CAT Quant Inequalities: Rules, Wave Curve Method and Solved Examples

CAT Quant Inequalities is a core Algebra topic where answers come as ranges, not single values. It covers linear and quadratic inequalities, perfect squares, fractions, higher powers, AM-GM, cubics, exponents and logarithms.cracku+2

Master three golden rules: flip the sign with negatives, avoid cancelling or cross‑multiplying unknowns, and use the wave curve method for polynomials. Here, we explain each rule with simple examples and solved CAT‑style questions.

You can also view the Top Quant Inequalities Video Classes by PW to clear your content more effectively

What Are CAT Quant Inequalities?

CAT Quant Inequalities is a wide Algebra topic. It joins linear inequalities, quadratic inequalities, perfect squares, fractions, higher powers, AM-GM, cubics, exponents and logarithms.

You should know linear equations, quadratic equations and two-variable equations first. Then the topic becomes a revision of all of Algebra, with one new idea: the answer is a range, not a single value.

How Do You Solve Linear Inequalities?

A linear inequality is solved like a linear equation, but the answer is a range of values. Take 2(x − 1) > 5. Open the bracket to get 2x > 7, so x > 7/2. On the number line, this is everything after 7/2.

The endpoint tells you if equality is allowed. The table below shows the two types.

For example, x > 7/2 is written as (7/2, ∞). And x ≥ 13 is written as [13, ∞), as 13 is allowed.

What Are the Three Golden Rules of Inequalities?

Some steps that work in equations do not work here. The trouble is always the negative number.

1. Cancelling or multiplying by a negative number reverses the sign: We know −3 > −9. Cancel the minus on both sides, and you get 3 > 9, which is false. So write 3 < 9. Likewise, −x < 3 becomes x > −3.

2. Do not cross-multiply with an unknown variable: Take x/y > 2. You cannot write x > 2y, as y may be negative. If y is positive, x > 2y. If y is negative, x < 2y.

3. Do not cancel an unknown variable from both sides: In x² > x, you cannot cancel x. Try x = −2: 4 > −2 is true, but cancelling gives −2 > 1, which is false.

You may cancel or cross multiply by a positive number freely. For a negative number, flip the sign. For an unknown, bring all terms to one side and use the wave curve method.

How Do You Solve Range and Word Problems?

Learn how to solve range, word-based, and two-variable inequality problems by translating conditions into mathematical expressions and checking the given constraints carefully.

• Range of x²: Suppose x lies between −2 and 3, with both ends excluded. Many students square the ends and say 4 to 9. This is a trap. Since x can be 0, the smallest value of x² is 0. The largest comes from x near 3, but 3 is not allowed, so 9 is not reached. The range is 0 ≤ x² < 9. Always check if an endpoint is allowed before using it.

• Chairs and tables: A chair costs Rs. 500, and a table costs Rs. 800, with a budget under Rs. 6,000. At least 3 tables are needed, and chairs must be more than tables. Use c and t. The budget gives 5c + 8t < 60. To get the most chairs, keep tables at the minimum, which is 3. Then 5c < 36, so c < 7.2. Chairs are whole numbers, so the maximum is 7.

• Boys and girls: Say 40% of the girls and 60% of the boys leave. Write the fractions: 2/5 and 3/5. Since the denominator is 5, take boys as 5x and girls as 5y. After leaving, 2x boys and 3y girls remain. If girls then exceed boys by 8, then 3y = 2x + 8. Try the smallest x that gives a whole y, with boys more than 10. For x = 5, y = 6. So boys are 25, girls are 30, and the total is 55. Do not assume exact numbers when only conditions are given.

• Two-variable inequalities: If one expression is above y and another is below y, compare the two and drop y. For example, 25 − x ≤ y < 40 − 4x gives 25 − x < 40 − 4x, so x ≤ 5.

How Does the Wave Curve Method Work for Quadratic Inequalities?

The wave curve (sign chart) method solves most quadratic inequalities. Follow these steps:

1. Move everything to one side so that the other side is zero.

2. Factorise by splitting the middle term.

3. Set each factor to zero. These are the critical points.

4. Mark them on the number line.

5. Find the starting sign. Multiply the coefficients of x in all the linear factors. If the product is positive, start from positive at the right end. If it is negative, start from negative. Then the sign alternates.

6. Pick the regions you need: negative regions for < 0 and positive regions for > 0.

Take x² − 11x + 18 ≤ 0, which is (x − 2)(x − 9) ≤ 0. The critical points are 2 and 9. The product of the coefficients is positive, so the curve starts positive. The negative region is between 2 and 9. The sign includes equality, so the answer is [2, 9]. For a strict sign, use open brackets.

What Changes with Higher Powers and Fractions?

Learn how higher powers, fractions, and cancelled variables affect the solution set, with special attention to sign changes and critical points. 

• Higher powers: An odd power acts like a normal linear factor, so the sign changes at its point. An even power does not change the sign, so the curve bounces back there. Mark the even-power points. When you find the starting sign, ignore the even-power factors, as they are always positive.

• Fractions: Bring all terms to one side, take the LCM, factorise the top and mark every critical point. The rule: a denominator's critical point is never included, even if the sign has equality. A numerator's critical point can be included if equality is allowed. For example, 1/(x + 5) ≤ 1/(2x − 3) becomes (x − 8)/((x + 5)(2x − 3)) ≤ 0. The answer is x < −5 or 3/2 < x ≤ 8, with 8 included and −5 and 3/2 not included.

If a variable cancels completely, you get a plain statement. If it is true, every x works. If it is false, no x works.

When both a fraction and an even power appear, check every critical point at the end, as some may still be valid when equality is allowed.

How Do Perfect Squares and the Discriminant Help?

Learn how perfect squares, the discriminant, and root relationships can simplify quadratic inequalities and help determine their solution sets. 

• Perfect squares: A square cannot be negative. So x² + y² + z² ≤ 0 forces x = y = z = 0. And x² − 2x + 1 ≤ 0 is (x − 1)² ≤ 0, so x = 1.

When the square is hidden, complete it using a² ± 2ab + b² = (a ± b)². Take x² − 4x + y² + 2y + 5 ≤ 0. Write it as (x − 2)² + (y + 1)² ≤ 0, so x = 2 and y = −1. With natural numbers, use the smallest values of each square to see when equality can happen.

• Discriminant: If a quadratic does not factorise, find D = b² − 4ac. If D < 0, it has no real roots, so the graph stays fully above or fully below the x-axis. If the x² coefficient is positive, the quadratic is always positive. If it is negative, it is always negative. This tells you if an inequality is true for all x or for no x.

• Reverse questions: If the solution range is given, its endpoints are the roots. Use the sum and product of roots to find the coefficients. For example, if the solution is −2 < x < 7, the roots are −2 and 7. The sum is 5, and the product is −14.

How Does AM-GM Help in Inequalities?

For positive numbers, AM ≥ GM, and equality holds when the numbers are equal. This gives a very useful result, shown in the table below.

So x + 1/x can never lie between −2 and 2.

Use AM-GM when a question has sums, products, and a maximum or minimum. For example, if x + y = 10 with positive x and y, then xy ≤ 25, with equality at x = y = 5. If 3x + 3y + z = 9, take the three numbers 3x, 3y and z to get xyz ≤ 3.

For terms with different powers, divide each term by its power and set them equal. For x + 2y = 12 and the maximum of x²y⁴, set x/2 = 2y/4, which gives x = y. So x = y = 4.

Hidden forms also appear. Writing (6x − x² − 1)/x as 6 − (x + 1/x) shows it cannot lie between 4 and 8. Also, 4x/y + 16y/x has a GM of 8, so it cannot take values between −16 and 16. A substitution such as t = 2ˣ also exposes AM-GM, as 2ˣ is always positive.

How Do You Solve Cubic, Exponential and Log Inequalities?

Learn how to solve cubic, exponential, and logarithmic inequalities by factoring expressions, applying base rules, checking domains, and finding the final solution range.

• Cubics: Factorising is the hard part. Use hit and trial with the factors of the constant term to find one root. Let the other two roots be α and β. Use the sum and product of roots of the cubic to find them. For x³ − 8x² − 5x + 84, the root 4 works. Then α + β = 4 and αβ = −21, so the roots are 7 and −3. After that, use the wave curve.

• Exponents: The base decides everything:

First, make the bases equal. Use a substitution when the same power repeats, such as t = 3ˣ. For 9ˣ + 81 < 10 × 3ˣ⁺¹, put t = 3ˣ to get t² − 30t + 81 < 0, so 3 < t < 27 and 1 < x < 3.

• Logarithms. The same base rule applies. But the argument of a log must be positive, so always check the domain. Take log₄(2x − 5) < 2. The base is more than 1, so 2x − 5 < 16, giving x < 10.5. The domain gives x > 2.5. So the integers are 3 to 10, which is 8 values. For a base between 0 and 1, the sign flips, and the domain still applies. In CAT questions, always intersect your range with the domain, and then count the integers inside the final interval.

Quick Summary of Inequality Rules

The table below puts the main rules in one place for quick revision.

Practise CAT PYQs on this topic. After each question, note if the mistake was a sign flip, a missed critical point or a missed domain.

Inequalities questions are easier when you practise a few questions of each type and keep the rules in one place. You can practice it with other CAT Quant aptitude questions, CAT Quant Algebra notes and the CAT Quant Arithmetic guides. A few CAT Quant mock tests and the  PW CAT online course can help you check how well you spot the right method.