Class 11 Maths Chapter 5 Linear Inequalities NCERT Solutions | Graphs, Examples, Step-by-Step Guide

Class 11 Maths Chapter 5 – Linear Inequalities

Introduction

Linear inequalities are an extension of linear equations. Instead of finding one exact value, here we find a range of values that satisfy a condition.

This chapter is important because it connects algebra with graphs and real-life situations like budgeting, limits, and constraints.

What is a Linear Inequality?

A linear inequality is an expression like:

ax + b > 0
ax + b < 0
ax + b ≥ 0
ax + b ≤ 0

It represents a set of values, not just one answer.

Important Rules

1. Add or subtract the same number on both sides

2. Multiply or divide by a positive number → sign stays same

3. Multiply or divide by a negative number → sign reverses

Example:

-2x > 6
x < -3

Formulas and Concepts Used

General Form

ax + b > 0

Two Variable Inequality

ax + by + c > 0

Graph Example

$$x > 2$$

Solution means all real numbers greater than 2.

Solved NCERT Examples (Exercise Based)

Example 1 (Basic Inequality)

Solve: x + 7 > 10

Solution:
x > 3

Example 2 (With Negative Coefficient)

Solve: -3x < 9

Solution:
x > -3

(Sign reversed)

Example 3 (Fraction Form)

Solve: x/2 + 3 ≥ 5

Solution:
x/2 ≥ 2
x ≥ 4

Example 4 (Variable on Both Sides)

Solve: 2x + 3 > x + 5

Solution:
2x - x > 5 - 3
x > 2

Example 5 (Another Variation)

Solve: 5 - x ≤ 2

Solution:
-x ≤ -3
x ≥ 3

Example 6 (Double Inequality)

Solve: 2 < x + 1 ≤ 5

Solution:
1 < x ≤ 4

Example 7 (Word Type NCERT Style)

A number increased by 5 is greater than 9

Let number = x

x + 5 > 9
x > 4

Example 8 (Two Variable Inequality)

Solve: x + y ≤ 4

Solution:
Draw line x + y = 4
Shade region below line

Example 9 (Graph Region)

Solve: 2x + y > 6

Solution:
Draw line 2x + y = 6
Shade upper region

Example 10 (Combined Inequalities)

Solve: x > 2 and x < 6

Solution:
2 < x < 6

Key Learning from NCERT Exercises

• Handling negative signs carefully

• Representing answers on number line

• Graphical solutions for two variables

• Understanding solution region

15 Important FAQs

1. What is a linear inequality?

An inequality involving a linear expression.

2. What symbols are used?

<, >, ≤, ≥

3. What happens when multiplying by a negative number?

The inequality sign reverses.

4. What is solution set?

All values satisfying the inequality.

5. What is number line representation?

Graphical way to show solution.

6. Open circle represents?

Strict inequality (< or >).

7. Closed circle represents?

Inclusive inequality (≤ or ≥).

8. What is boundary line?

Line separating solution regions.

9. What is feasible region?

Region satisfying all inequalities.

10. What is graph of inequality?

Shaded region representing solution.

11. Example of inequality?

x + 3 > 5

12. Solve x > 3?

All numbers greater than 3.

13. Solve x ≤ 4?

All numbers less than or equal to 4.

14. What is two variable inequality?

Inequality with x and y.

15. Why are inequalities important?

They represent real-life constraints.