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hut72632_ch02_C.qxd 2 9/1/05 19:11 Page 203 Summary D EFINITION /P ROCEDURE E XAMPLE R EFERENCE Introduction to Integers Section 2.1 Positive Integers Integers used to name whole numbers to the right of the origin on the number line. p. 122 Negative Integers Integers used to name the opposites of whole numbers. Negatives are found to the left of the origin on the number line. Integers Whole numbers and their opposites. The integers are The origin 3 2 1 0 Negative integers 1 2 3 Positive integers {. . . , 3, 2, 1, 0, 1, 2, 3, . . .} Absolute Value The distance (on the number line) between the point named by a signed number and the origin. The absolute value of x is written x. 7 7 10 10 Addition of Integers Adding Integers 1. If two integers have the same sign, add their absolute values. Give the result the sign of the original integers. 2. If two integers have different signs, subtract their absolute values, the smaller from the larger. Give the result the sign of the integer with the larger absolute value. Section 2.2 9 7 16 (9) (7) 16 Section 2.3 16 8 16 (8) 8 8 15 8 (15) 7 9 (7) 9 7 2 Multiplication of Integers Multiplying Integers Multiply the absolute values of the two integers. 1. If the integers have different signs, the product is negative. © 2007 McGraw-Hill Companies 2. If the integers have the same sign, the product is positive. p. 133–134 15 (10) 5 (12) 9 3 Subtraction of Integers Subtracting Integers 1. Rewrite the subtraction problem as an addition problem by a. Changing the subtraction symbol to an addition symbol b. Replacing the integer being subtracted with its opposite 2. Add the resulting integers as before. p. 124 p. 139 Section 2.4 5(7) 35 (10)(9) 90 8 7 56 (9)(8) 72 (2)2 (2)(2) 4 22 2 2 4 p. 145–146 Continued 203 204 9/1/05 CHAPTER 2 19:11 Page 204 INTEGERS AND INTRODUCTION TO ALGEBRA D EFINITION /P ROCEDURE E XAMPLE Division of Integers Dividing Integers Divide the absolute values of the two integers. 1. If the integers have different signs, the quotient is negative. 2. If the integers have the same sign, the quotient is positive. Section 2.5 32 8 4 18 2 9 Introduction to Algebra: Variables and Expressions Multiplication xy (x)(y) These all mean the product of x and y or x times y. xy The product of m and n is mn. The product of 2 and the sum of a and b is 2(a b). Combining Like Terms To combine like terms: 1. Add or subtract the coefficients (the numbers multiplying the variables). 2. Attach the common variable. Evaluate 2x 3y if x 5 and y 2. 2x 3y Solution A value for a variable that makes an equation a true statement. Section 2.8 3xy is a term The Addition Property of Equality If a b, then a c b c p. 177 p. 179 5x 2x 7x 8a 5a 3a Section 2.9 2x 3 5 is an equation p. 186 4 is a solution for the above equation because 2(4) 3 5 p. 187 The Addition Property of Equality Equivalent Equations Equations that have exactly the same solutions. p. 167 2 5 (3)(2) 10 (6) 4 Introduction to Linear Equations Equation A statement that two expressions are equal. p. 159 Section 2.7 Simplifying Algebraic Expressions Term A number or the product of a number and one or more variables. p. 151 Section 2.6 Evaluating Algebraic Expressions Evaluating Algebraic Expressions To evaluate an algebraic expression: 1. Replace each variable or letter with its number value. 2. Do the necessary arithmetic, following the rules for the order of operations. R EFERENCE Section 2.10 2x 3 5 and x 4 are equivalent equations p. 192 If 2x 3 7, then 2x 3 3 7 3 p. 192 © 2007 McGraw-Hill Companies hut72632_ch02_C.qxd hut72632_ch02_C.qxd 9/20/05 16:18 Page 205 Summary Exercises This summary exercise set is provided to give you practice with each of the objectives of this chapter. Each exercise is keyed to the appropriate chapter section. When you are finished, you can check your answers to the odd-numbered exercises against those presented in the back of the text. If you have difficulty with any of these questions, go back and reread the examples from that section. The answers to the even-numbered exercises appear in the Instructor’s Solutions Manual. Your instructor will give you guidelines on how to best use these exercises in your instructional setting. [2.1] Represent the integers on the number line shown. 1. 6, 18, 3, 2, 15, 9 20 10 0 10 20 Place each of the groups of integers in ascending order. 2. 4, 3, 6, 7, 0, 1, 2 3. 7, 8, 8, 1, 2, 3, 3, 0, 7 Find the opposite of each number. 5. 63 4. 17 Evaluate. 6. 9 7. 9 8. 9 9. 9 10. 12 8 11. 812 12. 8 12 13. 18 12 14. 73 15. 95 © 2007 McGraw-Hill Companies [2.2] Add. 16. 3 (8) 17. 10 (4) 18. 6 (6) 19. 16 (16) 20. 18 0 [2.3] Subtract. 21. 8 13 22. 7 10 23. 10 (7) 24. 5 (1) 25. 9 (9) 26. 0 (2) 205 hut72632_ch02_C.qxd 206 9/1/05 CHAPTER 2 [2.4] 19:11 Page 206 INTEGERS AND INTRODUCTION TO ALGEBRA Multiply. 27. (10)(7) 28. (8)(5) 29. (3)(15) 30. (1)(15) 31. (0)(8) Evaluate each of the expressions. 32. 18 3 5 33. (18 3) 5 34. 5 42 35. (5 4)2 36. 5 32 4 37. 5(32 4) 38. 5(4 2)2 39. 5 4 22 40. (5 4 2)2 41. 3(5 2)2 42. 3 5 22 43. (3 5 2)2 [2.5] Divide. 44. 80 16 45. 63 7 46. 81 9 47. 0 5 48. 32 8 49. 7 0 51. 6 1 5 (2) 50. 8 6 8 (10) 52. 25 4 5 (2) [2.6] Write, using symbols. 53. 5 more than y 54. c decreased by 10 55. The product of 8 and a 56. The quotient when y is divided by 3 © 2007 McGraw-Hill Companies Perform the indicated operations. hut72632_ch02_C.qxd 9/20/05 16:18 Page 207 SUMMARY EXERCISES 57. 5 times the product of m and n 58. The product of a and 5 less than a 59. 3 more than the product of 17 and x 60. The quotient when a plus 2 is divided by a minus 2 Identify which are expressions and which are not. 61. 4(x 3) 62. 7 8 63. y 5 9 64. 11 2(3x 9) [2.7] Evaluate the expressions if x 3, y 6, z 4, and w 2. 65. 3x w 66. 5y 4z 67. x y 3z 68. 5z 2 69. 3x2 2w2 70. 3x3 71. 5(x2 w2) 72. 6z 2w 73. 2x 4z yz 74. 3x y wx 75. x(y2 z2) (y z)(y z) 76. y(x w)2 x 2xw w2 [2.8] 2 List the terms of the expressions. 77. 4a3 3a2 78. 5x2 7x 3 Circle like terms. © 2007 McGraw-Hill Companies 79. 5m 2, 3m, 4m 2, 5m 3, m 2 80. 4ab2, 3b2, 5a, ab2, 7a2, 3ab2, 4a2b Combine like terms. 81. 5c 7c 82. 2x 5x 83. 4a 2a 84. 6c 3c 207 hut72632_ch02_C.qxd 208 9/1/05 CHAPTER 2 19:11 Page 208 INTEGERS AND INTRODUCTION TO ALGEBRA 85. 9xy 6xy 86. 5ab2 2ab2 87. 7a 3b 12a 2b 88. 6x 2x 5y 3x 89. 5x3 17x2 2x3 8x2 90. 3a3 5a2 4a 2a3 3a2 a [2.9] Tell whether the number shown in parentheses is a solution for the given equation. 91. 7x 2 16 92. 5x 8 3x 2 (2) 93. 7x 2 2x 8 [2.10] 94. 4x 3 2x 11 (2) 95. x 5 3x 2 x 23 (4) (6) 96. 1 x 2 10 3 (7) (21) Solve the equations and check your results. 97. x 5 7 99. 5x 4x 5 98. x 9 3 100. 3x 9 2x 101. 5x 3 4x 2 102. 9x 2 8x 7 103. 7x 5 6x 4 104. 3 4x 1 x 7 2x © 2007 McGraw-Hill Companies 105. 4(2x 3) 7x 5 hut72632_ch02_C.qxd 9/1/05 19:11 Page 209 Self-Test for Chapter 2 Name Section The purpose of this self-test is to help you check your progress so that you can find sections and concepts that you need to review before the next in-class exam. Allow yourself about an hour to take this test. At the end of that hour, check your answers against those given in the back of this text. If you missed a question, notice the section reference that accompanies the answer. Go back to that section and reread the examples until you have mastered that particular concept. Represent the integers on the number line shown. 1. 5, 12, 4, 7, 18, 17 Date ANSWERS 1. 2. 3. 4. 20 10 0 10 20 5. 2. Place the following group of integers in ascending order: 4, 3, 6, 5, 0, 2, 2 7. Evaluate. 3. 7 6. 4. 7 8. 9. 5. 18 7 6. 187 10. 11. 7. 24 5 12. Find the opposite of each integer. 8. 40 9. 19 13. 14. Add. 10. 8 (5) 15. 11. 6 (9) 16. 17. 12. (9) (12) 18. © 2007 McGraw-Hill Companies Subtract. 19. 13. 9 15 14. 9 15 15. 5 (4) 16. 7 (7) Multiply. 17. (8)(5) 18. (9)(7) 19. (6)(4) 209 hut72632_ch02_C.qxd 9/1/05 19:11 Page 210 ANSWERS 20. Evaluate each expression. 21. 20. 75 3 21. 36 9 9 22. (15)(3) 9 23. 9 0 22. 23. 24. 25. 24. 23 4 5 25. 4 52 35 26. 26. 4(2 4)2 27. 16 (4) (5) 27. 28. 28. If x 2, y 1, and z 3, evaluate the expression 29. Write, using symbols. 30. 29. 5 less than a 9x2y . 3z 30. The product of 6 and m 31. 31. 4 times the sum of m and n 32. 32. The quotient when the sum of a and b is divided by 3 33. Identify which are expressions and which are not. 34. 33. 5x 6 4 34. 4 (6 x) 35. 36. Combine like terms. 37. 35. 8a 7a 38. 37. Subtract 9a2 from the sum of 12a2 and 5a2. 39. 38. Number Problem 36. 10x 8y 9x 3y Tom is 8 years younger than twice Moira’s age. Write an expression for Tom’s age. Let x represent Moira’s age. 40. The length of a rectangle is 4 more than twice the width. Write an expression for the length of the rectangle. 42. 43. Tell whether the number shown in parentheses is a solution for the given equation. 40. 7x 3 25 (5) 41. 8x 3 5x 9 44. Solve the equations and check your results. 42. x 7 4 44. 9x 2 8x 5 210 43. 7x 12 6x (4) © 2007 McGraw-Hill Companies 39. Geometry 41. hut72632_ch02_C.qxd 9/1/05 19:11 Page 211 Cumulative Review for Chapters 1 and 2 Name Section Date ANSWERS The following exercises are presented to help you review concepts from earlier chapters that you may have forgotten. This section is meant as review material and not as a comprehensive exam. The answers are presented in the back of the text. If you have difficulty with any of these exercises, be certain to at least read through the summary related to that section. 1. 2. 3. 1. Give the place value of 7 in 3,738,500. 4. 2. Give the word name for 302,525. 5. 3. Write two million, four hundred thirty thousand as a numeral. 6. 7. In exercises 4 to 6, name the property of addition that is illustrated. 4. 5 12 12 5 5. 9 0 9 6. (7 3) 8 7 (3 8) 8. 9. 10. 11. In exercises 7 and 8, perform the indicated operations. 7. 593 275 98 8. Find the sum of 58, 673, 5,325, and 17,295. In exercises 9 and 10, round the numbers to the indicated place value. © 2007 McGraw-Hill Companies 9. 5,873 to the nearest hundred 10. 953,150 to the nearest ten thousand In exercise 11, estimate the sum by rounding to the nearest hundred. 11. 943 3,281 778 2,112 570 211 hut72632_ch02_C.qxd 9/1/05 19:11 Page 212 ANSWERS 12. 13. Evaluate. 14. 12. 5 14 15. 13. 514 14. What is the opposite of 3? 16. 17. 18. In exercises 15 to 24, evaluate each expression. 15. 12 (6) 16. 7 7 17. 5 (7) 18. 9 (4) (7) 19. 8 (8) 20. 5 (5) 21. (8)(12) 22. (6)(15) 23. 14 (7) 24. 25 0 19. 20. 21. 22. 23. 24. 25. 26. In exercises 25 to 28, evaluate the expressions if a 5, b 3, c 4, and d 2. 27. 25. 6ad 26. 3b2 27. 3(c 2d) 28. 28. 29. 2a 7d ab 30. In exercises 29 and 30, combine like terms. 31. 29. 6x 14 3x 5 30. 4x 8y 2x 7y 32. 31. x 5 17 212 32. 3x 5 2x 3 © 2007 McGraw-Hill Companies In exercises 31 and 32, solve each equation and check your solution.