Module 2 · Algebra
Chapter 19: Inequalities
Algebra
4 topics
What this chapter covers
The 4 topics below make up Inequalities on the CAT. Click any topic to open its notes, examples and practice test.
Study material & tests
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Notes, worked examples and hint-guided practice for every topic.
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🎯 Topic tests
8 auto-graded questions per topic, pulled from the question bank.
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📝 Chapter test
25 fresh mixed-difficulty questions, timed, with full solutions.
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Formula Reference Sheet
This chapter
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 |
CAT reference
🖩 Graphing Calculator