Worksheet Solutions: Linear Inequalities | Mathematics Class 10 ICSE PDF Download

Section A: Very Short Answer Questions 

Q1. Solve: 5x – 7 < 18, where x ∈ N.
Solution:
5x – 7 < 18
→ 5x < 25
→ x < 5
Since x ∈ N, solution set = {1, 2, 3, 4}

Q2. Solve: 3x + 2 ≥ 11, x ∈ W.
Solution:
3x + 2 ≥ 11
→ 3x ≥ 9
→ x ≥ 3
Since x ∈ W, solution set = {3, 4, 5, …}

Q3. Solve: -2x > -10, x ∈ Z.
Solution:
Divide by -2 (reverse sign):
x < 5
Solution set = {…, −3, −2, −1, 0, 1, 2, 3, 4}

Q4. Solve: 7 – x ≤ 3, x ∈ N.
Solution:
7 – x ≤ 3
→ -x ≤ -4
→ x ≥ 4
Solution set = {4, 5, 6, …}

Q5. Solve: 2x – 1 ≤ 7, x ∈ N.
Solution:
2x – 1 ≤ 7
→ 2x ≤ 8
→ x ≤ 4
Solution set = {1, 2, 3, 4}

Section B: Short Answer Questions 

Q6. Solve: 4x – 9 < 11, x ∈ N.
Solution:
4x – 9 < 11
→ 4x < 20
→ x < 5
Solution set = {1, 2, 3, 4}

Q7. Solve : x – 3 ≥ 2, x ∈ Z.
Solution:
x – 3 ≥ 2
→ x ≥ 5
Solution set = {5, 6, 7, …}
Number line: Filled circle at 5, arrow to the right.

Q8. Solve: 12 – 3x ≥ 0, x ∈ N.
Solution:
12 – 3x ≥ 0
→ -3x ≥ -12
→ x ≤ 4
Solution set = {1, 2, 3, 4}

Q9. Solve: -4x + 6 < -2, x ∈ Z.
Solution:
-4x + 6 < -2
→ -4x < -8
→ x > 2
Solution set = {3, 4, 5, …}

Q10. Solve: 3x – 5 ≤ 10 and 2x + 1 ≥ 3, x ∈ N.
Solution:

First inequality:
3x – 5 ≤ 10 → 3x ≤ 15 → x ≤ 5

Second inequality:
2x + 1 ≥ 3 → 2x ≥ 2 → x ≥ 1

Combined: 1 ≤ x ≤ 5
Solution set = {1, 2, 3, 4, 5}

Section C: Long Answer Questions

Q11. Solve: -2 < 2x – 5 ≤ 7, x ∈ Z.
Solution:
Split into two parts:

(i) -2 < 2x – 5
→ 2x > 3
→ x > 1.5
→ x ≥ 2

(ii) 2x – 5 ≤ 7
→ 2x ≤ 12
→ x ≤ 6

Combined: 2 ≤ x ≤ 6
Solution set = {2, 3, 4, 5, 6}

Q12. Solve and graph: (x – 2)(x – 6) < 0, x ∈ R.
Solution:
Product < 0 means one factor positive and the other negative.

Case 1: x – 2 > 0 and x – 6 < 0
→ x > 2 and x < 6
→ 2 < x < 6

Case 2: x – 2 < 0 and x – 6 > 0 → not possible.

Final solution: 2 < x < 6
Graph: Hollow circles at 2 and 6, shaded line in between.

Q13. Solve: 3x + 4 ≥ 10 and 5 – x > 0, x ∈ N.
Solution:

First inequality:
3x + 4 ≥ 10
→ 3x ≥ 6
→ x ≥ 2

Second inequality:
5 – x > 0
→ x < 5

Combined: 2 ≤ x < 5
Solution set = {2, 3, 4}