How many Inequalities & Modulus questions come in CAT Quant?

About 1.3 questions in a 22-question Quant section, worth roughly 4 marks. Inequalities & Modulus appeared in 21 of the 24 CAT papers from 2017 to 2025, 34 questions in all, which ranks it 4 of 25 Quant chapters.

Are Inequalities & Modulus questions in CAT MCQ or TITA?

38% of the Inequalities & Modulus questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.

What types of Inequalities & Modulus questions does CAT ask?

CAT keeps returning to 5 question shapes in Inequalities & Modulus: A modulus equation — split at the critical points; Two moduli added — read it as distance; Find a maximum or a minimum; A rational or quadratic inequality — chart the signs; Count the integer solutions.

Inequalities & Modulus for CAT: Formulas, Weightage & Question Types

Modulus equations, maxima and minima, and counting integer solutions.

Inequalities & Modulus formula sheet

Everything CAT needs from Inequalities & Modulus, in one place. Bookmark it for revision; the lesson in the app explains where each line comes from and when to reach for it.

Everything for this chapter, in one place. $x$ is real throughout.

Modulus — the basics


Definition	$\lvert x \rvert = x$ if $x \ge 0$ , and $-x$ if $x < 0$
Always	$\lvert x \rvert \ge 0$ and $\lvert x \rvert^2 = x^2$
Products	$\lvert ab \rvert = \lvert a \rvert \lvert b \rvert$
What it means	$\lvert x - c \rvert$ is the distance from $x$ to $c$

Solving with a modulus


$\lvert x \rvert = a$ , $a > 0$	$x = a$ or $x = -a$
$\lvert x \rvert < a$ , $a>0$	$-a < x < a$
$\lvert x \rvert > a$ , $a>0$	$x < -a$ or $x > a$
$\lvert x \rvert = a$ , $a < 0$	no solution
$\lvert f(x) \rvert = g(x)$	needs $g(x) \ge 0$ — check every root against this

Two moduli


Triangle inequality	$\lvert a+b \rvert \le \lvert a \rvert + \lvert b \rvert$ , equal exactly when $a$ and $b$ have the same sign
Reverse form	$\bigl\lvert \lvert a \rvert - \lvert b \rvert \bigr\rvert \le \lvert a - b \rvert$
Sum of two distances	$\lvert x-p \rvert + \lvert x-q \rvert \ge \lvert p-q \rvert$ , equal exactly when $x$ lies between $p$ and $q$
Worth memorising	$\lvert x+y \rvert + \lvert x-y \rvert = 2\max(\lvert x \rvert, \lvert y \rvert)$

Sign charts — how every rational inequality is solved


Method	factor into linear pieces, mark every root and every break, test one point, alternate
Odd power of a factor	the sign flips across that root
Even power of a factor	the sign does not flip
Denominator roots	always excluded, drawn hollow, even when the inequality is $\le$ or $\ge$
Never do this	cross-multiply by a quantity whose sign you do not know

AM–GM and friends — for positives


Two terms	$\dfrac{a+b}{2} \ge \sqrt{ab}$ , equal only when $a=b$
The workhorse	$t + \dfrac{1}{t} \ge 2$ for $t>0$ , and $\le -2$ for $t<0$
General two-term	$kt + \dfrac{m}{t} \ge 2\sqrt{km}$ for $t>0$
Squares	$x^2 + y^2 \ge 2\lvert xy \rvert$
Fixed sum	the product is largest when the numbers are equal
Fixed product	the sum is smallest when the numbers are equal
Cauchy–Schwarz	$(ax+by)^2 \le (a^2+b^2)(x^2+y^2)$ , equal exactly when $ay = bx$

Quadratic maxima and minima — see the Quadratic chapter


Extreme value of $ax^2+bx+c$	$-\dfrac{D}{4a}$ , reached at $x = -\dfrac{b}{2a}$

The 5 Inequalities & Modulus question types CAT repeats

Past CAT papers return to the same few shapes. Learn to recognise them and most questions become a known routine. Beyond these, CAT has asked 3 rarer one-off shapes here.

01
A modulus equation — split at the critical points
02
Two moduli added — read it as distance
03
Find a maximum or a minimum
04
A rational or quadratic inequality — chart the signs
05
Count the integer solutions

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Inequalities & Modulus in CAT: quick answers

How many Inequalities & Modulus questions come in CAT Quant?: About 1.3 questions in a 22-question Quant section, worth roughly 4 marks. Inequalities & Modulus appeared in 21 of the 24 CAT papers from 2017 to 2025, 34 questions in all, which ranks it 4 of 25 Quant chapters.
Are Inequalities & Modulus questions in CAT MCQ or TITA?: 38% of the Inequalities & Modulus questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.
What types of Inequalities & Modulus questions does CAT ask?: CAT keeps returning to 5 question shapes in Inequalities & Modulus: A modulus equation — split at the critical points; Two moduli added — read it as distance; Find a maximum or a minimum; A rational or quadratic inequality — chart the signs; Count the integer solutions.

Inequalities & Modulus for CAT: Formulas, Weightage & Question Types

Inequalities & Modulus formula sheet

The 5 Inequalities & Modulus question types CAT repeats

Is Inequalities & Modulus costing you marks?

Inequalities & Modulus in CAT: quick answers

More Algebra for CAT