Linear Inequalities Practice Questions (original) (raw)

Last Updated : 23 Jul, 2025

Linear inequalities are fundamental concepts in algebra that help us understand and solve a wide range of real-world problems. These inequalities express the relationship between two expressions using inequality symbols like <, ≤, >, and ≥. Learning to solve linear inequalities involves understanding how to manipulate these expressions to isolate the variable and determine the range of possible solutions.

In this article, we will explore various practice questions on linear inequalities to enhance your understanding and problem-solving skills. We will cover single-variable inequalities, systems of linear inequalities, and graphical representations of these solutions. Each section includes step-by-step explanations to help you grasp the methods used in solving different types of inequalities.

What are Linear Inequalities?

Linear inequalities are mathematical expressions involving linear functions that use inequality symbols (such as <, ≤, >, or ≥) instead of an equality sign. These inequalities describe a relationship where one side of the expression is not strictly equal to the other but is either less than, greater than, or equal to (depending on the inequality used).

Examples of Linear Inequalities

Solved Problems on Linear Inequalities

**Problem 1: Solve the inequality: x - 5 ≤ 3.

**Solution:

**Given: x - 5 ≤ 3

Add 5 to both sides to isolate x:

x - 5 + 5 ≤ 3 + 5

⇒ x ≤ 8

Thus, x ∈ (-∞, 8]

**Problem 2: Solve the inequality: 4x - 7 > 9

**Solution:

**Given: 4x - 7 > 9

Add 7 to both sides:

4x - 7 + 7 > 9 + 7

_⇒ 4x > 16

Divide both sides by 4:

x > 4

Thus, x ∈ (4, ∞)

**Problem 3: Solve the inequality: -2x + 6 ≥ 8

**Solution:

**Given: -2x + 6 ≥ 8

Subtract 6 from both sides:

-2x + 6 - 6 ≥ 8 - 6

_⇒ -2x ≥ 2

Divide both sides by -2, reversing the inequality sign:

x ≤ -1

Thus, x ∈ (-∞, -1]

**Problem 4: Solve the inequality: -2 < 3x - 5 ≤ 7

**Solution:

**Given: -2 < 3x - 5 ≤ 7

Add 5 to all parts:

-2 + 5 < 3x - 5 + 5 ≤ 7 + 5

_⇒ 3 < 3x ≤ 12

Divide all parts by 3:

1 < x ≤ 4

Thus, x ∈ (1, 4]

**Problem 6: Solve the inequality: 5(2x - 1) ≤ 3(3x + 4)

**Solution:

**Given: 5(2x - 1) ≤ 3(3x + 4)

Distribute the constants:

10x - 5 ≤ 9x + 12

Subtract 9x from both sides:

10x - 9x - 5 ≤ 12

_⇒ x - 5 ≤ 12

Add 5 to both sides:

x ≤ 17

Thus, x ∈ (-∞, 17]

**Problem 7: Solve the inequality: 4x + 7 > 2x - 5

**Solution:

**Given: 4x + 7 > 2x - 5

Subtract 2x from both sides:

4x - 2x + 7 > 2x - 2x - 5

_⇒ 2x + 7 > -5

Subtract 7 from both sides:

2x > -12

Divide both sides by 2:

x > -6

Thus, x ∈ (-6, ∞)

Problem 8: Solve the inequality: \frac{2x - 3}{4} \leq 1.

**Solution:

**Given: \frac{2x - 3}{4} ≤ 1

Multiply both sides by 4 to eliminate the fraction:

4 \cdot \frac{2x - 3}{4} ≤ 1 \cdot 4

_⇒ 2x - 3 ≤ 4

Add 3 to both sides:

2x - 3 + 3 ≤ 4 + 3

_⇒ 2x ≤ 7

Divide both sides by 2:

x ≤ 7/2

_⇒ x ≤ 3.5

Thus, x ∈ (-∞, 3.5]

**Problem 10: Solve the inequality: 3 < 2(x + 1) ≤ 7

**Solution:

**Given: 3 < 2(x + 1) ≤ 7

Divide the compound inequality into two parts and solve each separately.

Solve 3 < 2(x + 1):

Divide by 2:

3/2 < x + 1

Subtract 1 from both sides:

3/2 - 1 < x

_⇒ 1/2 < x

Solve 2(x + 1) ≤ 7:

Divide by 2:

x + 1 ≤ 7/2

Subtract 1 from both sides:

x ≤ 5/2

Combine the solutions:

1/2 < x ≤ 5/2

Thus, x ∈ (1/2, 5/2]

Practice Problems on Linear Inequalities

**Problem 1: Solve the inequality:

3x - 4 ≤ 2x + 5

**Problem 2: Solve the compound inequality:

-2 ≤ 4 - 3x < 10

**Problem 3: Solve the inequality and graph the solution:

(2x + 3)/5 > 1

**Problem 4: Solve the system of inequalities:

**Problem 5: Solve the inequality:

5 - 2(x - 3) ≥ 3(x + 4) - 7

**Problem 6: Solve the compound inequality:

-1 < 2x + 1 ≤ 5

**Problem 7: Solve the inequality:

4(2x - 1) < 3(x + 6)

**Problem 8: Solve the inequality:

7 - 3(2x + 1) ≥ 2(3 - x) + 1

**Problem 9: Solve the inequality and represent the solution on a number line:

\frac{3x - 2}{4} ≤ \frac{x + 5}{2}

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