Question: 1.3
Identifying a Property of Real Numbers In
Exercises 49-56, identify the property of real
numbers illustrated by the statement.
49. 13 - 13 = 0
50. 7(1) = 7
51. 7(9 + 3) = 7\cdot 9 + 7\cdot 3
52. 15(4) = 4(15)
53. 5 + (4 - y) = (5 + 4) - y
54. 6(2) = (6 \cdot 2)x
55. xy \cdot 1 = xy
56. x - y = (x - y)
Identifying Properties of Real Numbers In
Exercises 57 and 58, identify the property of real
numbers that justifies each step.
57. x + 2 = 4
(x + 2) + (-2) = 4 + (-2)
x + [2 + (-2)] = 2
x + 0 = 2
x = 2
58. 3x - 8 = 1
(3x - 8) + 8 = 1 + 8
3x + (-8 + 8) = 9
3x + 0 = 9
3x = 9
(1/3)(3x) = (1/3)(9)
(1 \cdot 3)x = 3
1 \cdot x = 3
x = 3
1.4
Identifying Terms and Coefficients In Exercises
59-62, identify the terms and coefficients of the
algebraic expression.
59. 4y^2 - x^2 + \frac{17}{2} y
60. \frac{x}{3} - 2xy^2 + \frac{1}{5}
61. 52 - 12x^3 + \frac{1}{x}
62. 2a^2 + 6b^2 + \frac{1}{q}
Simplifying an Algebraic Expression In Exercises
63-70, simplify the expression.
63. 6x + 3x
64. 10y - 7y
65. 3x - 2x + 7x - 3x
66. 8m - 4m + m - 3m
67. 5(x - 4) + 10
68. 3x - (y - 2x)
69. 3b + 5(b - a)
70. -2[(6 - r)] + 5t
Review Exercises 47
69. 3b + 5b - a
70. -2[(6 - r)] + 5t
Evaluating an Algebraic Expression In Exercises
71-74, evaluate the algebraic expression for the
specified values of the variable(s). If not possible,
state the reason.
71. x^2 - 2x - 3
(a) x = 3
(b) x = 0
72. \frac{x}{y + 2}
(a) x = 0, y = 3
(b) x = 5, y = -2
73. y^2 - 2x + 4x
(a) x = 4, y = -1
(b) x = -2, y = 2
74. |2x - x^2| - 2y
(a) x = 6, y = 3
(b) x = 4, y = -5
1.5
Translating a Verbal Phrase In Exercises 75-78,
translate the verbal phrase into an algebraic expression.
75. Twelve decreased by twice the number n
76. One hundred increased by the product of 15 and a
number x
77. The sum of the square of a number j and 49
78. Three times the positive value of the difference of a
number n and 1, all divided by 5
Translating an Algebraic Expression In Exercises
79-82, write a verbal description of the algebraic
expression without using the variable.
79. 2y + 7
80. 5x - 3
81. \frac{x - 5}{4}
82. 4(a - 1)
Constructing a Mathematical Model In Exercises
83-86, write an algebraic expression that represents
the specified quantity, and simplify, if possible.
83. The amount of income tax due on a taxable income of
7 dollars when the tax rate is 18%
84. The distance traveled in 8 hours at an average speed of
r miles per hour
85. The area of a rectangle whose length is l units and
whose width is 5 units less than the length
86. The sum of two consecutive odd integers, the first of
which is 2n + 1
Graph shapes and math:
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1. **Problem:** Identify the property of real numbers illustrated by the statement $13 - 13 = 0$.
2. **Property Used:** This is the **Additive Inverse Property**, which states that for any real number $a$, $a + (-a) = 0$.
3. **Explanation:** Here, $13 - 13$ can be rewritten as $13 + (-13)$, which equals $0$ because $-13$ is the additive inverse of $13$.
4. **Intermediate Work:**
$$
13 - 13 = 13 + (-13) = 0
$$
5. **Final Answer:** The property illustrated is the **Additive Inverse Property**.