Quant
Number System
Every high-yield CAT Number System formula on one page — factors, HCF-LCM, unit-digit cycles, remainders with Fermat and Euler, factorials and base systems — with worked examples.
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Inequalities
CAT'26 QUANT CHEATSHEET
Every inequalities formula and CAT shortcut you need for CAT 2026 — on one page.
Inequalities is quiet, high-frequency CAT territory: it rarely gets its own flashy question, but it hides inside quadratics, modulus problems, maxima and minima, and the classic AM-GM and Cauchy setups. The whole topic runs on a few reflexes — flip the sign on a negative, read a sign chart with the wavy-curve rule, and never cross-multiply a rational expression. This sheet lays out every formula you need for the CAT quant section, from linear and quadratic inequalities through modulus, AM-GM, Cauchy-Schwarz and Titu’s Lemma, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is leaking marks.
Dev note: the canonical route
/cheatsheets/optima-learn-inequalities-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Inequalities: every formula you need
1Core Idea
The range of values for which an expression is true.
>, <, ≥, ≤ → the answer is an interval, not one value
Example: solve 3x − 7 > 2x + 5 ⇒ x > 12.
CAT Insight: Write every answer in interval notation whenever you can.
2Linear Inequalities
Move terms exactly like an equation.
Multiply or divide by a negative → reverse the sign
Example: −2x + 6 ≤ 10 → −2x ≤ 4 → x ≥ −2.
Common Mistake: Most errors come from forgetting to flip the sign when dividing by a negative.
3Quadratic Inequalities
Find the roots, then test the sign of the parabola.
a > 0: positive outside the roots, negative between
a < 0 reverses the pattern
a < 0 reverses the pattern
Example: x2 − 5x + 6 > 0, roots 2 and 3 → (−∞, 2) ∪ (3, ∞).
4Wavy Curve Method
A sign chart for polynomial inequalities.
Odd power → sign changes · even power → sign stays
Example: (x−1)(x−3)(x−5) > 0 → (1, 3) ∪ (5, ∞).
CAT Hack: Even power = the curve bounces off the axis; odd power = it crosses.
5Rational Inequalities
Handle numerator and denominator together.
Never cross-multiply · denominator roots are never included
Example: (x−2)/(x+3) > 0 → (−∞, −3) ∪ (2, ∞).
CAT Insight: Denominator = 0 is always invalid, so exclude those roots from the answer.
6Modulus Inequalities
Split the modulus into two cases.
|f(x)| < a ⇒ −a < f(x) < a
|f(x)| > a ⇒ f(x) < −a or f(x) > a
|f(x)| > a ⇒ f(x) < −a or f(x) > a
Example: |2x − 5| < 3 → 1 < x < 4.
7AM-GM Inequality
Bounds a sum below using the geometric mean.
AM ≥ GM: (a + b)/2 ≥ √(ab), equality at a = b
Example: minimum of x + 9/x for x > 0 is 6, at x = 3.
CAT Insight: Equality holds when the terms are equal — that is exactly where the optimum sits.
8Cauchy-Schwarz
Relates a product of sums to sums of squares.
(a1b1 + …)2 ≤ (a12 + …)(b12 + …)
Example: if x2 + y2 = 1, the maximum of 3x + 4y is 5.
CAT Favourite: Shows up in the harder CAT algebra and arithmetic sets — worth recognising on sight.
9Titu's Lemma
The ready-to-use form of Cauchy-Schwarz.
a2/x + b2/y ≥ (a + b)2/(x + y)
Example: if a + b + c = 1, the minimum of 1/a + 1/b + 1/c is 9.
CAT Hack: See a Σ(a2/b) expression? Reach straight for Titu's Lemma.
10System of Inequalities
Solve each inequality, then intersect.
Final answer = the common region (intersection)
Example: x > −3, x < 4, x ≥ −1 → [−1, 4).
11Integer Solutions
Count only the integers in the interval.
Closed interval [a, b] → count = b − a + 1
Example: 2 ≤ x ≤ 5 → integers 2, 3, 4, 5 → count = 4.
CAT exam shortcuts, traps & revision
12
CAT Exam Shortcuts
- Solve like an equation; flip the sign on a negative multiply or divide
- Quadratic (a > 0): positive outside the roots, negative between
- Wavy curve: odd power crosses, even power bounces
- Rational: never cross-multiply; exclude denominator zeros
- |f| < a ⇒ −a < f < a; |f| > a ⇒ f < −a or f > a
- AM ≥ GM; Titu: a2/x + b2/y ≥ (a+b)2/(x+y)
13
Most Common CAT Traps
- Forgetting to flip the sign after dividing by a negative.
- Cross-multiplying a rational inequality.
- Including a denominator root in the answer.
- Assuming an even-multiplicity root changes the sign (it does not).
- Missing that an odd-multiplicity root does change the sign.
14
30-Second Revision Box
- Answers are intervals, not single values
- Flip the sign when multiplying or dividing by a negative
- Quadratic: roots first, then the sign of the parabola
- Wavy curve: odd = cross, even = bounce
- Rational: exclude denominator zeros, never cross-multiply
- AM ≥ GM (equal terms); min of x + k/x is 2√k
- Integers in [a, b]: count = b − a + 1
Inequalities rewards a clean setup over heavy algebra — spot whether a question wants a sign chart, an interval intersection, or a hidden AM-GM, and the answer usually falls out in a couple of lines. Drill this sheet until flipping the sign and reading the wavy curve are automatic, then test it on full sets and track progress with the CAT score predictor. It pairs naturally with the Functions cheat sheet, since modulus and optimisation appear in both. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
