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4. Contents Lesson 10-1Simplifying Algebraic Expressions Lesson 10-2Solving Two-Step Equations Lesson 10-3Writing Two-Step Equations Lesson 10-4Solving Equations with Variables on Each Side Lesson 10-5Inequalities Lesson 10-6Solving Inequalities by Adding or Subtracting Lesson 10-7Solving Inequalities by Multiplying or Dividing

5. Lesson 1 Contents Example 1Write Equivalent Expressions Example 2Write Equivalent Expressions Example 3Write Expressions with Subtraction Example 4Write Expressions with Subtraction Example 5Identify Parts of an Expression Example 6Simplify Algebraic Expressions Example 7Simplify Algebraic Expressions Example 8Simplify Algebraic Expressions Example 9Translate Phrases into Expressions

6. Example 1-1a Simplify. Answer: Use the Distributive Property to rewrite.

7. Example 1-1b Answer: Use the Distributive Property to rewrite

8. Example 1-2a Answer: Use the Distributive Property to rewrite Simplify.

9. Example 1-2b Answer: Use the Distributive Property to rewrite

10. Example 1-3a Answer: Distributive Property Simplify. Definition of subtraction Use the Distributive Property to rewrite Rewrite

11. Example 1-3b Answer: Use the Distributive Property to rewrite

12. Example 1-4a Answer: Distributive Property Simplify. Rewrite Use the Distributive Property to rewrite

13. Example 1-4b Answer: Use the Distributive Property to rewrite

14. Example 1-5a Definition of subtraction Identity Property; Identify the terms, like terms, coefficients, and constants in Answer: The like terms are The coefficients are The constant is –5.

15. Example 1-5b Identify the terms, like terms, coefficients, and constants in Answer: The terms are The like terms are The coefficients are The constant is –2.

16. Example 1-6a Distributive Property Identity Property; Answer: Simplify. 6n and n are like terms. Simplify 6n – n.

17. Example 1-6b Answer: Simplify 7n  n.

18. Example 1-7a Answer: Distributive Property Commutative Property Simplify 5s  3 – 12s. are like terms.

19. Example 1-7b Answer: Simplify 6s  2 – 10s.

20. Example 1-8a Commutative Property Definition of subtraction Distributive Property Answer: –3 Simplify. Simplify 8z  z – 5 – 9z  2. are like terms. –5 and 2 are also like terms.

21. Example 1-8b Answer: –z Simplify 6z  z – 2 – 8z  2.

22. Example 1-9a THEATER Tickets for the school play cost $5 for adults and $3 for children. A family has the same number of adults as children. Write an expression in simplest form that represents the total amount of money spent on tickets. If x represents the number of adult tickets, then x also represents the number of children tickets. To find the total amount spent, multiply the cost of each ticket by the number of tickets purchased. Then add the expressions.

23. Example 1-9a Answer: The expression $8x represents the total amount of money spent on tickets, where x is the number of adults or children. Distributive Property Simplify.

24. Example 1-9b MUSEUM Tickets for the museum cost $10 for adults and $7.50 for children. A group of people have the same number of adults as children. Write an expression in simplest form that represents the total amount of money spent on tickets to the museum. Answer: $17.50x

25. End of Lesson 1

26. Lesson 2 Contents Example 1Solve a Two-Step Equation Example 2Solve Two-Step Equations Example 3Solve Two-Step Equations Example 4Equations with Negative Coefficients

27. Example 2-1a Method 1 Use a model. Remove 1-tile from the mat. Solve

28. Example 2-1a Separate the remaining tiles into 5 equal groups. There are 5 tiles in each group.

29. Example 2-1a Method 2 Use symbols. Use the Subtraction Property of Equality. Use the Division Property of Equality. Write the equation. Subtract 1 from each side. Simplify. Divide each side by 5. Answer: The solution is 5.

30. Example 2-1b Answer: 6 Solve

31. Example 2-2a Method 1 Vertical Method Write the equation. Add 8 to each side. Simplify. Divide each side by 2. Simplify. Check your solution. Solve

32. Example 2-2a Answer: The solution is 21. Method 2 Horizontal Method Write the equation. Add 8 to each side. Simplify. Divide each side by 2. Simplify. The sentence is true. Write the equation. Check Replace n with 21.

33. Example 2-2b Answer: 9 Check your solution. Solve

34. Example 2-3a Write the equation. Subtract 2 from each side. Simplify. Multiply each side by 3. Simplify. Answer: The solution is –18. Solve

35. Example 2-3b Answer: –26 Solve

36. Example 2-4a Answer: The solution is –2. Write the equation. Subtract 8 from each side. Simplify. Divide each side by –3. Simplify. Definition of subtraction Solve

37. Example 2-4b Answer: –3 Solve

38. Example 2-4b Check your solution. Write the equation. Combine like terms; Simplify. Divide each side by 2. Add 2 to each side. Identity Property; Simplify. Solve

39. Example 2-4b Answer: The solution is 8. Write the equation. Multiply. The statement is true. Check Replace k with 8.

40. Example 2-4b Answer: 5 Solve

41. End of Lesson 2

42. Lesson 3 Contents Example 1Translate Sentences into Equations Example 2Translate Sentences into Equations Example 3Translate Sentences into Equations Example 4Translate and Solve an Equation Example 5Write and Solve a Two-Stop Equation

43. Example 3-1a Translate three more than half a number is 15 into an equation. Answer:

44. Example 3-1b Translate five more than one-third a number is 7 into an equation. Answer:

45. Example 3-2a Translate nineteen is two more than five times a number into an equation. Answer:

46. Example 3-2b Translate fifteen is three more than six times a number into an equation. Answer:

47. Example 3-3a Translate eight less than twice a number is –35 into an equation. Answer:

48. Example 3-3b Translate six less than three times a number is –22 into an equation. Answer:

49. Example 3-4a Two more than of a number is 6. Find the number. Two more than of a number is 6. Words Variable Equation

50. Example 3-4a Answer: The number is 12. Simplify. Mentally multiply each side by 3. Subtract 2 from each side. Write the equation.

51. Example 3-4b Three more than six times a number is 15. Find the number. Answer: 2

52. Example 3-5a TRANSPORTATION A taxi ride costs $3.50 plus $2 for each mile traveled. If Jan pays $11.50 for the ride, how many miles did she travel? Her cost starts at $3.50 and adds $2 until it reaches $11.50. Organize the data for the first few miles into a table and look for a pattern. 2 1 0 CostMiles 3

53. Example 3-5a Write an equation to represent the situation. Let m represent the number of miles. 3.50 + 2m2m 11.50 flat rate plus m miles at $2 per mile equals $11.50 Simplify. Write the equation. Subtract 3.50 from each side.

54. Example 3-5a Answer: Jan traveled 4 miles. Divide each side by 2. Simplify.

55. Example 3-5b TRANSPORTATION A rental car costs $100 plus $0.25 for each mile traveled. If Kaya pays $162.50 for the car, how many miles did she travel? Answer: 250 miles

56. End of Lesson 3

57. Lesson 4 Contents Example 1Equations with Variables on Each Side Example 2Equations with Variables on Each Side Example 3Use an Equation to Solve a Problem

58. Example 4-1a Check your solution. Write the equation. Simplify by combining like terms. Replace x with 2. Mentally divide each side by 2. Subtract 7x from each side. Write the equation. Check To check your solution, replace x with 2 in the original equation. The sentence is true. Answer: The solution is 2. Solve

59. Example 4-1b Check your solution. Answer: ─3 Solve

60. Example 4-2a Write the equation. Simplify. Mentally divide each side by –5. Add 2 to each side. Subtract 8x from each side. Simplify. Answer: The solution is –3. Solve

61. Example 4-2b Answer: –10 Solve

62. Example 4-3a GRID-IN TEST ITEM Find the value of x so that the polygons have the same perimeter. Read the Test Item You need to find the value of x that will make the perimeter of the triangle equal to the perimeter of the rectangle.

63. Example 4-3a Triangle Rectangle Solve the Test Item Write expressions for the perimeter of each figure. Then set the two expressions equal to each other and solve for x.

64. Example 4-3a

65. Answer:

66. Example 4-3b GRID-IN TEST ITEM Find the value of x so that the polygons have the same perimeter.

67. Example 4-3b Answer:

68. End of Lesson 4

69. Lesson 5 Contents Example 1Write Inequalities with Example 2Write Inequalities with Example 3Write Inequalities with  or  Example 4Write Inequalities with  or  Example 5Determine the Truth of an Inequality Example 6Determine the Truth of an Inequality Example 7Graph an Inequality Example 8Graph an Inequality

70. Example 5-1a SPORTS Members of the little league team must be under 14 years old. Write an inequality for the sentence. Answer:

71. Example 5-1b SPORTS Members of the peewee football team must be under 10 years old. Write an inequality for the sentence. Answer:

72. Example 5-2a CONSTRUCTION The ladder must be over 30 feet tall to reach the top of the building. Write an inequality for the sentence. Answer:

73. Example 5-2b CONSTRUCTION The new building must be over 300 feet tall. Write an inequality for the sentence. Answer:

74. Example 5-3a POLITICS The president of the United States must be at least 35. Write an inequality for the sentence. Answer:

75. Example 5-3b VOTING To vote, you must be at least 18 years old. Write an inequality for the sentence. Answer:

76. Example 5-4a CAPACITY A theater can hold a maximum of 300 people. Write an inequality for the sentence. Answer:

77. Example 5-4b CAPACITY A football stadium can hold a maximum of 10,000 people. Write an inequality for the sentence. Answer:

78. Example 5-5a For the given value, state whether the inequality is true or false. Answer: Since –4 is less than 6, Write the inequality. Simplify. Replace x with 0.

79. Example 5-5b For the given value, state whether the inequality is true or false. Answer: false

80. Example 5-6a Write the inequality. Simplify. Replace x with 1. For the given value, state whether the inequality is true or false. Answer: Since 3 is not greater than or equal to 4, the sentence is false.

81. Example 5-6b For the given value, state whether the inequality is true or false. Answer: true

82. Example 5-7a Answer: Place a closed circle at –1. Then draw a line and an arrow to the left. Graph n –1 on a number line. The closed circle means the number –1 is included in the graph.

83. Example 5-7b Answer: Graph n –3 on a number line.

84. Example 5-8a Answer: Place an open circle at –1. Then draw a line and an arrow to the right. Graph n –1 on a number line. The open circle means –1 is not included in the graph.

85. Example 5-8b Answer: Graph n –3 on a number line.

86. End of Lesson 5

87. Lesson 6 Contents Example 1Solve an Inequality Using Addition Example 2Solve an Inequality Using Subtraction Example 3Graph the Solutions of an Inequality Example 4Use an Inequality to Solve a Problem

88. Example 6-1a Write the inequality. Simplify. Add 4 to each side. Replace n with a number greater than 10, such as 11. Write the inequality. Check The statement is true. Answer: Any number greater than 10 will make the statement true, so the solution is Check your solution. Solve

89. Example 6-1b Check your solution. Answer: Solve

90. Example 6-2a Check your solution. Write the inequality. Simplify. Subtract 8 from each side. Check Replace x in the original inequality with –15 and then with a number less than –15. Answer: The solution is Solve

91. Example 6-2b Check your solution. Answer: Solve

92. Example 6-3a The solution is Graph the solution. Then graph the solution on a number line. Solve

93. Example 6-3a Answer: Place an open circle at Draw a line and an arrow to the left.

94. Example 6-3b Answer: Solve Then graph the solution on a number line.

95. Example 6-4a TOWING COMPANY A pickup truck is towing a trailer that weighs 3,525 pounds. The maximum towing capacity of the truck is 4,700 pounds. Determine how much more weight can be added to the trailer and still be towed by the truck.

96. Example 6-4a Words Variable Inequality The phrase maximum capacity means less than or equal to. So, the current weight being towed plus any more weight must be less than or equal to 4,700 pounds. 3,525 + w 4,700 current weight plus weight added must be less than or equal to 4,700 pounds

97. Example 6-4a Answer: Up to 1,175 more pounds can be added to the trailer. Write the inequality. Simplify. Subtract 3,525 from each side.

98. Example 6-4b SPORTS A weightlifter can lift up to 375 pounds. He is currently lifting 255 pounds. Determine how much more weight can be added and still be lifted by the weightlifter. Answer: Up to 120 more pounds can be added.

99. End of Lesson 6

100. Lesson 7 Contents Example 1Divide by a Positive Number Example 2Multiply by a Positive Number Example 3Multiply or Divide by a Negative Number Example 4Multiply or Divide by a Negative Number Example 5Solve a Two-Step Inequality

101. Example 7-1a Check your solution. Answer: The solution is You can check this solution by substituting numbers less than –5 into the inequality. Write the inequality. Simplify. Divide each side by 6. Solve

102. Example 7-1b Check your solution. Answer: Solve

103. Example 7-2a and check your solution. Then graph Write the inequality. Simplify. Multiply each side by 2. the solution on a number line. The solution is You can check this solution by substituting 18 and a number greater than 18 into the inequality. Solve

104. Example 7-2a Answer: Graph the solution,

105. Example 7-2b and check your solution. Then graph the solution on a number line. Answer: Solve

106. Example 7-3a Check your solution. Answer: The solution is You can check this solution by replacing b in the original inequality with –20 and a number greater than –20. Write the inequality. Simplify. Multiply each side by –4 and reverse the inequality symbol. Solve

107. Example 7-3b Check your solution. Answer: Solve

108. Example 7-4a Write the inequality. Check this result. Divide each side by –4 and reverse the inequality symbol. Answer: Graph the solution, Solve Then graph the solution on a number line.

109. Example 7-4b Answer: Then graph the solution on a number line. Solve

110. Example 7-5a PACKAGES A box weighs 1 pound. It is filled with books that weigh 2 pounds each. Jesse can carry at most 20 pounds. Assuming space is not an issue, write and solve an inequality to find how many books he can put in the box and still carry it. The phrase at most means less than or equal to. of books he puts in the box. Then write an inequality. 1 pound plus 2 pounds per book is less than or equal to 20 pounds 1 2p 20

111. Example 7-5a Answer: Since he can not put half a book in the box, Jesse can put at most 9 books in the box. Write the inequality. Simplify. Subtract 1 from each side. Divide each side by 2. Simplify.

112. Example 7-5b PACKAGES A box weighs 2 pounds. It is filled with toys that weigh 1 pound each. Danielle can carry at most 30 pounds. Assuming space is not an issue, write and solve an inequality to find how many toys she can put in the box and still carry it. Answer: She can put at most 28 toys in the box.

113. End of Lesson 7

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