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Lesson 2.1 Exercises, pages 89–94
A
3. Write the absolute value of each number.
a) 15 円15 円 ⴝ 15
b) -7.6
c) 0 円0円 ⴝ 0
d) - 2
7
円ⴚ7.6円 ⴝ 7.6
2
ⴚ
72
7
ⴝ
2
2
4. Determine the absolute value of the number represented by each
point on the number line.
D
⫺3
C
⫺2
⫺1
A: 円1 円 ⴝ 1
B
A
0
1
B: 円0 円 ⴝ 0
2
3
C: 円ⴚ1.5円 ⴝ 1.5
D: 円ⴚ3円 ⴝ 3
5. Use absolute value to determine the distance between each pair of
numbers on a number line. Sketch a number line to illustrate the
solution.
a) -5 and 7
b) -3 and -10
円7 ⴚ (ⴚ 5)円 ⴝ 円12円
ⴝ 12
円ⴚ10 ⴚ (ⴚ3)円 ⴝ 円ⴚ7円
ⴝ7
7 units
12 units
⫺6
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⫺4
⫺2
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0
2
4
6
8
⫺12 ⫺10 ⫺8
⫺6
⫺4
⫺2
2.1 Absolute Value of a Real Number—Solutions
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6. Evaluate.
a)
√
52
Since
b)
√2
x ⴝ x,
√ 2
√
1 -422
Since
√
√
5 ⴝ 5
x2 ⴝ x ,
(ⴚ4)2 ⴝ ⴚ4 ⴝ5
c)
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ⴝ4
√
10.12
√
Since x2 ⴝ x,
√
2
d)
√
1 -11.422
Since
√
x2 ⴝ x ,
√
10.1 ⴝ 10.1
(ⴚ11.4)2 ⴝ ⴚ11.4
ⴝ 10.1
ⴝ 11.4
B
7. a) Determine each absolute value.
ƒ -11 ƒ , ƒ 12 ƒ , ƒ 0 ƒ , ƒ 4 ƒ , ƒ -5 ƒ
12 ⴝ 12
ⴚ 11 ⴝ 11
4 ⴝ 4
0 ⴝ 0
ⴚ5 ⴝ 5
b) Order the numbers in part a from least to greatest.
Show the solution on a number line.
From least to greatest:
0<4<5<11<12
That is, 0 <4 < ⴚ 5< ⴚ11 < 12 5
0
11
4
0
2
4
12
6
8
10
12
14
8. For each pair of numbers, write two expressions to represent the
distance between the numbers on a number line, then determine
this distance.
3
1
a) -2 4 and 5 4
3
1
1
3
1
3
ⴚ 5 2 and 2 5 ⴚ aⴚ2 b 2 , or 2 5 ⴙ 2 2
4
4
4
4
4
4
2 ⴚ2
2
3
4
12
11
21 2
ⴝ 2ⴚ ⴚ
4
4
4
32
ⴝ 2ⴚ 2
4
ⴚ2 ⴚ 5
ⴝ ⴚ8 ⴝ8
The numbers are 8 units apart on a number line.
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2.1 Absolute Value of a Real Number—Solutions
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7
b) -5 5 and -3 10
2
ⴚ5 ⴚ aⴚ3 b 2 , or 2 ⴚ5 ⴙ 3
3
5
7
10
3
5
7 2
,
10
7
3
7
3
ⴚ aⴚ5 b 2 , or 2 ⴚ3 ⴙ 5 2
10
5
10
5
and 2 ⴚ3
7 2
28
37 2
ⴝ 2ⴚ ⴙ
10
5
10
56
37 2
ⴝ 2ⴚ ⴙ
10
10
19 2
2
ⴝ ⴚ
10
19
9
ⴝ , or 1
10
10
9
The numbers are 1 units apart on a number line.
10
2
3
5
ⴚ5 ⴙ 3
c) 12.47 and -8.23
12.47 ⴚ ( ⴚ8.23) , or 12.47 ⴙ 8.23 , and ⴚ8.23 ⴚ 12.47
12.47 ⴙ 8.23 ⴝ 20.7
ⴝ 20.7
The numbers are 20.7 units apart on a number line.
9. Order the absolute values of the numbers in this set from greatest to
least. Describe the strategy you used.
1
8.5, -8 3, -10.2, -0.2, 0.1, 9.8
12
1
ⴝ8
3
3
8.5 ⴝ 8.5
2
ⴚ8
ⴚ0.2 ⴝ 0.2
0.1 ⴝ 0.1
ⴚ10.2 ⴝ 10.2
9.8 ⴝ 9.8
From greatest to least:
1
3
10.2>9.8>8.5>8 >0.2>0.1
1
3
That is, ⴚ10.2 , 9.8, 8.5, 2 ⴚ8 2 , ⴚ0.2 , 0.1
I determined the absolute value of each number, then ordered
the results from greatest to least.
10. The square of a number is 36. What is the absolute value of the
number? Show your reasoning.
6
√36 ⴝ √
2
(ⴚ6)
√36 ⴝ √
2
36 ⴝ 6
36 ⴝ (ⴚ6)2
ⴝ 6 ⴝ ⴚ 6
ⴝ6
ⴝ6
So, the absolute value of the number is 6.
2
11. Evaluate.
a)
√
1 2
a3 b
8
12
8
1
ⴝ3
8
ⴝ 23
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b)
√
a- 2 b
5
6
2
52
6
ⴝ 2 ⴚ2
5
6
ⴝ2
2.1 Absolute Value of a Real Number—Solutions
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c)
√
19 - 1122
√
ⴝ (ⴚ 2)2
ⴝ ⴚ2 ⴝ2
d)
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√
113.5 - 16.122
√
ⴝ (ⴚ2.6)2
ⴝ ⴚ2.6
ⴝ 2.6
12. Evaluate.
a) ƒ 5142 - 3 ƒ
ⴝ 20 ⴚ 3
ⴝ 17
ⴝ 17
c) 10 - ƒ 5 - 8 ƒ
ⴝ 10 ⴚ ⴚ3 ⴝ 10 ⴚ 3
ⴝ7
e) 3 ƒ -2 ƒ - 4 ƒ 3 ƒ
ⴝ 3(2) ⴚ 4(3)
ⴝ 6 ⴚ 12
ⴝ ⴚ6
b) 3 ƒ 10 - 15 ƒ
ⴝ 3 ⴚ5
ⴝ 3(5)
ⴝ 15
d) ƒ 6 + 1 -102 ƒ - ƒ 5 - 7 ƒ
ⴝ ⴚ4 ⴚ ⴚ2
ⴝ4ⴚ2
ⴝ2
f) - 3 ƒ 4 - 1 ƒ + 2 ƒ 1 - 4 ƒ
ⴝ ⴚ33 ⴙ 2ⴚ3 ⴝ ⴚ3(3) ⴙ 2(3)
ⴝ ⴚ9 ⴙ 6
ⴝ ⴚ3
13. Evaluate.
a)
√
15.1 - 2.322 - ƒ 5.1 - 2.3 ƒ
√
ⴝ 2.82 ⴚ 2.8
ⴝ 2.8 ⴚ 2.8
ⴝ0
c)
ƒ 2 - 1 - 82 ƒ
ƒ 3 ƒ - ƒ -2 ƒ
10 3ⴚ2
10
ⴝ
1
ⴝ
ⴝ 10
4
b) 5.1 - 2.3 - ƒ 5.1 - 2.3 ƒ
ⴝ 2.8 ⴚ 2.8
ⴝ 2.8 ⴚ 2.8
ⴝ0
d) ƒ 5 - 4 ƒ 15 + 42 - 215 + 42
ⴝ 1 (9) ⴚ 2(9)
ⴝ 1(9) ⴚ 18
ⴝ 9 ⴚ 18
ⴝ ⴚ9
2.1 Absolute Value of a Real Number—Solutions
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2
5
1
e) 3 2 - 8 - 4 2
22 5
2
ⴚ ⴚ 2
3 8
8
22 72
ⴝ ⴚ
3 8
2 7
ⴝ
a b
3 84
7
ⴝ
12
ⴝ
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3
1
3
1
f) 2 1 4 - 2 2 2 - 2 4 - 2 2
7
5
3
2
ⴚ 2 ⴚ 2 ⴚ 2
4
2
4
4
7
10
1
2 ⴚ 2 2
ⴝ 2 ⴚ
4
4
4
32
1
2
ⴝ ⴚ ⴚ
4
4
3
1
ⴝ ⴚ
4
4
2
1
ⴝ , or
4
2
ⴝ
2
14. Evaluate each expression for the given value of x.
a) ƒ x2 + 6x - 5 ƒ , x = 2
ⴝ
ⴝ
ⴝ
ⴝ
22 ⴙ 6(2) ⴚ 5 4 ⴙ 12 ⴚ 5 11
11
c) ƒ x3 - 5x - 2 ƒ , x = - 3
ⴝ
ⴝ
ⴝ
ⴝ
( ⴚ3)3 ⴚ 5( ⴚ3) ⴚ 2
ⴚ 27 ⴙ 15 ⴚ 2 ⴚ14 14
b) ƒ - 3x2 + 7x + 1 ƒ , x = 3
ⴝ
ⴝ
ⴝ
ⴝ
ⴚ3(3)2 ⴙ 7(3) ⴙ 1
ⴚ27 ⴙ 21 ⴙ 1
ⴚ5 5
d) ƒ 5x3 + 2x + 7 ƒ , x = -1
ⴝ
ⴝ
ⴝ
ⴝ
5(ⴚ1)3 ⴙ 2(ⴚ1) ⴙ 7 ⴚ5 ⴚ 2 ⴙ 7 0 0
C
15. Two integers, a and b, are between -5 and 5. The sum of the
absolute values of the integers is 4. What might the integers be?
How many answers are possible?
Integers between ⴚ5 and 5 are: ⴚ4, ⴚ3, ⴚ2, ⴚ1, 0, 1, 2, 3, 4
|a | ⴙ | b| ⴝ 4
So, I test all pairs of these integers using the equation above.
Possible pairs of integers are: a ⴝ 4, b ⴝ 0; a ⴝ 3, b ⴝ 1; a ⴝ 2, b ⴝ 2;
a ⴝ 1, b ⴝ 3; a ⴝ 0, b ⴝ 4; a ⴝ ⴚ4, b ⴝ 0; a ⴝ ⴚ3, b ⴝ ⴚ1; a ⴝ ⴚ2,
b ⴝ ⴚ2; a ⴝ ⴚ1, b ⴝ ⴚ3; a ⴝ 0, b ⴝ ⴚ4; a ⴝ ⴚ3, b ⴝ 1; a ⴝ ⴚ1,
b ⴝ 3; a ⴝ 3, b ⴝ ⴚ1; a ⴝ 1, b ⴝ ⴚ3; a ⴝ ⴚ2, b ⴝ 2; a ⴝ 2, b ⴝ ⴚ2
There are 16 possible answers.
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16. When 7 is added to an integer, x, the absolute value of the sum is 12.
Determine a value for x. How many values of x are possible? Show
what you did to solve the problem.
Write, then solve an equation: x ⴙ 7 ⴝ 12
Since 12 ⴝ 12 and ⴚ12 ⴝ 12,
then, x ⴙ 7 ⴝ 12
or
x ⴙ 7 ⴝ ⴚ12
xⴝ5
x ⴝ ⴚ19
So, two values of x are possible: 5 or ⴚ19
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2.1 Absolute Value of a Real Number—Solutions
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