Inequalities and Modulus: Linear and Quadratic Inequalities, Absolute Value and AM-GM Maximum Minimum for SSC CGL Maths

Inequalities and the modulus (absolute value) are short but scoring topics in the SSC Tier I and Tier II maths papers. The questions are based on a few rules and the AM–GM inequality. This guide on inequalities and modulus for SSC explains the rules, the methods for solving linear and quadratic inequalities and modulus equations and the use of AM–GM for maximum and minimum values, with solved examples, practice questions and FAQs.

Properties of Inequalities

• If a > b, then a + c > b + c and a − c > b − c.
• If a > b and c > 0, then ac > bc; if c < 0, then ac < bc (the sign reverses when multiplying or dividing by a negative number).
• If a > b and b > c, then a > c (transitive).
• If a > b > 0, then 1/a < 1/b.
• For positive numbers, a > b gives a² > b² and √a > √b.
• The square of any real number is not negative: x² ≥ 0.

Linear Inequalities

Solve like an equation, but reverse the sign when multiplying or dividing by a negative number. For example, 3x − 5 < 10 gives x < 5; −2x < 6 gives x > −3. The solution is written as an interval: (a, b) is open, [a, b] is closed.

Quadratic Inequalities

For a rational inequality such as (x − a)/(x − b) > 0, the solution is the same as (x − a)(x − b) > 0 with x ≠ b.

Modulus (Absolute Value)

• |x| = x if x ≥ 0 and |x| = −x if x < 0; |x| is the distance from zero and is never negative.
• |x| = a (a > 0) gives x = a or x = −a.
• |x| < a gives −a < x < a; |x| ≤ a gives −a ≤ x ≤ a.
• |x| > a gives x < −a or x > a.
• |x − a| < b gives a − b < x < a + b.
• |ab| = |a||b|; |a/b| = |a|/|b|; |a|² = a².
• Triangle inequality: |a + b| ≤ |a| + |b|; |a − b| ≥ ||a| − |b||.
• The minimum value of |x − a| + |x − b| is |a − b|, attained for x between a and b.

AM–GM Inequality

For two positive numbers: (a + b)/2 ≥ √(ab); the equality holds only when a = b. For n positive numbers: the arithmetic mean ≥ the geometric mean ≥ the harmonic mean.

Solved Examples

Example 1. Solve |2x − 3| = 5.
2x − 3 = 5 or 2x − 3 = −5, so x = 4 or −1.

Example 2. Solve |x − 2| < 3.
−3 < x − 2 < 3, so −1 < x < 5.

Example 3. Solve x² − 5x + 6 < 0.
(x − 2)(x − 3) < 0, so 2 < x < 3.

Example 4. Find the maximum value of x(10 − x).
x + (10 − x) = 10 is constant, so the product is the maximum at x = 5: 25.

Example 5. Find the minimum value of x + 16/x for x > 0.
By AM–GM: x + 16/x ≥ 2√16 = 8 (at x = 4).

Example 6. Solve −3x + 7 > 1.
−3x > −6, so x < 2 (sign reversed).

Example 7. Find the minimum value of |x − 2| + |x − 7|.
|2 − 7| = 5.

Quick Tips

• Reverse the sign of the inequality when multiplying or dividing by a negative number.
• For quadratic inequalities, find the roots first and draw the number line.
• Remember AM ≥ GM for the maximum and minimum questions.
• Use the “between the roots” and “outside the roots” rule.

Quick Revision Points

• |x| < a means −a < x < a.
• (x − a)(x − b) < 0 gives a value between the roots.
• AM ≥ GM.
• x + 1/x ≥ 2 for x > 0.
• The sign reverses when multiplying by a negative number.

Practice Questions With Answers

1. Solve |x| = 7.
   Answer: x = 7 or −7
2. Solve x² − 9 < 0.
   Answer: −3 < x < 3
3. Solve 2x + 3 > 11.
   Answer: x > 4
4. Find the minimum value of x + 25/x for x > 0.
   Answer: 10
5. What is the maximum value of xy if x + y = 20?
   Answer: 100
6. Solve |x + 1| < 4.
   Answer: −5 < x < 3
7. Solve x² − 7x + 12 > 0.
   Answer: x < 3 or x > 4
8. What is the minimum value of |x − 1| + |x − 6|?
   Answer: 5
9. If a > b and c < 0, which is true: ac > bc or ac < bc?
   Answer: ac < bc
10. What is the minimum value of x + 1/x for x > 0?
    Answer: 2

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Mathematics notes page, read the previous topic Polynomials, Remainder and Factor Theorem.

Frequently Asked Questions

What is an inequality?

A statement that compares two expressions using >, <, ≥ or ≤.

When does the sign of the inequality reverse?

When both sides are multiplied or divided by a negative number.

What is a modulus?

The absolute value of a number, its distance from zero.

How do I solve |x| < a?

Write −a < x < a.

How do I solve |x| > a?

Write x < −a or x > a.

How do I solve a quadratic inequality?

Find the roots and use the rule: between the roots when the product is less than zero and outside when it is more than zero.

What is the AM–GM inequality?

The arithmetic mean of positive numbers is not less than their geometric mean.

When is the product of two numbers the maximum for a fixed sum?

When the two numbers are equal.

What is the triangle inequality?

|a + b| is less than or equal to |a| + |b|.

Is this topic asked in SSC?

Yes. Inequalities and the modulus are asked in Tier I and Tier II.

What is the minimum value of x + 1/x?

2, for positive x.

Can a modulus be negative?

No. It is always zero or positive.