Download Absolute Value Inequalities

Solve Absolute Value Inequalities
When an inequality contains an absolute value, the absolute value
must be removed in order to graph the solution and/or give
interval notation. The way the absolute value is removed depends
on the direction of the inequality symbol.
Consider |x| < 2.
Since absolute value is the distance from zero, this would be read
as “the distance from zero is less than 2”. So on a number line,
the solution area is all points less than 2 units away from zero.
-2 < x < 2
Notice that the graph looks just like a conjunction. The interval
notation is (-2, 2).
Consider |x| > 2
Since absolute value is the distance from zero, this would be read
as “the distance from zero is greater than 2”. So on a number line,
the solution area is all points more than 2 units away from zero.
x < -2 or x > 2
Notice that the graph looks just like a disjunction. The interval
notation is (-∞, -2) ∪ (2, ∞).
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
To solve absolute value inequalities, the procedure is similar to
that used when solving absolute value equations:
Isolate the absolute value.
If the inequality is greater than (>) or greater than or equal
to (>), solve the separate inequalities.
If the inequality is less than (<) or less than or equal to (<),
make a three-part inequality (tripartite), and solve.
Remember, when multiplying or dividing by a negative, the
inequality symbol will switch directions.
Example 1: Solve, graph, and write interval notation for the solution
|4x − 5| > 6
> so use separate inequalities
4x – 5 < -6
+5 +5
4x
< -1
4
4
x<-
or
4x – 5 > 6
+5+5
4x > 11
4
4
1
x>
4
1
11
4
4
(−∞, − ] ∪ [
Solve each inequality
Add 5 to all sides
Divide all sides by 4
11
4
, ∞)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 2: Solve, graph, and write interval notation for the solution
-4 - 3|x| < -16
Isolate the absolute value
+4
+4
Add 4 to both sides
- 3|x| < -12
Divide both sides by -3
-3
-3
Switch the inequality sign
|x| > 4
> so use separate inequalities
x < -4 or x > 4
(-∞, -4] ∪ [4, ∞)
Example 3: Solve, graph, and write interval notation for the solution
9 - 2|4x + 1| > 3
Isolate the absolute value
-9
-9
Subtract 9 from both sides
- 2|4x + 1| > -6
Divide both sides by -2
-2
-2
Switch the inequality sign
|4x + 1| < 3
< so use the tripartite
-3 < 4x + 1 < 3
-1
-1 -1
-4 < 4x < 2
4
4
4
-1 <
x
<
Subtract 1 from all sides
Divide all sides by 4
1
2
1
(−1, )
2
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
It is important to remember that when solving these equations, the
absolute value is always positive – greater than a negative. An
absolute value less than a negative number indicates no solution.
Similarly, an absolute value greater than a negative indicates a
solution of all real numbers.
Example 4: Solve, graph, and write interval notation for the solution
12 + 4|6x − 1| < 4 Isolate the absolute value
- 12
- 12 Subtract 12 from both sides
4|6x − 1| < -8
4
4 Divide both sides by 4
|6x − 1| < -2
Since absolute value cannot be less than a negative, there is no solution
or ∅.
Example 5: Solve, graph, and write interval notation for the solution
5 - 6|x + 7| < 17
Isolate the absolute value
-5
-5
Subtract 5 from both sides
-6|x + 7| < 12
-6
-6
Divide both sides by -6
|x + 7| > -2
Switch the inequality sign
Since absolute value is always greater than a negative, the solution is all
real numbers.
(-∞, ∞)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)