Solving Absolute Value Inequalities in 2026
Absolute value inequalities compare a distance expression $|ax + b|$ against a boundary $c$. The fundamental mnemonic for solving these is:
- "Less thAND" ($<, \le$): Forms a compound conjunction (AND), generating a bounded interior segment.
- "GreatOR" ($>, \ge$): Forms a compound disjunction (OR), generating two exterior rays extending toward infinity.
Inequality Rules Reference Table
| Inequality Form | Logical Translation | Solution Structure | Interval Notation ($c > 0$) | ||
|---|---|---|---|---|---|
| **$ | u | < c$** | $-c < u < c$ | Bounded Open Segment | $(-c, c)$ |
| **$ | u | \le c$** | $-c \le u \le c$ | Bounded Closed Segment | $[-c, c]$ |
| **$ | u | > c$** | $u < -c \lor u > c$ | Two Disjoint Open Rays | $(-\infty, -c) \cup (c, \infty)$ |
| **$ | u | \ge c$** | $u \le -c \lor u \ge c$ | Two Disjoint Closed Rays | $(-\infty, -c] \cup [c, \infty)$ |
Mathematical Formulations
\text{For } |ax + b| < c: \quad -c < ax + b < c \implies \frac{-c - b}{a} < x < \frac{c - b}{a}
\text{For } |ax + b| > c: \quad ax + b < -c \;\lor\; ax + b > c \implies x < \frac{-c - b}{a} \;\lor\; x > \frac{c - b}{a}