Frequently in algebra you are required to simplify
expressions with negative as well as positive numbers. When adding numbers
it is best to think in terms of MONEY $$$$$! A positive number is
like money coming in to your possession, or income; a negative
number is like money going out, like expenditures or debts. When
writing a negative number, it is helpful to write the negative number in
parentheses.
For example, 8 + (-5) means you have 8 dollars in your
possession and you spend 5 dollars. The result is +3 or 3, which means you
have 3 dollars.
What if the larger magnitude is negative? For example,
5+(-12) means you have 5 dollars and spend 12 dollars. The result is (-7)
or a debt of 7 dollars.
What if both numbers are negative? If you have two
debts, like (-7) + (-12), the result is (-19) or a total debt of 19
dollars.
In summary, it is obvious that when you add positive
numbers, you get a positive number. When you add positive and negative
numbers, you subtract the numbers and the sign of the answer is the same
as the sign of the larger magnitude. When you add negative numbers, you
always add the numbers and the sign is always negative.
ADDITION
RULES
RULE SIGN OF ANSWER
WHAT TO DO
EXAMPLE
(+)+(+)
+
Add the numbers
(+8)+(+4) = +12
(-)+(-)
-
Add the numbers
(-8)+(-4) = -12
(+)+(-)
Sign of the larger
Subtract the numbers
(+12)+(-8) = +4
(-)+(+)
Sign of the larger
Subtract the numbers
(-12)+(+8) = -4
|
EXERCISES:
1.
(-8) + 12 = ____
2. (-12) + 8 = ____
3. (-10) + 8 = ____
4. 10 + (-4) =
____
5. (-2) + (-6) = ____
6. (-14) + (-6) = ____
7. (-32) + 28 =
____
8. (-18) + 30 = ____
9. (-12) + (-14) =
____
10.
(-6) + 30 = ____
11. (-20) + 8 = ____
12.
(-32) + (-8) = ____
13.
26 + (-14) = ____
14. (-24) + (-16) =
____
15. (-14) + (-14) =
____
16.
(-15) + 28 = ____
17. (-22) + 64 = ____
18. (-12) + (-88) =
____
19.
(-12) + 20 = ____
20. (-20) + 12 = ____
21. (-12) + (-8) =
____
22.
16 + (-4) = ____
23. (-34) + (-16) =
____
24. (-14) + (-28) =
____
25.
(-125) + 28 = ____ 26. (-52) + 120 = ____
27. (-12) + (-87) =
____
28.
(-12) + 34 + (-26) = _______
29. (-28) +
(-125) + 95 = _______
30.
200 + (-120) + (-85) = ______
31. (-200) + (-135) +
75 = ______
32.
(-40) + 62 + (-62) +(-73) + 73 + 726 + (-726) = _______
When
subtracting negative numbers, remember that the negative of a negative is
the positive of the number. For
examples, -(-8) is a (+8) or just 8; -(-10) is +10 or 10; -(-x) is x; etc.
Of course, the negative of a positive is a negative.
As examples, -(+8)
is (-8); ‑(+10) is -10; etc.
SUBTRACTION
RULES
- (- x)
= +x
- (+x) = - x
|
33.
-(-2) = _____
34. -(-6) = _____
35.
-(-10) = _______
36.
-(-12) = _____
37. -(-18) = _____
38. -(-120) = _______
39.
8 - (-12) = 8 + ___
40. -8 - (-12) = -8 + ____
=
________
= _________
41.
-12 -(-12) = ____ + ____
42. -8 - (-8) = ____ +
____
= ________
= _________
43.
12 - (-8) = ________
44. -12 - (-8) =
_________
= ________
= _________
45.
(-38) - (-12) = ________
46. 38 - (-12) =
_________
= ________
= _________
RULES
FOR MULTIPLICATION AND DIVISION
RULE
SIGN OF ANSWER
RULE
SIGN OF ANSWER
(+)
• (+) +
(+)
÷ (+)
+
(+)
• (-)
-
(+) ÷ (-)
-
(-) • (+)
-
(-) ÷ (+)
-
(-)
• (-)
+
(-) ÷ (-) +
|
When When multiplying or dividing positive and negative numbers, remember
that multiplying is actually a short way to add.
For example, 4 x 3 means 4 threes.
In this way 4 x (-3) means 4 three dollar debts or (-12).
Likewise, (-3) x 4 is also (-12).
In the same way that a negative
of a negative is a positive, a negative
times a negative is a positive.
The following are rules for multiplication and division of signed
numbers. Remember also, that
the word “of” means “times”.
EXERCISES:
47.
4(-5) = ______
48. (-3)(-4) =
_____
49. (-6) 4 =
_____
50.
(-7)(-3) = ______
51. (-6)(7) =
_____
52. (-7) (-9) =
____
53.
7(-8) = ______
54. (-6)(-9) =
_____
55. (-23) 4 =
_____
56.
(-17)(-3) = ______
57. (-16)(9) = _____
58. (-5) (-24) =
____
59.
7(-8)(-2) = _____
60. 5(-9)(2) = _____
61. (-2)(-4)(-25)
= _____
62.
(-7)(-8)(-2) = ____
63. 5(-9)(-2) = _____
64. (-14)(-4)(-25)
= ____
When raising to a power, did you notice that a
positive number raised to any power is always positive?
Did you notice that a negative number raised to an even power is
always positive? Perhaps you
noticed that a negative number raised to an odd power is always negative.
POWER
RULES
(POSITIVE)ANY
POWER = POSITIVE
(NEGATIVE)EVEN
POWER = POSITIVE
(NEGATIVE)ODD
POWER = NEGATIVE
|
EXERCISES:
65.
(-2)2 = _____
66. (-5)2 = _____
67. (-4)2 = _____
68.
(-5)3 = _____
69. (-3)3 = _____
70. (-4)3 = _____
71.
(-1)12 = _____
72. (-1)9 = _____
73. (-1)15 = _____
74.
(-1)24 = _____
75. (-10)3 = _____
76. (-3)4 = _____
77.
(-2)3(-1) = ____
78. (-3)4 (-1)8 = ____
79. (-2)4 (-1)3 = ____
80.
(-2)5 (-1)2 = ____
81.
(-5)2 (-2)3 = ____ 82. (-3)2 (-2)3
= ____
83.
(-2)2 (-3)3
= ____
84. (-2)3 (-3)3 = ____
85. (-3)2 (-2)2
= ____
However, be
careful when the negative number does not have parentheses.
Is there a difference between
(-2)2 and -2 2 ? The quantity (-2)2 means (-2)
times (-2), which is +4. However,
-22 means the negative of (two to the second power).
By order of operations agreement, this means to raise two to the
second power, and then take the negative.
This result is -4. The important question to ask is this: “What is it that you
are raising to the power?” In
the case of (-2)2 , you are raising (-2) to the second power. However, with -22 only the 2 is squared, not the “‑”.
Therefore,
(-2)2 = 4, but -22 = -4
Also notice
that in
(-22), only the 2 (not the negative) is squared.
Therefore
(-22)= -4. Notice that
(-2)2
(-22 ).
EXERCISES.
Complete the following:
86. -24 = _____
87. (-2)4 = _____
88. (-5)2 = _____
89.
-52 = _____
90. (-2)3 = _____
91. -23 = _____
92.
-34 = _____
93. (-3)4 = _____
94. (-5)3 = _____
95.
-53 = _____
96.
-110 = _____
97. -113 = _____
98. -22 (-3)3 99.
-23 (-3)2
100. (-3)3 (-2)2
ABSOLUTE VALUE
The absolute
value of a number, denoted by vertical bars on each side of a number
or a quantity, represents the size or
magnitude of that number.
Another way to think of absolute value of a number is the distance
on the number line of that number from zero.
As examples, |‑3| is 3; |‑7| is 7; |10| is 10; |0| is 0. Notice
that the absolute value definition does not apply to what is outside the
absolute value bars. For
example, -|3| is -3; -|‑3| is -3; -|7| is -7; -|‑7| is
-7.
EXERCISES:
101. |-4|
102. |-6|
103. -|4|
104. - | - 6|
105. - |-6|
106. - |-4|
107.
|-
4| + 3 |- 3|
108. |‑
5| - 3|‑ 3|
109.
- |‑ 8| + 3 |‑ 9|
110.
-3 |‑ 5| - 5 |‑ 6|
111. - 4 |4 - 6|
- 8 |‑ 8 + 3|
112. 9 |‑ 7 + 1| - 3 |5 ‑ 9|
113. |-11| - | - 5|
2 114.
|-5 - 3|
2 - 4|7
- 2|
115.
| - 8 - 5|
2 + |3 - 12|
2
116. |- 82-
5 | + |3
2 – 12|
117. |- 82
+
5 | - |3
2 – 12|
118.
- | - 5 2 - 3 2|
ANSWERS 1.03
p. 13 - 20:
1. 4; 2. -4; 3. -2;
4. 6; 5. -8; 6. -20; 7. -4; 8. 12;
9. -26; 10. 24; 11. -12; 12. -40;
13. 12; 14. -40; 15. -28; 16.
13; 17. 42; 18. -100; 19. 8; 20. -8;
21. -20; 22. 12; 23. -50; 24. -42;
25. -97; 26. 68; 27. -99; 28. -4;
29. -58; 30. -5; 31. -260; 32. -40; 33. 2;
34. 6; 35. 10; 36. 12; 37. 18; 38.
120; 39. 20; 40. 4; 41. 0; 42. 0; 43.
20; 44. -4; 45. -26; 46. 50; 47. -20;
48. 12; 49. -24; 50. 21; 51. -42; 52.
63; 53. -56; 54. 54; 55. -92; 56.
51; 57. -144; 58. 120; 59. 112; 60.
-90; 61. -200; 62. -112; 63. 90; 64.
-1400; 65. 4; 66. 25; 67. 16; 68. -125;
69. -27; 70. -64; 71. 1; 72. -1;73. -1; 74. 1; 75. -1000;
76.
81; 77. 8; 78. 81; 79. -16; 80. -32; 81. -200;
82. -72; 83. -108; 84. 216; 85.
36; 86. -16; 87. 16; 88. 25; 89. -25;
90. -8;
91. -8; 92. -81; 93. 81;
94. -125; 95.
-125; 96. -1; 97. -1; 98. 108; 99. -72;
100.-108; 101. 4; 102. 6; 103. -4;
104. -6; 105. -6; 106. -4; 107. 13; 108.
-4; 109. 19; 110. -45; 111. -48;
112. 42; 113. -14; 114. 44; 115.
250; 116. 72; 117. 56; 118. -34.
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