50 221 Adding Real Numbers ALGEBRA 1 LESSON 1-4 1-4

  49. Adding Real Numbers ALGEBRA 1 LESSON 1-4 77. Answers may vary. Sample: 78. The matrices are not the same size, so they can’t be added. 79. No; time and temperature are different quantities and can’t be added. 80.a. 82.a. 4 b. –4 80. (continued) b. c. 4 employees d. 10 employees e. Answers may vary. Sample: Multiply the entries in each column by the appropriate hourly wage, then by 8, and then add all entries to find the total wages. f. $3230 81. $7 2 0 1 –1 3 0.5 13 5 6 2 18 4 2 2 6 2 0 2 8 3 5 1 10 2 2 1 4 1 0 1 5 2 1 1 8 2 0 1 2 1 0 1 1-4

  50. 90. – 91. 92. 93. 94. Pos.; if m is neg., –m is pos. and the sum of two pos. is pos. 95. Neg.; if n is pos., –n is neg. and the sum of two neg. is neg. 96. Pos.; if m is neg., –m is pos. and the sum of two pos. is pos. 97. Zero; sum of neg. and pos. is the difference of the abs. values. |n| = |m| so |n| – |m| = 0. 98. zero; n + (–m) = n + (–n) = 0 x 12 83. 84. 85. 86. – 87. 88. – 89. 1 2 1 4 1 –1 2 1 t 6 1 2 –3m + 1 4 m 9 1 2 8 32 –27 20 6 0 1 2 w 10 c 2 107. < 108. > 109. > 110. = 111. 9 112. 2.2 113. 18 114. 21 99. B 100. F 101. D 102. F 103. C 104. H 105. < 106. = 58a 21 2b 9 x 12 Adding Real Numbers ALGEBRA 1 LESSON 1-4 1-4

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