CAT 2026 Inequalities: 4 Shortcuts, Solved Examples & 5 Practice Questions

Why Practise Inequalities for CAT 2026?

Algebra accounted for 20 of the 66 Quantitative Aptitude questions in CAT 2025, approximately 30.3% of the section. 

Inequalities featured once in each of the three slots, while CAT 2024 Slot 2 included a rational inequality question.

 These trends make inequalities worth CAT 2026 Last 2 Months Revision, although the number of questions may vary.

4 Golden Shortcuts to Solve Inequalities for CAT 2026

The fastest method depends on the form of the inequality. Use these shortcuts to handle sign changes, rational expressions, interval conditions and absolute values.

Shortcut 1: Reverse the Inequality When Multiplying or Dividing by a Negative Number

When you multiply or divide both sides by a negative number, reverse the inequality sign. Forgetting this rule is one of the easiest ways to lose marks.

Example 1: Solve -3x+7>16.

-3x>9

Dividing by -3 reverses the sign.

Answer: x<-3

Example 2: Solve 5-2x11.

-2x6

Dividing by -2 gives:

Answer: x-3

Shortcut to remember: The sign reverses only when multiplying or dividing by a negative number, not when adding or subtracting a number.

Shortcut 2: Use Critical Points to Solve Rational Inequalities

For inequalities involving fractions, do not cross-multiply blindly because the denominator may be positive or negative. Bring all terms to one side, combine them into a single fraction and mark the points where the numerator or denominator becomes zero.

Example 1: CAT 2024 Slot 2 previous-year question

Solve 1x+512x-3.

Combining the fractions gives:

x-8(x+5)(2x-3)0

The critical points are -5, 32 and 8. The values -5 and 32 are excluded because they make the denominator zero. Checking the intervals gives:

Answer: x<-5 or 32<x8. 

Example 2: Solve x-2x+3>0.

The critical points are 2 and -3. The fraction is positive when the numerator and denominator have the same sign.

Answer: x<-3 or x>2.

Shortcut to remember: Include a numerator's zero only when equality is allowed; never include a value that makes the denominator zero.

Shortcut 3: Combine Compound Inequalities Before Solving

When two conditions must hold simultaneously, find their common interval instead of solving each condition separately and stopping there.

Example 1: Solve 2<3x-111.

Add 1 throughout:

3<3x12

Divide throughout by 3:

Answer: 1<x4

Example 2: Find the integer values satisfying -32x+1<8.

Subtract 1 throughout:

-42x<7

Divide by 2:

-2x<72

The integers in this interval are:

Answer: -2,-1,0,1,2,3

Shortcut to remember: When the question asks for integers, list only the integers inside the final interval.

Shortcut 4: Convert Absolute-Value Inequalities into Simple Bounds

Absolute-value inequalities become easier when you interpret |x-a| as the distance between x and a on the number line.

For |x-a|<b, where b>0, write a-b<x<a+b. For |x-a|>b, the solution lies outside those bounds.

Example 1: Solve |x-4|3.

Write the double inequality:

-3x-43

Add 4 throughout:

Answer: 1x7

Example 2: Solve |2x+1|>5.

Split the inequality into two cases:

2x+1>5 or 2x+1<-5

Solving gives x>2 or x<-3.

Answer: x<-3 or x>2

Shortcut to remember: For a strict absolute-value inequality using >, the solution lies outside the two boundary points; for <, it lies between them.

CAT 2026 Inequalities Practice Questions

Test the four shortcuts with these five questions. Try solving them without looking at the answer key:

Question 1: Quadratic Inequality

The inequality (x-2)(x-5)x-30 is satisfied by which of the following sets of real values of x?

A. x[2,3)[5,)

B. x(-,2](3,5]

C. x(-,2](3,5][5,)

D. x[2,5], excluding x=3

Question 2: Linear Inequality

If x is a real number satisfying 2x-35-x+13x-42, then the set of all possible values of x is:

A. x3213

B. x-317

C. x317

D. x317

Question 3: Quadratic Inequality

The number of integers satisfying x2-7x+100 is:

A. 3

B. 4

C. 5

D. 6

Question 4: Absolute Value Inequality

The set of real values of x satisfying |2x-3|+|x+1|7 is an interval of the form [a,b]. The value of b-a is:

A. 4

B. 5

C. 6

D. 7

Question 5: Rational Inequality

The solution set of x2-4x+3x2-5x+6>0 is:

A. (-,1)(2,3)(3,)

B. (-,1)(3,)

C. (1,2)(2,3)

D. (-,1)[3,)

Answer Key with Quick Solutions

Use the answer key to check both your final answer and the method you applied.

How Much Time Can Inequality Shortcuts Save in CAT 2026?

For familiar, straightforward inequalities, sign rules and interval methods can reduce unnecessary calculations. The estimates below are illustrative practice timings, not measured results or guaranteed exam outcomes.

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CAT 2026 Quant Other Useful Resources

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1. MBA Core Functional Specialisations - Free eBooks