Inequalities Worksheets with Answer Key - Free Printable

I'll solve each inequality step by step, working through all 12 problems carefully.

Problem 1: 4x - 27 > 17
Add 27 to both sides: 4x > 44
Divide by 4: x > 11

Problem 2: -2/7(1 - 4x) + 8 < 18
Subtract 8 from both sides: -2/7(1 - 4x) < 10
Multiply both sides by -7/2 (flip the inequality): 1 - 4x > -35
Subtract 1: -4x > -36
Divide by -4 (flip again): x < 9

Problem 3: 22 < 3(x + 7) < 23
Divide all parts by 3: 22/3 < x + 7 < 23/3
Subtract 7 from all parts: 22/3 - 7 < x < 23/3 - 7
Convert: 22/3 - 21/3 < x < 23/3 - 21/3
Result: 1/3 < x < 2/3

Problem 4: 2v + 11 > 4v - 7 - 16
Simplify right side: 2v + 11 > 4v - 23
Subtract 2v from both sides: 11 > 2v - 23
Add 23: 34 > 2v
Divide by 2: 17 > v, or v < 17

Problem 5: 1/2 ≤ (4x + 5)/2
Multiply both sides by 2: 1 ≤ 4x + 5
Subtract 5: -4 ≤ 4x
Divide by 4: -1 ≤ x, or x ≥ -1

Problem 6: -3x - 19 < -18
Add 19 to both sides: -3x < 1
Divide by -3 (flip inequality): x > -1/3

Problem 7: (t + 3)/2 < -8
Multiply both sides by 2: t + 3 < -16
Subtract 3: t < -19

Problem 8: 0.5x + 3.6 > 8.1
Subtract 3.6: 0.5x > 4.5
Divide by 0.5: x > 9

Problem 9: -10 < 6x + 2 ≤ 32
Subtract 2 from all parts: -12 < 6x ≤ 30
Divide by 6: -2 < x ≤ 5

Problem 10: -1 < -x/9 + 2
Subtract 2: -3 < -x/9
Multiply by -9 (flip inequality): 27 > x, or x < 27

Problem 11: 1 ≤ 4x - 7 < 11
Add 7 to all parts: 8 ≤ 4x < 18
Divide by 4: 2 ≤ x < 4.5

Problem 12: -24 ≤ 3x - 9 ≤ -14
Add 9 to all parts: -15 ≤ 3x ≤ -5
Divide by 3: -5 ≤ x ≤ -5/3

Let me verify my calculations:

Problem 2 check: -2/7(1-4x) + 8 < 18
If x = 8: -2/7(1-32) + 8 = -2/7(-31) + 8 = 62/7 + 8 = 62/7 + 56/7 = 118/7 ≈ 16.86 < 18 ✓
If x = 10: -2/7(1-40) + 8 = -2/7(-39) + 8 = 78/7 + 8 = 78/7 + 56/7 = 134/7 ≈ 19.14 > 18
So x < 9 is correct ✓

Problem 10 check: -1 < -x/9 + 2
If x = 26: -26/9 + 2 = -26/9 + 18/9 = -8/9, and -1 < -8/9 ✓
If x = 28: -28/9 + 2 = -28/9 + 18/9 = -10/9, and -1 < -10/9 is false ✗
So x < 27 is correct ✓

Final Answer:
1. x > 11
2. x < 9
3. 1/3 < x < 2/3
4. v < 17
5. x ≥ -1
6. x > -1/3
7. t < -19
8. x > 9
9. -2 < x ≤ 5
10. x < 27
11. 2 ≤ x < 4.5
12. -5 ≤ x ≤ -5/3