Worksheet Solutions: Linear Inequalities

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}