2. Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 1-1Expressions and Formulas Lesson 1-2Properties of Real Numbers Lesson 1-3Solving Equations Lesson 1-4Solving Absolute Value Equations Lesson 1-5Solving Inequalities Lesson 1-6Solving Compound and Absolute Value Inequalities

5. Lesson 1 Contents Example 1Simplify an Expression Example 2Evaluate an Expression Example 3Expression Containing a Fraction Bar Example 4Use a Formula

6. Example 1-1a Find the value of First, subtract 2 from 7. Then cube 5. Multiply 125 by 3. Subtract 375 from 384. Finally, divide 9 by 3. Answer:The value is 3.

7. Example 1-1b Answer: 9 Find the value of

8. Example 1-2a Evaluateifand Replace s with 2 and t with 3.4. Find 2 2. Subtract 3.4 from 4. Multiply 3.4 and 0.6. Subtract 2.04 from 2. Answer:The value is –0.04.

9. Example 1-2b Answer: –110 Evaluateifand

10. Example 1-3a Evaluate the numerator and the denominator separately. Multiply 40 by –2. Evaluateif,, and

11. Example 1-3b Answer:The value is –9. Simplify the numerator and the denominator. Then divide.

12. Example 1-3c Evaluateifand Answer: –23

13. Add 13 and 25. Example 1-4a Find the area of a trapezoid with base lengths of 13 meters and 25 meters and a height of 8 meters. Area of a trapezoid Replace h with 8, b 1 with 13, and b 2 with 25. Multiply 4 and 38. Answer:The area of the trapezoid is 152 square meters. Multiply 8 by.

14. Example 1-4b Answer: 50 cm 3 The formula for the volume V of a pyramid is, where B represents the area of the base and h is the height of the pyramid. Find the volume of the pyramid shown below.

15. End of Lesson 1

16. Lesson 2 Contents Example 1Classify Numbers Example 2Identify Properties of Real Numbers Example 3Additive and Multiplicative Inverses Example 4Use the Distributive Property to Solve a Problem Example 5Simplify an Expression

17. Example 2-1a Name the sets of numbers to which belongs. Answer:rationals (Q) and reals (R)

18. Example 2-1b Answer:rationals (Q) and reals (R) Name the sets of numbers to which belongs. The bar over the 9 indicates that those digits repeat forever.

19. Example 2-1c Answer:irrationals (I) and reals (R) Name the sets of numbers to which belongs. lies between 2 and 3 so it is not a whole number.

20. Example 2-1d Answer:naturals (N), wholes (W), integers (Z), rationals (Q) and reals (R) Name the sets of numbers to which belongs.

21. Example 2-1e Answer:rationals (Q) and reals (R) Name the sets of numbers to which –23.3 belongs.

22. Example 2-1f Name the sets of numbers to which each number belongs. a. b. c. d. e. 32.1 Answer: rationals (Q) and reals (R) Answer: irrationals (I) and reals (R) Answer: naturals (N), wholes (W), integers (Z) rationals (Q) and reals (R) Answer: rationals (Q) and reals (R)

23. Example 2-2a Name the property illustrated by. The Additive Inverse Property says that a number plus its opposite is 0. Answer: Additive Inverse Property

24. Example 2-2b The Distributive Property says that you multiply each term within the parentheses by the first number. Answer: Distributive Property Name the property illustrated by.

25. Example 2-2c Name the property illustrated by each equation. a. b. Answer: Identity Property of Addition Answer: Inverse Property of Multiplication

26. Example 2-3a Identify the additive inverse and multiplicative inverse for –7. Since –7 + 7 = 0, the additive inverse is 7. Sincethe multiplicative inverse is Answer: The additive inverse is 7, and the multiplicative inverse is

27. Example 2-3b Identify the additive inverse and multiplicative inverse for. Sincethe additive inverse is Sincethe multiplicative inverse is Answer: The additive inverse is and the multiplicative inverse is 3.

28. Identify the additive inverse and multiplicative inverse for each number. a. 5 b. Example 2-3c Answer: additive: –5 ; multiplicative: Answer: additive: multiplicative:

29. Example 2-4a Postage Audrey went to a post office and bought eight 34-cent stamps and eight 21-cent postcard stamps. How much did Audrey spend altogether on stamps? There are two ways to find the total amount spent on stamps. Method 1 Multiply the price of each type of stamp by 8 and then add.

30. Example 2-4b Method 2 Add the prices of both types of stamps and then multiply the total by 8. Answer: Audrey spent a total of $ 4.40 on stamps. Notice that both methods result in the same answer.

31. Example 2-4c Chocolate Joel went to the grocery store and bought 3 plain chocolate candy bars for $0.69 each and 3 chocolate- peanut butter candy bars for $0.79 each. How much did Joel spend altogether on candy bars? Answer: $ 4.44

32. Example 2-5a Simplify Distributive Property Multiply. Commutative Property (+) Distributive Property Answer:Simplify.

33. Example 2-5b Simplify. Answer:

34. End of Lesson 2

35. Lesson 3 Contents Example 1Verbal to Algebraic Expression Example 2Algebraic to Verbal Sentence Example 3Identify Properties of Equality Example 4Solve One-Step Equations Example 5Solve a Multi-Step Equation Example 6Solve for a Variable Example 7Apply Properties of Equality Example 8Write an Equation

36. Example 3-1a Write an algebraic expression to represent 3 more than a number. Answer:

37. Example 3-1b Write an algebraic expression to represent 6 times the cube of a number. Answer:

38. Example 3-1c Write an algebraic expression to represent the square of a number decreased by the product of 5 and the same number. Answer:

39. Example 3-1d Write an algebraic expression to represent twice the difference of a number and 6. Answer:

40. Write an algebraic expression to represent each verbal expression. a. 6 more than a number b. 2 less than the cube of a number c. 10 decreased by the product of a number and 2 d. 3 times the difference of a number and 7 Example 3-1e Answer:

41. Example 3-2a Write a verbal sentence to represent. Answer: The sum of 14 and 9 is 23.

42. Example 3-2b Write a verbal sentence to represent. Answer: Six is equal to –5 plus a number.

43. Example 3-2c Write a verbal sentence to represent. Answer: Seven times a number minus 2 is 19.

44. Example 3-2d Write a verbal sentence to represent each equation. a. b. c. Answer: The difference between 10 and 3 is 7. Answer: Three times a number plus 2 equals 11. Answer: Five is equal to the sum of 2 and a number.

45. Example 3-3a Name the property illustrated by the statement if xy = 28 and x = 7, then 7y = 28. Answer: Substitution Property of Equality

46. Example 3-3b Name the property illustrated by the statement. Answer: Reflexive Property of Equality

47. Name the property illustrated by each statement. a. b. Answer: Transitive Property of Equality Example 3-3c Answer: Symmetric Property of Equality

48. Example 3-4a Solve. Check your solution. Original equation Add 5.48 to each side. Simplify. Check: Original equation Answer: The solution is 5.5. Simplify. Substitute 5.5 for s.

49. Example 3-4b Solve. Check your solution. Original equation Simplify. Multiply each side bythe multiplicative inverse of

50. Example 3-4c Answer: The solution is 36. Check: Original equation Simplify. Substitute 36 for t.

51. Solve each equation. Check your solution. a. b. Example 3-4d Answer: –2 Answer: 15

52. Example 3-5a Solve Original equation Distributive and Substitution Properties Commutative, Distributive, and Substitution Properties Addition and Substitution Properties Division and Substitution Properties Answer: The solution is –19.

53. Example 3-5b Answer: –6 Solve

54. Example 3-6a Geometry The area of a trapezoid is where A is the area, b 1 is the length of one base, b 2 is the length of the other base, and h is the height of the trapezoid. Solve the formula for h.

55. Example 3-6b Area of a trapezoid Multiply each side by 2. Simplify. Divide each side by. Simplify.

56. Example 3-6c Answer:

57. Example 3-6d Geometry The perimeter of a rectangle is where P is the perimeter, is the length, and w is the width of the rectangle. Solve the formula for w. w Answer:

58. Example 3-7a Multiple-Choice Test Item what is the value of AB CD

59. Example 3-7b Read the Test Item You are asked to find the value of the expression 4g – 2. Your first thought might be to find the value of g and then evaluate the expression using this value. However, you are not required to find the value of g. Instead, you can use the Subtraction Property of Equality on the given equation to find the value of 4g – 2.

60. Example 3-7c Solve the Test Item Original equation Subtract 7 from each side. Answer: B

61. Example 3-7d Multiple-Choice Test Item what is the value of A 12 B 6 C –6 D –12 Answer: D

62. Example 3-8a Home Improvement Carl wants to replace the 5 windows in the 2nd-story bedrooms of his home. His neighbor Will is a carpenter and he has agreed to help install them for $250. If Carl has budgeted $1000 for the total cost, what is the maximum amount he can spend on each window? Explore Let c represent the cost of each window. Plan The number of windows times the cost per window plus the cost for a carpenter equals the total cost. 5c+250=1000

63. Example 3-8b Solve Original equation Subtract 250 from each side. Simplify. Answer: Carl can afford to spend $ 150 on each window. Divide each side by 5.

64. Example 3-8c ExamineThe total cost to replace five windows at $150 each is 5(150) or $750. Add the $250 cost of the carpenter to that, and the total bill to replace the windows is 750 + 250 or $1000. Thus, the answer is correct.

65. Home Improvement Kelly wants to repair the siding on her house. Her contractor will charge her $300 plus $150 per square foot of siding. How much siding can she repair for $1500? Example 3-8d Answer: 8 ft 2

66. End of Lesson 3

67. Lesson 4 Contents Example 1Evaluate an Expression with Absolute Value Example 2Solve an Absolute Value Equation Example 3No Solution Example 4One Solution

68. Example 4-1a Evaluate Answer: The value is 4.7. Replace x with 4. Simplify –2(4) first. Subtract 8 from 6. Add.

69. Example 4-1b Answer: –13.7 Evaluate

70. Example 4-2a SolveCheck your solutions. Case 1 or Case 2 Check: Answer: The solutions are 5 or –11. Thus, the solution set is

71. Example 4-2b SolveCheck your solutions. Answer:

72. Example 4-3a Answer: The solution set is . Solve Original equation Subtract 5 from each side. This sentence is never true.

73. Example 4-3b Solve Answer: 

74. Example 4-4a SolveCheck your solutions. Case 1 or Case 2 There appear to be two solutions, 11 or

75. Example 4-4b Check: Answer: Since, the only solution is 11. The solution set is {11}.

76. Example 4-4c Answer: {6} Solve

77. End of Lesson 4

78. Lesson 5 Contents Example 1Solve an Inequality Using Addition or Subtraction Example 2Solve an Inequality Using Multiplication or Division Example 3Solve a Multi-Step Inequality Example 4Write an Inequality

79. Example 5-1a SolveGraph the solution set on a number line. Original inequality Subtract 4y from each side. Simplify. Subtract 2 from each side. Simplify. Rewrite with y first.

80. Example 5-1b Answer: Any real number greater than –5 is a solution of this inequality. A circle means that this point is not included in the solution set.

81. Example 5-1c Answer: SolveGraph the solution set on a number line.

82. Example 5-2a SolveGraph the solution set on a number line. –40  p Simplify. p  –40 Rewrite with p first. Original inequality Divide each side by –0.3, reversing the inequality symbol.

83. Example 5-2b Answer: The solution set is A dot means that this point is included in the solution set.

84. Example 5-2c SolveGraph the solution set on a number line. Answer:

85. Example 5-3a SolveGraph the solution set on a number line. Original inequality Multiply each side by 2. Add –x to each side. Divide each side by –3, reversing the inequality symbol.

86. Example 5-3b Answer: The solution set isand is graphed below.

87. Example 5-3c SolveGraph the solution set on a number line. Answer:

88. Example 5-4a Consumer Costs Alida has at most $10.50 to spend at a convenience store. She buys a bag of potato chips and a can of soda for $1.55. If gasoline at this store costs $1.35 per gallon, how many gallons of gasoline can Alida buy for her car, to the nearest tenth of a gallon? ExploreLet the number of gallons of gasoline that Alida buys. PlanThe total cost of the gasoline is 1.35g. The cost of the chips and soda plus the total cost of the gasoline must be less than or equal to $10.50. Write an inequality.

89. Example 5-4b The cost of chips & soda plus the cost of gasoline is less than or equal to $10.50. 1.55+1.35g  10.50 Divide each side by 1.35. Solve Original inequality Subtract 1.55 from each side. Simplify.

90. Example 5-4c Answer: Alida can buy up to 6.6 gallons of gasoline for her car. ExamineSince is actually greater than 6.6, Alida will have enough money if she gets no more than 6.6 gallons of gasoline.

91. Example 5-4d Rental Costs Jeb wants to rent a car for his vacation. Value Cars rents cars for $25 per day plus $0.25 per mile. How far can he drive for one day if he wants to spend no more that $200 on car rental? Answer: up to 700 miles

92. End of Lesson 5

93. Lesson 6 Contents Example 1Solve an “and” Compound Inequality Example 2Solve an “or” Compound Inequality Example 3Solve an Absolute Value Inequality (<) Example 4Solve an Absolute Value Inequality (>) Example 5Solve a Multi-Step Absolute Value Inequality Example 6Write an Absolute Value Inequality

94. Example 6-1a Solve Graph the solution set on a number line. Method 1Write the compound inequality using the word and. Then solve each inequality. and Method 2Solve both parts at the same time by adding 2 to each part. Then divide each part by 3.

95. Example 6-1b Graph the solution set for each inequality and find their intersection. y  4y  4

96. Example 6-1c Solve Graph the solution set on a number line. Answer:

97. Example 6-2a Solve each inequality separately. or Answer: The solution set is Solveor Graph the solution set on a number line.

98. Example 6-2b Answer: Solve Graph the solution set on a number line.

99. You can interpretto mean that the distance between d and 0 on a number line is less than 3 units. To maketrue, you must substitute numbers for d that are fewer than 3 units from 0. Example 6-3a All of the numbers between –3 and 3 are less than 3 units from 0. SolveGraph the solution set on a number line. Answer: The solution set is Notice that the graph of is the same as the graph of d > –3 and d < 3.

100. Example 6-3b Answer: SolveGraph the solution set on a number line.

101. Example 6-4a You can interpretto mean that the distance between d and 0 on a number line is greater than 3 units. To maketrue, you must substitute values for d that are greater than 3 units from 0. All of the numbers not between –3 and 3 are greater than 3 units from 0. Answer: The solution set is Notice that the graph of is the same as the graph of SolveGraph the solution set on a number line.

102. Example 6-4b SolveGraph the solution set on a number line. Answer:

103. Example 6-5a Solve Graph the solution set on a number line. Solve each inequality. or is equivalent to Answer: The solution set is.

104. Example 6-5b Solve Graph the solution set on a number line. Answer:

105. Example 6-6a Housing According to a recent survey, the average monthly rent for a one-bedroom apartment in one city is $750. However, the actual rent for any given one- bedroom apartment might vary as much as $250 from the average. Write an absolute value inequality to describe this situation. Let the actual monthly rent. The rent for an apartment can differ from the average by as much as$250. Answer:  250

106. Answer: The solution set is The actual rent falls between $500 and $1000, inclusive. Solve the inequality to find the range of monthly rent. Example 6-6b Rewrite the absolute value inequality as a compound inequality. Then solve for r. –r–r r –r–r

107. Health The average birth weight of a newborn baby is 7 pounds. However, this weight can vary by as much as 4.5 pounds. a. Write an absolute value inequality to describe this situation. b. Solve the inequality to find the range of birth weights for newborn babies. Example 6-6c Answer: Answer:The birth weight of a newborn baby will fall between 2.5 pounds and 11.5 pounds, inclusive.

108. End of Lesson 6

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