Inequalities for CAT 2026: 16 Properties + Wavy Curve Method
Inequalities for CAT 2026 sit at the centre of recent Algebra-heavy CAT papers. CAT 2024 showed a notably stronger Algebra presence, with inequalities now co-equal with quadratics as a CAT algebra anchor. Yet dedicated inequality content for CAT is almost non-existent online. This cheatsheet pins 16 properties, walks through the wavy curve method step by step, and closes with 15 CAT-level questions across linear, quadratic, modulus, and rational inequalities.
The reason most aspirants leak marks on CAT inequalities is that they learn the topic as ten disconnected algebra rules. CAT 2026 will test the topic as one unified framework: 16 properties plus the wavy curve method for any polynomial or rational expression. The framework below is built around that unified view.
Why the Wavy Curve Method Is the Hero of CAT Inequalities
The wavy curve method solves any polynomial inequality of any degree in three steps: mark the critical points on the number line, label signs alternating from the rightmost positive interval, and read off the intervals that match the inequality direction. The method scales from a quadratic to a degree-5 polynomial without changes. It handles rational inequalities by treating denominator zeros as excluded points. It handles strict and non-strict inequalities by including or excluding endpoints.
The split between aspirants who finish CAT inequality questions in 30 seconds and those who take three minutes comes down to whether they reach for wavy curve or for case-by-case algebra. The 16 properties below cover the algebraic foundation; the wavy curve method covers the geometric pattern. Together they form the complete CAT inequalities toolkit.
The 16 Inequality Properties and the Wavy Curve Method for CAT 2026
Block 1 — Basic Algebraic Properties (5 properties)
Recognition cue: any direct inequality manipulation.
| # | Property | Use case |
|---|---|---|
| 1 | If a > b, then a + c > b + c and a − c > b − c | Addition / subtraction preserves. |
| 2 | If a > b and c > 0, then ac > bc and a/c > b/c | Positive multiplier preserves. |
| 3 | If a > b and c < 0, then ac < bc and a/c < b/c | Negative multiplier flips. |
| 4 | Transitive: a > b and b > c imply a > c | Chain of inequalities. |
| 5 | If a > b and c > d, then a + c > b + d | Adding same-direction inequalities. |
Block 2 — Reciprocal and Squaring Properties (4 properties)
| # | Property | Recognition cue |
|---|---|---|
| 6 | For a, b > 0 with a > b: 1/a < 1/b | Reciprocal of positives flips. |
| 7 | For a, b same sign with a > b: a2 > b2 iff |a| > |b| | Squaring rule. |
| 8 | If a, b > 0 and a > b, then root a > root b | Square root preserves for positives. |
| 9 | For positives a > b, c > d: ac > bd | Multiplying positive inequalities. |
Block 3 — AM-GM-HM and Cauchy-Schwarz (3 properties)
Recognition cue: optimisation, minimum, maximum, or constrained sum-product question.
| # | Property | Use case |
|---|---|---|
| 10 | For positives: AM ≥ GM ≥ HM, i.e., (a+b)/2 ≥ √(ab) ≥ 2ab/(a+b) | Two-variable optimisation. |
| 11 | For n positives: (a1+...+an)/n ≥ (a1×...×an)1/n | n-variable AM-GM. |
| 12 | Cauchy-Schwarz: (a12+a22)(b12+b22) ≥ (a1b1+a2b2)2 | Sum-of-products bound. |
Block 4 — Modulus Inequalities (4 properties)
Recognition cue: any |x| or |expression| in the inequality.
| # | Property | Use case |
|---|---|---|
| 13 | |x| < a iff −a < x < a | Single-modulus bounded. |
| 14 | |x| > a iff x < −a or x > a | Single-modulus unbounded. |
| 15 | |x − c| < a iff c − a < x < c + a | Translated modulus. |
| 16 | Triangle inequality: |a + b| ≤ |a| + |b|; |a − b| ≥ ||a| − |b|| | Sum of moduli. |
The Wavy Curve Method Step by Step
The wavy curve method solves polynomial inequalities of any degree. Step 1: bring the inequality to f(x) > 0, f(x) >= 0, f(x) < 0, or f(x) <= 0 form with one side zero. Step 2: factor f(x) and write the critical points (zeros of factors). Step 3: arrange critical points in increasing order on the number line. Step 4: starting from the rightmost interval (assume positive if leading coefficient is positive), label alternating signs. Step 5: read off the intervals that match the inequality direction. For repeated roots, the sign does not alternate at that point; for distinct roots, it does.
For rational inequalities like (x - a)/(x - b) > 0, treat the rational like a polynomial: critical points are zeros of numerator AND denominator. The wavy curve method then applies, with the denominator zeros strictly excluded from the solution.
Three Inequality Traps That Recur in CAT Papers
Three traps recur in CAT algebra inequality topic for CAT 2026 questions. The first is multiplying or dividing by an unknown of unknown sign. If x could be positive or negative, multiplying both sides by x can flip or preserve the inequality. The fix is to consider both cases or to bring everything to one side and use wavy curve. The second trap is the modulus expansion error: |x - c| < a expands to c - a < x < c + a, not c - a < x < c. Forgetting the upper bound is a common slip.
The third trap is repeated roots in the wavy curve. If a factor appears squared like (x - 2) squared, the sign does not flip at x = 2. CAT setters embed this to test wavy curve fluency.
15 CAT-Level Inequality Questions With Solutions
Solve 3x − 7 > 5x + 1.
−2x > 8, so x < −4. Answer: x < −4
Solve −3(x − 4) ≤ 6.
x − 4 ≥ −2, so x ≥ 2. Answer: x ≥ 2
Solve x2 − 5x + 6 > 0.
Factor (x − 2)(x − 3) > 0. Wavy curve: positive outside [2, 3]. Answer: x < 2 or x > 3
Solve (x − 1)(x + 2)(x − 4) > 0.
Critical points −2, 1, 4. Signs from right: + − + −. Positive in −2 < x < 1 or x > 4. Answer: −2 < x < 1 or x > 4
Solve (x − 1)2(x + 3) > 0.
No sign-flip at 1 (squared). Sign flip at −3. Positive: x > −3 except x = 1. Answer: x > −3, x ≠ 1
Solve (x − 2)/(x + 3) < 0.
Critical points −3, 2 (with −3 strictly excluded). Negative in −3 < x < 2. Answer: −3 < x < 2
Solve |2x − 5| < 3.
−3 < 2x − 5 < 3. So 2 < 2x < 8, giving 1 < x < 4. Answer: 1 < x < 4
Solve |x − 7| ≥ 4.
x − 7 ≤ −4 or x − 7 ≥ 4. So x ≤ 3 or x ≥ 11. Answer: x ≤ 3 or x ≥ 11
Solve |x − 2| + |x − 5| < 7.
Case 1 (x < 2): −(x−2) − (x−5) < 7, so 7 − 2x < 7, x > 0. Range: 0 < x < 2. Case 2 (2 ≤ x ≤ 5): (x−2) − (x−5) = 3 < 7, always true. Range: 2 ≤ x ≤ 5. Case 3 (x > 5): (x−2) + (x−5) < 7, so 2x < 14, x < 7. Range: 5 < x < 7. Combined: 0 < x < 7. Answer: 0 < x < 7
If x > 0, find the minimum of x + 4/x.
AM-GM: (x + 4/x)/2 ≥ root(x × 4/x) = 2. Min value = 4. Achieved at x = 2. Answer: 4
If a, b, c > 0 with abc = 8, find the minimum of a + b + c.
AM ≥ GM: (a+b+c)/3 ≥ cube-root(8) = 2. So a + b + c ≥ 6. Minimum = 6 when a = b = c = 2. Answer: 6
For what values of k does x2 − 6x + k > 0 hold for all real x?
Discriminant < 0: 36 − 4k < 0, so k > 9. Answer: k > 9
Solve |x2 − 4| < 5.
−5 < x2 − 4 < 5. So −1 < x2 < 9. Since x squared is always non-negative, 0 ≤ x2 < 9 gives −3 < x < 3. Answer: −3 < x < 3
If 2 < x < 8, find the range of 1/x.
Reciprocal flips for positives: 1/8 < 1/x < 1/2. Answer: 1/8 < 1/x < 1/2
Solve (x − 1)(x − 2)(x − 3)(x − 4) > 0.
Critical points 1, 2, 3, 4. From right: + − + − +. Positive intervals: x < 1, 2 < x < 3, x > 4. Answer: x < 1 or 2 < x < 3 or x > 4
Wire Inequalities Into Your CAT 2026 Algebra Block
Inequalities is co-equal with quadratics in CAT 2026 Algebra. A diagnostic-driven plan wires this block right after Quadratics so the wavy curve reflex compounds with the discriminant analysis.
Wire My Wavy Curve ReflexWhere Inequalities Fit in the CAT 2026 Algebra Cluster
Inequalities is the second Algebra topic after Quadratics, with the two often combined in CAT questions. A focused 3 to 4 day study block, scheduled in week six of any 9-month CAT plan, covers the topic. For working professionals, 6 to 8 days works equally well. The Optima Learn CAT exam guide sequences the Algebra cluster, and the CAT 2026 waitlist details page explains how the personalised planner sequences inequalities and quadratics together.
Three Reflexes That Compress Inequality Solves to Under 60 Seconds
Once 16 properties are memorised, three reflexes separate aspirants who finish these questions in 60 seconds from those who take three minutes. Reflex one: critical-points-first. For any polynomial or rational inequality, write critical points before anything else. Reflex two: modulus to interval translation. Convert every modulus form to its interval representation immediately. Reflex three: AM-GM trigger. Whenever the question asks for a min or max of a sum or product, reach for AM-GM. The CAT preparation blogs library has companion cheatsheets on Quadratics, Logarithms, and Functions.
Common Doubts About Inequalities Preparation for CAT 2026
Is the wavy curve method really faster than case analysis?
Yes, for any polynomial of degree 2 or higher. Case analysis (multiplying both sides, considering positive/negative cases) gets unmanageable past quadratic. The wavy curve method handles degree 5 polynomials with the same effort as quadratics. For CAT 2026, default to wavy curve and skip case analysis.
How tricky are recent CAT inequality questions?
CAT 2024 had a parameter-range inequality and CAT 2025 had a modulus-plus-quadratic combination. Both rewarded the wavy curve and modulus-to-interval reflexes from this cheatsheet.
How do I revise inequalities one week before CAT 2026?
A one-week revision plan: day one, re-read the 16-property cheatsheet. Day two, drill basic and reciprocal properties. Day three, drill wavy curve on quadratics and cubics. Day four, drill modulus and rational inequalities. Day five, attempt 15 mixed PYQs under timed conditions. Day six, review every error. Day seven, scan the cheatsheet for 15 minutes only before the exam.
Final note. Inequalities CAT 2026 reduces to 16 properties plus the wavy curve method. The topic rewards interval-thinking over case-by-case algebra. Drill block by block, build the three reflexes, and the CAT score predictor alongside mocks will track the lift across Algebra.
