Solve and graph inequalities on number lines in this math worksheet.

Let's go through each inequality problem step by step, solve them algebraically, and verify the graphed solutions. The goal is to solve the inequalities and graph the solution sets on a number line.

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1) $ \frac{c}{3} > 8 $ and $ \frac{c}{6} < 5 $

Step 1: Solve each inequality
- $ \frac{c}{3} > 8 $ → Multiply both sides by 3:
$ c > 24 $
- $ \frac{c}{6} < 5 $ → Multiply both sides by 6:
$ c < 30 $

Step 2: Combine with "and"
We need values that satisfy both:
$ 24 < c < 30 $

✔ Graph: Open circles at 24 and 30, shaded between.
✔ Answer: $ 24 < c < 30 $ — Correct!

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2) $ 1 \leq p + 11 \leq 5 $

This is a compound inequality. Subtract 11 from all parts:

$ 1 - 11 \leq p \leq 5 - 11 $
→ $ -10 \leq p \leq -6 $

✔ Graph: Closed circles at -10 and -6, shaded in between.
✔ Answer: $ -10 \leq p \leq -6 $ — Correct!

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3) $ -4 \leq n + 11 \leq 3 $

Subtract 11 from all parts:

$ -4 - 11 \leq n \leq 3 - 11 $
→ $ -15 \leq n \leq -8 $

✔ Graph: Closed circles at -15 and -8, shaded in between.
✔ Answer: $ -15 \leq n \leq -8 $ — Correct!

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4) $ g - 2 \leq 0 $ or $ g + 7 \geq 17 $

Solve each:
- $ g - 2 \leq 0 $ → $ g \leq 2 $
- $ g + 7 \geq 17 $ → $ g \geq 10 $

"Or" means either condition is acceptable.

So: $ g \leq 2 $ or $ g \geq 10 $

✔ Graph: Two rays: one from left to 2 (closed), and one from 10 to right (closed).
✔ Answer: $ g \leq 2 $ or $ g \geq 10 $ — Correct!

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5) $ \frac{s}{3} > 8 $ and $ \frac{s}{6} < 5 $

Same as #1, but variable is $ s $:

- $ \frac{s}{3} > 8 $ → $ s > 24 $
- $ \frac{s}{6} < 5 $ → $ s < 30 $

So: $ 24 < s < 30 $

✔ Graph: Open circles at 24 and 30, shaded between.
✔ Answer: $ 24 < s < 30 $ — Correct!

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6) $ 10 \leq 2d \leq 16 $

Divide all parts by 2:

$ 5 \leq d \leq 8 $

✔ Graph: Closed circles at 5 and 8, shaded between.
✔ Answer: $ 5 \leq d \leq 8 $ — Correct!

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7) $ 6v > 54 $ and $ v + 4 < 16 $

Solve each:
- $ 6v > 54 $ → $ v > 9 $
- $ v + 4 < 16 $ → $ v < 12 $

So: $ 9 < v < 12 $

✔ Graph: Open circles at 9 and 12, shaded between.
✔ Answer: $ 9 < v < 12 $ — Correct!

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8) $ \frac{k}{3} \leq 9 $ or $ \frac{k}{7} \geq 5 $

Solve each:
- $ \frac{k}{3} \leq 9 $ → $ k \leq 27 $
- $ \frac{k}{7} \geq 5 $ → $ k \geq 35 $

"Or" means either is acceptable.

So: $ k \leq 27 $ or $ k \geq 35 $

✔ Graph: Ray from left to 27 (closed), and ray from 35 to right (closed).
✔ Answer: $ k \leq 27 $ or $ k \geq 35 $ — Correct!

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9) $ 2z > 12 $ and $ z + 5 < 16 $

Solve each:
- $ 2z > 12 $ → $ z > 6 $
- $ z + 5 < 16 $ → $ z < 11 $

So: $ 6 < z < 11 $

✔ Graph: Open circles at 6 and 11, shaded between.
✔ Answer: $ 6 < z < 11 $ — Correct!

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10) $ 9y < 45 $ or $ 7y > 91 $

Solve each:
- $ 9y < 45 $ → $ y < 5 $
- $ 7y > 91 $ → $ y > 13 $

So: $ y < 5 $ or $ y > 13 $

✔ Graph: Ray from left to 5 (open), and ray from 13 to right (open).
✔ Answer: $ y < 5 $ or $ y > 13 $ — Correct!

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✔ Final Summary

All solutions are correctly solved and graphed.

| Problem | Solution |
|--------|----------|
| 1) | $ 24 < c < 30 $ |
| 2) | $ -10 \leq p \leq -6 $ |
| 3) | $ -15 \leq n \leq -8 $ |
| 4) | $ g \leq 2 $ or $ g \geq 10 $ |
| 5) | $ 24 < s < 30 $ |
| 6) | $ 5 \leq d \leq 8 $ |
| 7) | $ 9 < v < 12 $ |
| 8) | $ k \leq 27 $ or $ k \geq 35 $ |
| 9) | $ 6 < z < 11 $ |
| 10) | $ y < 5 $ or $ y > 13 $ |

All graphs match the solutions. Well done! 🎉

> 🔍 Note: When solving compound inequalities:
> - Use "and" → intersection (overlap)
> - Use "or" → union (combine)

And remember:
- Closed circle = ≤ or ≥
- Open circle = < or >

Let me know if you'd like this printed or turned into a study sheet!