The discussion of absolute value inequalities in one variable will be finalized with some examples.
Example: Solve the inequality |3x + 5| < 2.
Discussion and Solution:
We begin by converting |3x + 5| < 2 to the equivalent compound inequality by wedging 3x + 5 between 2 and its opposite.
If set builder notation is used to write this solution set we have
The graph of this solution set is shown in Fig. 8.
Observe that our discussion involves four equivalent ways to describe the solution set
i. compound inequality
ii. interval notation
iii. set builder notation
As an easy bonus we observe that no further computation is required to deduce the solution sets for |3x + 5| = 2 and |3x + 5| > 2
The solution of the boundary equation |3x + 5| = 2 consists of the endpoints of the interval .
Therefore, the solution set for |3x + 5| = 2 is
The graph of this solution set is in Fig. 9.
Furthermore, we can conclude that the solution set for the “greater than” inequality |3x + 5| > 2 is everything else. The solution set for |3x + 5| > 2 is
The graph of this solution set is in Fig. 10.
It is instructive to graph all three solution sets on the same number line to obtain Fig. 11.
It is very important to observe that the solution sets do not overlap and their union is the entire real number line.