Modulus Inequalities
The modulus |x| measures distance from zero, and two clean rules unlock most CAT questions. For a > 0: |x| < a means x lies within a of zero, so −a < x < a; |x| > a means x lies farther than a from zero, so x < −a or x > a. Shift these to a centre: |x − c| < a means c − a < x < c + a, an interval centred at c with radius a. So |x − 3| ≤ 5 gives −2 ≤ x ≤ 8. For the greater-than form, |x − 3| > 5 splits into x < −2 or x > 8. When the inequality mixes a modulus with a linear term, or has two modulus terms, fall back to a case analysis on the sign inside each modulus, solving each case on its own interval and taking the union. Always check the boundary points, since ≤ and < change whether endpoints are included.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Core rules & linear
| Sign-flip rule | Multiply/divide both sides by a negative ⇒ reverse the inequality |
|---|---|
| Adding a constant | a > b ⇒ a + c > b + c (direction unchanged) |
| Multiply by positive k | a > b, k > 0 ⇒ ka > kb |
| Reciprocal (same sign) | 0 < a < b ⇒ 1/a > 1/b |
| Transitivity | a > b and b > c ⇒ a > c |
CAT power-tools
| Modulus less-than | |x| < a ⇔ −a < x < a (a > 0) |
|---|---|
| Modulus greater-than | |x| > a ⇔ x < −a or x > a (a > 0) |
| Quadratic sign | a(x−p)(x−q) with a > 0: negative between roots, positive outside |
| AM-GM (n positives) | (a₁+…+aₙ)/n ≥ (a₁…aₙ)^(1/n), equality when all equal |
| AM-GM corollary | For x > 0, x + 1/x ≥ 2; x + k/x ≥ 2√k |