1-5 and 1-6 Operations With Real Numbers

Quick Review

To add numbers with different signs, find the difference of their absolute values. Then use the sign of the addend with the greater absolute value.

3 + ( − 4 ) = − ( 4 − 3 ) = − 1 3 plus open negative 4 close equals negative open 4 minus 3 close equals negative 1

To subtract, add the opposite.

9 − ( − 5 ) = 9 + 5 = 14 9 minus open negative 5 close equals 9 plus 5 equals 14

The product or quotient of two numbers with the same sign is positive: 5 · 5 = 25 ( − 5 ) · ( − 5 ) = 25 5 middle dot 5 equals 25 open negative 5 close middle dot open negative 5 close equals 25

The product or quotient of two numbers with different signs is negative: 6 · ( − 6 ) = − 36 − 36 ÷ 6 = − 6 6 middle dot open negative 6 close equals negative 36 minus 36 divides 6 equals negative 6

Example

Cave explorers descend to a site that has an elevation of − 1.3  mi. negative 1.3 , mi.  (Negative elevation means below sea level.) The explorers descend another 0.6 mi before they stop to rest. What is the elevation at their resting point?

− 1.3 + ( − 0.6 ) = − 1.9 negative 1.3 plus open negative 0.6 close equals negative 1.9

The elevation at their resting point is − 1.9  mi. negative 1.9 , mi.

Exercises

Find each sum. Use a number line.

56. 1 + 4
57. 3 + ( − 8 ) 3 plus open negative 8 close
58. − 2 + ( − 7 ) negative 2 plus open negative 7 close

Simplify each expression.

59. − 5.6 + 7.4 negative 5.6 plus 7.4
60. − 12 2 negative , 12 squared
61. − 5 ( − 8 ) negative 5 open negative 8 close
62. 4.5 ÷ ( − 1.5 ) 4.5 divides open negative 1.5 close
63. − 13 + ( − 6 ) negative 13 plus open negative 6 close
64. − 9 − ( − 12 ) negative 9 minus open negative 12 close
65. ( − 2 ) ( − 2 ) ( − 2 ) open negative 2 close open negative 2 close open negative 2 close
66. − 54 ÷ ( − 0.9 ) negative 54 divides open negative 0.9 close

Evaluate each expression for p = 5 and q = − 3 . q equals negative 3 .

67. − 3 q + 7 negative 3 q plus 7
68. − ( 4 q ) negative open 4 q close
69. q − 8 q minus 8
70. 5 p − 6 5 p minus 6
71. − ( 2 p ) 2 negative . open 2 p close squared
72. 7 q − 7 p 7 q minus 7 p
73. ( p q ) 2 open p q close squared
74. 2 q ÷ 4 p 2 q divides 4 p

1-7 The Distributive Property

Quick Review

Terms with exactly the same variable factors are like terms. You can combine like terms and use the Distributive Property to simplify expressions.

Distributive Property

a(b + c) = ab + ac

a ( b − c ) = a b − a c eh open b minus c close equals eh b minus eh c

Example

Simplify 7 t + ( 3 − 4 t ) . 7 bold italic t plus open 3 minus 4 bold italic t close .

7 t + ( 3 − 4 t ) = 7 t + ( − 4 t + 3 ) Commutative Property   = ( 7 t + ( − 4 t ) ) + 3 Associative Property   = ( 7 + ( − 4 ) ) t + 3 Distributive Property   = 3 t + 3 Simplify. table with 4 rows and 3 columns , row1 column 1 , 7 t plus open 3 minus 4 t close , column 2 equals 7 t plus open negative 4 t plus 3 close , column 3 cap commutative cap property , row2 column 1 , , column 2 equals open 7 t plus open negative 4 t close close plus 3 , column 3 cap associative cap property , row3 column 1 , , column 2 equals open 7 plus open negative 4 close close t plus 3 , column 3 cap distributive cap property , row4 column 1 , , column 2 equals 3 t plus 3 , column 3 cap simplify. , end table

Exercises

Simplify each expression.

75. 5 ( 2 x − 3 ) 5 open 2 x minus 3 close
76. − 2 ( 7 − a ) negative 2 open 7 minus eh close
77. ( − j + 8 ) 1 2 open , negative j plus 8 , close . 1 half
78. 3 v 2 − 2 v 2 3 , v squared , minus 2 , v squared
79. 2 ( 3 y − 3 ) 2 open 3 y minus 3 close
80. ( 6 y − 1 ) 1 4 open , 6 y minus 1 , close . 1 fourth
81. ( 24 − 24 y ) 1 4 open . 24 minus 24 y . close . 1 fourth
82. 6 y − 3 − 5 y 6 y minus 3 minus 5 y
83. 1 3 y + 6 − 2 3 y 1 third , y plus 6 minus , 2 thirds , y
84. − a b 2 − a b 2 negative eh , b squared , minus eh , b squared
85. Music All 95 members of the jazz club pay $30 each to go see a jazz performance. What is the total cost of tickets? Use mental math.
86. Reasoning Are 8 x 2 y 8 , x squared , y  and − 5 y x 2 negative 5 y , x squared  like terms? Explain.

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