Inequalities
Inequalities are where CAT separates the careful from the careless. The rules look like ordinary equation-solving, but a single overlooked step — flipping the sign when you multiply or divide by a negative — turns a correct method into a wrong answer. CAT loves this because it can test deep reasoning without heavy arithmetic: a question on the range of x, the number of integer solutions, or the minimum value of an expression rewards a clean method and punishes guesswork. This chapter builds that discipline in four stages. First, linear inequalities and the sign-flip rule that governs everything. Then quadratic and rational inequalities, solved fast with the wavy-curve (sign-scheme) method rather than by testing points blindly. Next, modulus inequalities, where |x| < a and |x| > a unfold into clean intervals. Finally, the graphical view and the AM-GM inequality, the single most useful tool for finding the minimum or maximum of an expression without calculus. Each topic carries worked examples, a hint ladder, and the traps that quietly cost marks, so you leave knowing not just the answer but the fastest safe route to it.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Multiply or divide by a negative ⇒ flip the inequality. Never flip when adding/subtracting or multiplying by a positive.
- Quadratics: factor, mark roots, and use the wavy curve — for a positive leading coefficient the sign is + outside the roots and − between them.
- Rational inequalities: treat numerator and denominator roots as critical points, but always EXCLUDE denominator roots (undefined).
- |x| < a ⇔ −a < x < a (one interval); |x| > a ⇔ x < −a or x > a (two rays). Shift the centre for |x − c|.
- |x − a| < |x − b| ⇒ x is nearer a, i.e. on a’s side of the midpoint (a + b)/2 — no algebra needed.
- AM-GM: for x > 0, x + k/x ≥ 2√k, equality at x = √k. The go-to minimum tool when no calculus is allowed.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Forgetting to flip the sign when multiplying or dividing both sides by a negative number.
- Cross-multiplying a rational inequality by a variable expression whose sign is unknown — it may be negative and silently flip the inequality.
- Including a denominator root in the solution set; the expression is undefined there and must be excluded.
- Reading |x| > a as a single interval instead of two separate rays (x < −a OR x > a).
- Applying AM-GM to terms that are not all positive, or quoting the minimum without checking the equality condition holds for a valid x.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Inequalities when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
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 |