You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x |? 8 has two solutions: 8 and ?8.

1
You can solve some absolute-value equations using
mental math. For instance, you learned that the
equation x ? 8 has two solutions 8 and ?8.
To solve absolute-value equations, you can use
the fact that the expression inside the absolute
value symbols can be either positive or negative.
2
Solve x ? 2 ? 5
Solve x ? 2 ? 5

SOLUTION
The expression x ? 2 can be equal to 5 or ?5.
x ? 2 IS POSITIVE
x ? 2 IS NEGATIVE
x ? 2 ? 5
x ? 2 ? 5
x ? 2 ? ?5
x ? 2 ? ?5
x ? 7
x ? ?3
The equation has two solutions 7 and 3.
7 ? 2 ? 5 ? 5
?3 ? 2 ? ?5 ? 5
3
Solve 2x ? 7 ? 5 ? 4
Solve 2x ? 7 ? 5 ? 4

SOLUTION
Isolate the absolute value expression on one side
of the equation.
2x ? 7 IS POSITIVE
2x ? 7 IS NEGATIVE
2x ? 7 ? 5 ? 4
2x ? 7 ? 5 ? 4
2x ? 7 ? 9
2x ? 7 ? 9
2x ? 7 ? 9
2x ? 7 ? ?9
2x ? 16
2x ? ?2
x ? 8
x ? ?1
4
Recall that ?x ?is the distance between x and 0.
If ?x ?? 8, then any number between ?8 and 8 is
a solution of the inequality.
Recall that x is the distance between x and
0. If x ? 8, then any number between ?8 and 8
is a solution of the inequality.
You can use the following properties to solve
absolute-value inequalities and equations.
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SOLVING ABSOLUTE-VALUE INEQUALITIES
a x ? b ? c and a x ? b ? ? c.
a x ? b ? c
a x ? b ? c
a x ? b ? c and a x ? b ? ? c.
When an absolute value is less than a number, the
inequalities are connected by and. When an
absolute
a x ? b ? c or a x ? b ? ? c.
a x ? b ? c
value is greater than a number, the inequalities
are
connected by or.
a x ? b ? c or a x ? b ? ? c.
a x ? b ? c
a x ? b ? c or a x ? b ? ? c.
a x ? b ? c
6
Solve x ? 4 lt 3
x ? 4 IS POSITIVE
x ? 4 IS NEGATIVE
x ? 4 ? 3
x ? 4 ? 3
x ? 4 ? ?3
x ? 4 ? ?3
x ? 7
x ? 1
Reverse inequality symbol.
The solution is all real numbers greater than 1
and less than 7.
This can be written as 1 ? x ? 7.
7
Solve 2x ? 1 ?3 ? 6 and graph the solution.
2x 1 IS NEGATIVE
2x 1 IS POSITIVE
2x ? 1 ? 3 ? 6
2x ? 1 ?3 ? 6
2x ? 1 ? 9
2x ? 1 ? 9
2x ? 1 ? ?9
2x ? 1 ? 9
2x ? ?10
2x ? 8
The solution is all real numbers greater than or
equal to 4 or less than or equal to ? 5. This
can be written as the compound inequality x ? ? 5
or x ? 4.
x ? ?5
x ? 4
4.
? 5
Reverse inequality symbol.
8
You work in the quality control department of a
manufacturing company. The diameter of a drill
bit must be between 0.62 and 0.63 inch.
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The diameter of a drill bit must be between 0.62
and 0.63 inch. a. Write an absolute-value
inequality to represent this requirement.
Let d represent the diameter (in inches) of the
drill bit.
0.62 ? d ? 0.63
Write a compound inequality.
Find the halfway point.
0.625
0.62 ? 0.625 ? d ? 0.625 ? 0.63 ? 0.625
Subtract 0.625 from each part of the compound
inequality.
?0.005 ? d ? 0.625 ? 0.005
Rewrite as an absolute-value inequality.
d ? 0.625 ? 0.005
10
The diameter of a drill bit must be between 0.62
and 0.63 inch. b. Does a bit with a diameter of
0.623 meet the requirement?
d ? 0.625 ? 0.005
0.623 ? 0.625 ? 0.005
?0.002 ? 0.005
0.002 ? 0.005