Absolute-Value Inequalities

abs(x) is just distance from 0 on the number line, with no direction attached. That single fact drives everything. If abs(x) < 4, you want every point within 4 units of 0, which is one continuous band: -4 < x < 4. If abs(x) > 4, you want every point farther than 4 units away in either direction, which breaks into two rays that never touch: x < -4 or x > 4. Less-than gives you an and statement (x is above -4 and below 4). Greater-than gives you an or statement (x is below -4 or above 4), because no single band can hold points shooting off both ways.

To solve, strip the bars by writing two inequalities. For abs(3x + 2) ≥ 8, set up 3x + 2 ≥ 8 and 3x + 2 ≤ -8, then solve each separately. The first gives x ≥ 2, the second gives x ≤ -10/3. Graph both as rays pointing outward from those points, nothing in between.

This topic sits next to linear inequalities and compound inequalities, since solving each branch is just linear-equation work you already know, repeated twice with a sign flip on one branch. It also sets up absolute-value equations and graphs of V-shaped functions, where the same distance idea explains the vertex and the two symmetric slopes.