Solving Two-Step Inequalities #2 | Worksheet - Free Printable

Let's solve each of the two-step inequalities step by step, and then graph the solution sets on the number lines provided.

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1. $ 3x + 6 < 42 $

Step 1: Subtract 6 from both sides
$ 3x < 42 - 6 $
$ 3x < 36 $

Step 2: Divide by 3
$ x < 12 $

✔ Solution: $ x < 12 $
Graph: Open circle at 12, shade to the left.

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2. $ 2x + 10 ≥ 4 $

Step 1: Subtract 10 from both sides
$ 2x ≥ 4 - 10 $
$ 2x ≥ -6 $

Step 2: Divide by 2
$ x ≥ -3 $

✔ Solution: $ x ≥ -3 $
Graph: Closed circle at -3, shade to the right.

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3. $ 6y - 14 ≤ 16 $

Step 1: Add 14 to both sides
$ 6y ≤ 16 + 14 $
$ 6y ≤ 30 $

Step 2: Divide by 6
$ y ≤ 5 $

✔ Solution: $ y ≤ 5 $
Graph: Closed circle at 5, shade to the left.

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4. $ -4z - 9 < 27 $

Step 1: Add 9 to both sides
$ -4z < 27 + 9 $
$ -4z < 36 $

Step 2: Divide by -4 → Reverse inequality sign
$ z > -9 $

✔ Solution: $ z > -9 $
Graph: Open circle at -9, shade to the right.

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5. $ 3m + 20 ≤ -8 $

Step 1: Subtract 20 from both sides
$ 3m ≤ -8 - 20 $
$ 3m ≤ -28 $

Step 2: Divide by 3
$ m ≤ -\frac{28}{3} $ ≈ $ -9.\overline{3} $

✔ Solution: $ m ≤ -\frac{28}{3} $ or $ m ≤ -9.\overline{3} $
Graph: Closed circle at $ -9.\overline{3} $, shade to the left.

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6. $ 3.2y - 2.9 ≥ 6.7 $

Step 1: Add 2.9 to both sides
$ 3.2y ≥ 6.7 + 2.9 $
$ 3.2y ≥ 9.6 $

Step 2: Divide by 3.2
$ y ≥ \frac{9.6}{3.2} = 3 $

✔ Solution: $ y ≥ 3 $
Graph: Closed circle at 3, shade to the right.

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7. $ -0.5t + 0.5 ≤ 8.5 $

Step 1: Subtract 0.5 from both sides
$ -0.5t ≤ 8.5 - 0.5 $
$ -0.5t ≤ 8 $

Step 2: Divide by -0.5 → Reverse inequality sign
$ t ≥ \frac{8}{-0.5} = -16 $

✔ Solution: $ t ≥ -16 $
Graph: Closed circle at -16, shade to the right.

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8. $ \left| n - \frac{1}{2} \right| > 1 $

This is an absolute value inequality.

We split into two cases:

Case 1: $ n - \frac{1}{2} > 1 $
→ $ n > 1 + \frac{1}{2} = \frac{3}{2} $

Case 2: $ n - \frac{1}{2} < -1 $
→ $ n < -1 + \frac{1}{2} = -\frac{1}{2} $

✔ Solution: $ n < -\frac{1}{2} $ or $ n > \frac{3}{2} $
Graph: Open circles at $ -\frac{1}{2} $ and $ \frac{3}{2} $, shade to the left of $ -\frac{1}{2} $ and to the right of $ \frac{3}{2} $.

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9. $ \left| j - 3 \right| ≥ 8 $

Again, absolute value inequality.

Case 1: $ j - 3 ≥ 8 $ → $ j ≥ 11 $

Case 2: $ j - 3 ≤ -8 $ → $ j ≤ -5 $

✔ Solution: $ j ≤ -5 $ or $ j ≥ 11 $
Graph: Closed circles at -5 and 11, shade to the left of -5 and to the right of 11.

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10. $ 0.25f + 0.5 > -1.75 $

Step 1: Subtract 0.5 from both sides
$ 0.25f > -1.75 - 0.5 $
$ 0.25f > -2.25 $

Step 2: Divide by 0.25
$ f > \frac{-2.25}{0.25} = -9 $

✔ Solution: $ f > -9 $
Graph: Open circle at -9, shade to the right.

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✔ Summary of Solutions:

| Problem | Solution |
|--------|---------|
| 1 | $ x < 12 $ |
| 2 | $ x ≥ -3 $ |
| 3 | $ y ≤ 5 $ |
| 4 | $ z > -9 $ |
| 5 | $ m ≤ -\frac{28}{3} $ (≈ -9.33) |
| 6 | $ y ≥ 3 $ |
| 7 | $ t ≥ -16 $ |
| 8 | $ n < -\frac{1}{2} $ or $ n > \frac{3}{2} $ |
| 9 | $ j ≤ -5 $ or $ j ≥ 11 $ |
| 10 | $ f > -9 $ |

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📌 How to Graph:

- Use open circle for $ < $ or $ > $
- Use closed circle for $ ≤ $ or $ ≥ $
- Shade the direction that satisfies the inequality.

You can now fill in the number lines accordingly using these solutions.

Let me know if you'd like a visual representation of any graph!