Solving Inequalities Worksheet with eight problems and number lines for graphing solutions.

Of course! Let's solve each inequality step by step and then describe how to graph the solution on the number line.

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① `2q - 4q ≤ -2`

Step 1: Combine like terms.
> `2q - 4q = -2q`
> So: `-2q ≤ -2`

Step 2: Divide both sides by -2.
⚠️ Important: When you divide or multiply an inequality by a *negative* number, you must flip the inequality sign.

> `q ≥ 1`

✔ Solution: `q ≥ 1`

Graphing:
On the number line (from -4 to 5), place a closed circle at `1` (because it’s “≥”, so 1 is included) and shade everything to the right.

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② `n - 8 < -8`

Step 1: Add 8 to both sides.

> `n < 0`

✔ Solution: `n < 0`

Graphing:
On the number line (from -5 to 7), place an open circle at `0` (because it’s “<”, so 0 is NOT included) and shade everything to the left.

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③ `k - 8 > 15`

Step 1: Add 8 to both sides.

> `k > 23`

✔ Solution: `k > 23`

Graphing:
On the number line (from 14 to 26), place an open circle at `23` and shade everything to the right.

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④ `0 ≤ 2p + 3p`

Step 1: Combine like terms.

> `0 ≤ 5p`

Step 2: Divide both sides by 5.

> `0 ≤ p` → or equivalently, `p ≥ 0`

✔ Solution: `p ≥ 0`

Graphing:
On the number line (from -5 to 2), place a closed circle at `0` and shade everything to the right.

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⑤ `-5(1 - 4s) > -5`

Step 1: Distribute the -5.

> `-5 + 20s > -5`

Step 2: Add 5 to both sides.

> `20s > 0`

Step 3: Divide both sides by 20.

> `s > 0`

✔ Solution: `s > 0`

Graphing:
On the number line (from -4 to 4), place an open circle at `0` and shade everything to the right.

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⑥ `5 + y ≤ -13`

Step 1: Subtract 5 from both sides.

> `y ≤ -18`

✔ Solution: `y ≤ -18`

Graphing:
On the number line (from -26 to -14), place a closed circle at `-18` and shade everything to the left.

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⑦ `-6 > 5n + 5 + 4`

Step 1: Combine constants on the right.

> `-6 > 5n + 9`

Step 2: Subtract 9 from both sides.

> `-15 > 5n`

Step 3: Divide both sides by 5.

> `-3 > n` → or equivalently, `n < -3`

✔ Solution: `n < -3`

Graphing:
On the number line (from -7 to 2), place an open circle at `-3` and shade everything to the left.

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⑧ `9 < b - 2`

Step 1: Add 2 to both sides.

> `11 < b` → or equivalently, `b > 11`

✔ Solution: `b > 11`

Graphing:
On the number line (from 9 to 21), place an open circle at `11` and shade everything to the right.

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## ✔ Final Answers Summary:

| Problem | Solution | Graph Description |
|---------|-----------------|--------------------------------------------------------|
| ① | `q ≥ 1` | Closed circle at 1, shade right |
| ② | `n < 0` | Open circle at 0, shade left |
| ③ | `k > 23` | Open circle at 23, shade right |
| ④ | `p ≥ 0` | Closed circle at 0, shade right |
| ⑤ | `s > 0` | Open circle at 0, shade right |
| ⑥ | `y ≤ -18` | Closed circle at -18, shade left |
| ⑦ | `n < -3` | Open circle at -3, shade left |
| ⑧ | `b > 11` | Open circle at 11, shade right |

You can now draw these graphs on your worksheet using the number lines provided!

Let me know if you want me to generate actual image graphs for each!