Solving Inequalities Involving |x − p|
Simple Explanation
For q > 0: |x − p| < q means x is within q units of p, giving a single interval, p − q < x < p + q. |x − p| > q means x is farther than q units from p, giving two separate intervals: x < p − q or x > p + q.
Why Do We Need It?
This distinguishes a "close to" condition (a single connected range, from < or ≤) from a "far from" condition (two separate ranges, from > or ≥) — a common real-world distinction, such as tolerance ranges versus alarm thresholds.
See It
A V-shaped graph with vertex (3, -5), crossing the x-axis at x = -2 and x = 8, illustrating where the graph is below versus above zero
Formula
Absolute Value Inequality Rules
|x − p| < q ⇔ p − q < x < p + q |x − p| > q ⇔ x < p − q or x > p + q (q > 0)
"Less than" absolute value inequalities describe a single band of values close to p; "greater than" absolute value inequalities describe two separate regions far from p.
- x
- — the unknown being solved for
- p
- — the reference point
- q
- — the distance threshold (q > 0)
When to use it: Whenever you need to solve an inequality with a single absolute value expression compared to a positive number.
Worked Example
Solve an absolute value inequality
Solve |x − 3| < 5.
Why Does This Work?
|x − p| < q means the distance from x to p is less than q, which describes every point strictly between p − q and p + q on the number line — a single band around p. The reverse, |x − p| > q, describes everywhere outside that band, which is naturally two separate rays.
Real-Life Example
Temperature control alarm
A greenhouse temperature system triggers an alarm whenever the temperature strays more than 4°C from the target 22°C.
|temperature − 22| > 4 models the alarm condition — the "greater than" pattern gives two separate danger zones (too hot or too cold), exactly matching the two-interval rule.
Practice
Solve |x + 1| > 4.
HardCommon mistake
Using the "less than" pattern (single interval) for a "greater than" inequality, or vice versa — always check the inequality direction first: < or ≤ gives one interval; > or ≥ gives two.
Quick Review
- |x − p| < q: one interval, p − q < x < p + q.
- |x − p| > q: two intervals, x < p − q or x > p + q.
- These rules only apply directly when q > 0 — check that first.