Linear Inequalities Practice Questions

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).

Solved Problems

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

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:

• 2x + y ≥ 3
• x - y < 2

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}