🔢 Quant

Inequalities

Linear, quadratic, and modulus inequalities; sign-scheme method.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~5%

of section questions

Est. hours⏱️

~8h

to master

~1/22

2021

~2/22

2022

~1/22

2023

~2/22

2024

~1/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~1.0 per slot (2021: 0.7 · 2022: 1.3 · 2023: 0.7 · 2024: 1.7 · 2025: 0.7). Every year (~1–2 per slot); modulus-plus-inequality hybrids are the modern default.

Inequalities & Modulus

Inequalities behave like equations with one rule added: multiplying or dividing by a negative flips the sign. Most CAT inequalities are polynomial (solved by the wavy curve) or modulus-based (solved by distance).

The wavy-curve method

To solve a factored polynomial inequality, mark the roots on a number line and, starting from the far right (always +), alternate signs across each simple root. A factor of even multiplicity does not flip the sign.

A worked example

Solve $(x-1)(x-4)<0$ .

The expression is negative between its roots, so the solution is

$\boxed{1<x<4}.$

For $(x-1)(x-4)>0$ instead, you'd take the two outer pieces: $x<1$ or $x>4$ .

Modulus as distance

$|x-a|$ is the distance of $x$ from $a$ . So:

$|x-a|<k\iff a-k<x<a+k$ (within $k$ of $a$ ).
$|x-a|>k\iff x<a-k$ or $x>a+k$ .
$|x-a|+|x-b|$ has minimum $|a-b|$ (anywhere on $[a,b]$ ).

🎯PYQ Evidence

Read modulus as distance and rational inequalities off a sign chart. : each |n−a| < |n−b| just says "n is closer to a than to b," so it splits at the midpoint (a+b)/2 — |n−60| < |n−100| gives n < 80 and |n−100| < |n−20| gives n > 60, leaving the 19 integers between. : mark where the numerator and denominator vanish (x = 8, −5, 3/2), then test one point per interval — the fraction can only change sign at those zeros and breaks, with −5 and 3/2 excluded since the bottom can't be zero. : the left side is the total distance from x to the points −a and 1, which stays constant only when x lies between them, so infinitely many solutions force the gap to equal 2, giving the largest a = 1. Distance and sign charts turn every modulus or fraction into a picture you can read.

Common traps

Forgetting to flip. Dividing by a negative (or an unknown that could be negative) reverses the inequality.
Squaring carelessly. Only square when both sides are known non-negative.
Open vs closed. $<$ excludes the endpoint; $\le$ includes it — matters for "number of integer solutions."