58. (D U F) (1 E

60. D U (F E)

62. (D fl E) U (F fl E)

64. (D U F) (D U E)

< 2-3 > Union and Intersection of Sets

Using the sets A, B, C, and N, list the elements in each set. If

the set is empty write 0. See Examples 2 and 3. A = {1,3,5,7,9) B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5}N={1,2,3,...}

17. A U C 18. A U B

Chapter 1 The Real Numbers 1-8

11. C = 13. A B 15. 99 E N

Use one of the symbols E, E, =, 0, U, or fl in each blank to make a true statement. See Example 4. A = {1, 3, 5, 7, 9) B= {2, 4, 6, 8)

{1, 2, 3, 4, 5} N = {1, 2, 3, ...}

29. A n B 0

30. A nc 0

31. A B = {1, 2, 3, 4, 5, 6, 7, 8, 9} 32. A B = 0

33. B C = {2, 4} 34. B C {1, 2, 3, 4, 5, 6, 8} 35. 3_Af1B 36. 3 Anc

37.4 Bnc 38. 8 B UC

66. {x I x is a natural number greater than 6}

67. {x x is an odd natural number greater than 111

68. {x x is an odd natural number less than 14}

vim 69. {x I x is an even natural number between 4 and 79}

70. Ixix is an odd natural number between 12 and 57}

Using the sets A, B, C, and D, list the elements in each set. If the set is empty write 0.

A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8, 9) C = {1, 3, 5, 7} D {2, 4, 6, 8}

20. A 11 B 22. B fl C Miscellaneous

List the elements in each set. 24. B U 0 26. B fl 0 65. {x I x is an even natural number less than 20}

28. A U N

<4> Subsets

Determine whether each statement is true or false. Explain your answer. See Example 5. A = 11, 3, 5, 7, 91 B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5} N = {1, 2, 3, ...}

39. A C N 40. B C N 41. {2, C C 42. C C A

43. B C 44. 0 A 45. 25 C B 46. 0 C C 47. A C0 48. B C 0 49.A11BCC 50. B C C {2, 4, 6, 8}

VIDEO

<5 > Combining Three or More Sets

Using the sets D, E, and F, list the elements in each set. If the set is empty write 0. See Example 6. D = {3, 5, 7} E_ {2, 4, 6, 8} F =11, 2, 3, 4, 5/

51. D U E VIDEO

71. A U B 72. A U C

73. A 11 B 74. A 11 C 75. (A (1 B) U D 76. (AfiC)UD 77. (A U B) U D 78. (B U C) (1 A 79. (B fl C) 11 A 80. (B f1D)f1C 81. (B f1 D) U (C fl D) 82. (A U B) fl (C U D)

Write each set using set-builder notation. Answers may vary.

83. {3, 4, 5, 6} 84. {1, 3, 5, 7} 85. {5, 7, 9, 11, . .}

12. A = N 14. CAN 16. 6 E C

19. A 11 C 21. B UC

23. A U 0 25. A 11 0 27. A 11N

63. (D U E) (D U F)

53. Df1F VIDEO

E U F54. D U F 56. Ef1F

52. D 11 E

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Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.

1. What are the integers?

2. What are the rational numbers?

3. What kinds of decimal numbers are rational numbers?

19. {a I a is an integer greater than -5}

20. {z I z is an integer between 2 and 12}

21. {w j w is a natural number between 0 and 5}

4. What kinds of decimal numbers are irrational? 22. {y y is a whole number greater than 0}

5. What are the real numbers?VIDEO

23. {x I x is an integer between -3 and 5}

6. What is the ratio of the circumference and diameter of any circle? 24. {), y is an integer between -4 and 7}

<1 > TheRational Numbers

Determine whether each statement is true or false. Explain your answer. See Example I.

7. -2 E N 8. -2 E Q

<4> The Real Numbers

Determine which elements of the set

9. 0 E Q

11. 0.95 E Q

13. Q CZ 1 15. x E Z

A = f-V10, -3, - 2 , -0.025, 0, V 2, 3-2 , I

I r- 1 8 10. 0 E N

12. 0.333 . . . E Q

14. QCN

16. -99 E Zare members of these sets. See Example 3.

25. Real numbers

26. Natural numbers < 2 > Graphing on the Number Line

List the elements in each set and graph each set on a number line. See Example 2.

17. {x I x is a whole number smaller than 6}

18. {x x is a natural number less than 7}

27. Whole numbers

28. Integers

0 29. Rational numbers VIDEO

30. Irrational numbers

Yg

,t4

18 Chapter 1 The Real Numbers 1 -18

Determine whether each statement is true or false. Explain. See Example 4.

31. Q C R 32. I C Q

33. I Q = 101 34. Z C Q

35. IU Q = R 36. znQ=0 37. 0.2121121112... E Q 38. 0.3333... E Q

72. The set of real numbers between -1 and 3

73. The set of real numbers between 0 and 2 inclusive

74. The set of real numbers between -1 and 1 inclusive 39. 3.252525 .. E I

40. 3.1010010001 ... E I VIDEO

41. 0.999...E I

42. 0.666... E Q

75. The set of real numbers greater than or equal to 1 and less 43. ir E I

44. 7r E Q than 3

75. The set of real numbers greater than -2

76. The set of real numbers less than -1

77. The set of real numbers less than 5

78. The set of real numbers between 3 and 4

76. The set of real numbers greater than 2 and less than or equal to 5.

Write the interval notation for the interval of real numbers shown in each graph.

<11[1111) 77 * 3 4 5 6 7 8 9

78. 3 4 5 6 7 8 9

79. -4-3-2-1 0 1 2

80. 3 4 5 6 7 8 9

81. 40 50 60 70 80 90 100

82. 3 4 5 6 7 8 9

83. -9 -8 -7 -6 -5I

-4H

-3

84. -12-11-10-9 -8 -7 -6

Write each union or intersection as a single interval. See Example 6.

85. (1, 5) U (4, 9)

86. (- 1, 2) U (0, 8)

87. (0, 3) 11 (2, 8)

88. (1, 8) fl (2, 10)

89. (- 2, 4) U (0, CO)

90. (-co, 4) U (1, 5)

Place one of the symbols C, , , E, or in each blank so that each statement is true.

45. N W 46. Z Q

47. Z N 48. Q W

49. Q R 50. 7 R

51.0 I 52.0 Q

a 53. N R 54. W R VIDEO

55. 5 Z 56. -6 Z

57. 7 Q 58. 8 Q

59. Vi R 60. N/2 1 61. 0 I 62. 0 Q

63. {2, 3} Q 64. {0, 1} N

65. {3, V2} R 66. {3, V-2- } Q

<5> Intervals of Real Numbers

Write each interval of real numbers in interval notation and graph it. See Example 5.

67. The set of real numbers greater than 1

Reading and Writing After reading this section, write out 1 the answers to these questions. Use complete sentences. + 1 33. ---

10 534.

1. What is absolute value? 35. +36.

23)

2. How do you add two numbers with the same sign? \+ 37. - 15 0.02 38. 39. -2.7 + (-0.01) 40.

3. How do you add two numbers with unlike signs and different absolute values?

41. 47.39 + (-44.587) 42.

43. 0.2351 + (-0.5) 44.

4. What is the relationship between subtraction and addition?<3> Subtraction

--1 + (--1) 8 \ 8)

-3

+ -1

4 2

0.45 + (-1.3)

0.8 + (-1)

0.65357 + (-2.375)

- 1.234 + (-4.756)

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5. How do you multiply signed numbers? Find each difference. See Example 4.

45. 7 - 10 46. 8 - 19

48. -5- 12 49. 7 - (-6) 6. What is the relationship between division and

multiplication? 51. -1-5 52. -4-6

<1 > Absolute Value 54. -15 - (-6) 55. 20 - (-3)

. Evaluate. See Examples 1 and 2.

7. 1 -9 1 a. 1 -12 1

9. 1 7 1 10.10.5 1

11. -1 -41 12. -119 1

13. -(-17) 14. -(-4)

57'

60.

58.

- 1 - -2 3

61.

1 1 -8- - -4

2 - 0.03

15. -(-(-5)) 16. -(-(-(-6)))

<2> Addition 63. 5.3 - (-2) 64.

Find each sum. See Example 3. 65. -2.44 - 48.29 66.

17. (-5) + 9 18. (-3) + 10 67. -3.89 - (-5.16) 68. • 19. (-4) + (-3) 20. (-15) + (-11)

-4- CI 22. 5 + ( - 1 51

47. -4- 7

50. 3 - (-9)

53. - 12 - (-3)

56. 50 - (-70)

59. 1 - -3

2

62. -0.02-3

-4.1 - 0.13 -8.8-9.164

0 - (-3.5)

3 1-29 1.3 Operations on the Set of Real Numbers 2 9

73. (0.3)( - 0.3) 75. (-0.02)(- 0)

117. -57 + 19 VIDEO

74. (-0.1)(-0.5) 76. (0.05)(-2.5)

<5> Division Find the multiplicative inverse of each number. See Example 6.

119. -171+ 3 1 VIDEO

77. 20 78. -5 6 79. --5-

1 80. - 8

118. 0 ± (- 36)

120. 64 - 1 -12 1

122. -20 + (1)

124. 1-341- 27

126. -3 + (--23)

121. 0 + (-0.15)

123. -63 + I 8 1

125. + (-12)

82. 0.125 81. -0.3

Evaluate. See Example 7.

83. -6 ÷ 3

85. 30 + (-0.8)

87. (-0.8)(0.1)

89. (-0.1) ÷ (-0.4)

127. --1 - 19 128. 1 - 22 84. 84 + (-2)

86. (-9)(-6)

88. 7 + (- 0.5)

90. (-18) + (-0.9)

129. 28 - 0.01

131. -29-0.3

133. (-2)(0.35)

130. 55 - 0.1

132. -0.241 - 0.3

134. (-3)(0.19) <6> Division by Zero Evaluate. If a quotient is undefined, say so. See Example 8.

91. 0 ÷ 19 93. -2+0

95.9 + (-24)

2( 9 ) 97. -0 99. (0.25)( - 365)

Use an operation with signed numbers to solve each problem and identify the operation used.

135. Net worth of a family. The Jones family has a house that is worth $85,000, but they still owe $45,000 on the mortgage. They have $2300 in credit card debt, $1500 in other debts, $1200 in savings, and two cars worth $3500 each. What is the net worth of the Jones family?

92. 0 + 99 94. 33 + 0

1 / 5\ 96,:

98.2(-5)

100. 7.5 + (-0.15)

101. (-51) + (-0.003) • 102. (-2.8)(5.9)

103. 0 + 1.3422 105. 339.4 + 0

104. 0 + 334.8 106. 0.667 + 0

Miscellaneous Perform these computations.

. 107. -62 + 13 109. -32 - (-25) 111. I - 151

VIDEO 113. -(-684)

136. Net worth of a bank. Just before the recession, First Federal Homestead had $15.6 million in mortgage loans, had $23.3 million on deposit, and owned $8.5 million worth of real estate. After the recession started, the value of the real estate decreased to $4.8 million. What was the net worth of First Federal before the recession and after the recession started? (To a financial institution a loan is an asset and a deposit is a liability.)

108. -88 + 39 110. -71 - (-19) 112. -1-751

1 114. 3-(-123)

116. -

137. Warming up. On January 11 the temperature at noon V1DEO

was 14°F in St. Louis and -6°F in Duluth. How much warmer was it in St. Louis? See the figure on the next page.

115. z - (- 4)

138. Bitter cold. On January 16 the temperature at midnight was -31°C in Calgary and -20°C in Toronto. How much wanner was it in Toronto?v

A

75.

69. a - b

a - c

71. (a - b)(a + b)

73. Vc2 - 2c + 1 2 b 1 a c c

77. ! a - 191

3 8 Chapter 1 The Real Numbers 8 Ark

< 5 > Algebraic Expressions Evaluate each expression for a = -1, b = 3, and c = -4. See Example 7.

67. b2 - 4ac 68. b2 - 2b - 3

Find the value of Y2 - Y1 for each choice of Yi, Y2, .3c1, and x2. See Example 8.

X2 - X1

79. y i = 4, y2 = -6, x i = 2, x2 = -7

80. yi = -3, y2 = -3, xi = 4, x2 = -5

81. yi = - 1, y2 = 2, x1 -= -3, x 2 = 1 .

82. y 1 = -2, y2 = 5, = 2, x 2 = 6

83. yi = 2.4, y2 = 5.6, x1 = 5.9, x 2 = 4.7

84. y i = - 5.7, y2 = 6.9, x 1 = 3.5, x2 = 4.2

Evaluate each expression without a calculator. Use a calculator to check.

85. -22 + 5(3)2 87. (-2 + 5)32

89. I52 - 4(1)(6)

91. [13 + 2(-5)]2

93. 4 - (-1) -3-2

95. 3(-2)2 - 5(-2) + 4 96. 3(-1) 2 + 5(-1) - 6

97.2 2

+ 3(-1-) - 2

98. 8(42 - 6(1) + 1

\ 2 / \ 2 / 99. 16 - 3 • 71+17 - 51

100. 1127-41-13 -4' 51 101. 3 - 7[4 - (2 - 5)] 102. 9 - 2[3 - (4 + 6)] 103. 3 - 4(2 - 4 - 6 1)

63._ 32 _ (_9)

64. -24 -5 2-32 32 24

65. 4 - 7 66. 42

-3 - (-3) -42 - (-16)

VIDEO

39. -(2 - 7)2

41. -52.23

43. (- 5)(- 2)3

45. -(32 - 4)2

47. 8 + 2V5 2 - 32

49. -60 + 10 3 + 2 • 5 + 6 1

50. 75 + (-5)(-3) + -2 • 6

40. -(1 - 3 2)3 42. 24 - 42 44. (-1)(2 - 8)3 46. -(6 - 23)4

48. 6 + 3V62 + 2 • 25

51. 5.5 - 2.34

52. 5.32 - 4 • 6.1

53. (1.3 - 0.31)(2.9 - 4.88)

54. (6.7 - 9.88)3

55. -388.8 + (13.5)(9.6)

56. ( -4.3)(5.5) + (3.2)(-1.2)

Evaluate. if an expression is undefined, say so. See Example 6.

86. _ 32 + 3(6)2

88. (-3 + 3)62

90. I62 - 4(2)(4)

92. [6 + 2(-4)]2

94. 2 - (-3)

3 - 5

<2> Exponential Expressions Evaluate each exponential expression. See Example 2.

13. 25 14. 34 15. (-1)4 16. (-1)5

17.3

)2 /

(-312 \

18.(-

2) 6 2)

< 3 > Square Roots Evaluate each radical. See Example 3.

19. -\ /44

21. 136 + 64 23. 1/4(7 + 9)

VIDEO

25. V3 + 13 + 9

27. -2V25 + 144

20. V100

22. 1/25 - 9 24. 1/(11 + 2)(18 - 5)

26. 125 - 16 - 5

28. -3V169 - 144

<4> Order of Operations Evaluate each expression. See Examples 4 and 5.

29. 4 - 6 • 2 30. 8 - 3 • 9

31. 6/2 • 3 32. -12/4 . 3

33. 5 - 6(3 - 5) 34. 8 - 3(4 - 6)

35. 2)(436.

4)

37. -32 + (-8)2 + 3 38. -62 + (-3)3

b - c 70.b - a

72. (a - c)(a + c)

74. Vat - 4bc

c c a 76. - + - - a b b

78. lb+c!

2 - 6 57.

9 - 7

. -3-5 59

6 - (-2)

61. 4 + 2 7 3 • 2 - 9

9 - 12 58.4 - 5

60. -14 - (-2) -3 - 3

62. -6 - 2(-3) 8 - 3(-3)