James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix A Numbers, Inequalities, and Absolute Values A7
Example 5 Express | 3x 2 2 | without using the absolute-value symbol.
SOLUTION
| 3x 2 2 | − H 3x 2 2
2s3x 2 2d
if 3x 2 2 > 0
if 3x 2 2 , 0
−H 3x 2 2
2 2 3x
if x > 2 3
if x , 2 3
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Recall that the symbol s means “the positive square root of.” Thus sr − s means
s 2 − r and s > 0. Therefore the equation sa 2 − a is not always true. It is true only
when a > 0. If a , 0, then 2a . 0, so we have sa 2 − 2a. In view of (3), we then
have the equation
4
sa 2 − | a |
which is true for all values of a.
Hints for the proofs of the following properties are given in the exercises.
5 Properties of Absolute Values Suppose a and b are any real numbers and
n is an integer. Then
1. | ab | − | a | | b | 2. Z
a
b
Z − | a |
| b |
sb ± 0d
3. | a n | − | a | n
For solving equations or inequalities involving absolute values, it’s often very helpful
to use the following statements.
_a
x
FIGURE 7
b
a
a
|x|
0 a
| a-b |
| a-b |
FIGURE 8
Length of a line segment − | a 2 b |
a
a
b
6 Suppose a . 0. Then
4. | x |
5. | x |
6. | x |
− a if and only if x − 6a
, a if and only if 2a , x , a
. a if and only if x . a or x , 2a
, a says that the distance from x to the origin is less
than a, and you can see from Figure 7 that this is true if and only if x lies between 2a
and a.
For instance, the inequality | x |
If a and b are any real numbers, then the distance between a and b is the absolute
value of the difference, namely, | a 2 b | , which is also equal to | b 2 a | . (See Figure 8.)
Example 6 Solve | 2x 2 5 | − 3.
SOLUTIOn By Property 4 of (6), | 2x 2 5 | − 3 is equivalent to
2x 2 5 − 3 or 2x 2 5 − 23
So 2x − 8 or 2x − 2. Thus x − 4 or x − 1.
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