Math worksheet for solving inequalities, featuring number line representations and algebraic problems.

Let's solve each inequality in Section B step by step and then explain how to represent the solutions on a number line.

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🔷 Section A: Show the inequalities on the number line

These are already given, so we'll just interpret them briefly:

1. x > 2
→ Open circle at 2, shaded to the right.

2. x ≤ -3
→ Closed circle at -3, shaded to the left.

3. -2.5 < x ≤ 12.5
→ Open circle at -2.5, closed circle at 12.5, shaded between.

4. -0.8 ≤ x < 1.4
→ Closed circle at -0.8, open circle at 1.4, shaded between.

✔ These are visual representations — no solving needed here.

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🔷 Section B: Solve the inequalities and show on number line

We’ll solve each one algebraically and describe how to graph it.

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#### 1) x + 3 ≥ 8

Subtract 3 from both sides:
$$
x ≥ 8 - 3 = 5
$$

✔ Solution: $ x \geq 5 $

👉 Number line: Closed circle at 5, shaded to the right.

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#### 2) x - 5 ≤ -1

Add 5 to both sides:
$$
x ≤ -1 + 5 = 4
$$

✔ Solution: $ x \leq 4 $

👉 Number line: Closed circle at 4, shaded to the left.

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#### 3) 4x < 36

Divide both sides by 4:
$$
x < 9
$$

✔ Solution: $ x < 9 $

👉 Number line: Open circle at 9, shaded to the left.

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#### 4) 8x + 3 > 27

Subtract 3:
$$
8x > 24
$$

Divide by 8:
$$
x > 3
$$

✔ Solution: $ x > 3 $

👉 Number line: Open circle at 3, shaded to the right.

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#### 5) 11x - 3 ≥ 41

Add 3:
$$
11x ≥ 44
$$

Divide by 11:
$$
x ≥ 4
$$

✔ Solution: $ x \geq 4 $

👉 Number line: Closed circle at 4, shaded to the right.

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#### 6) 12x + 6 < 90

Subtract 6:
$$
12x < 84
$$

Divide by 12:
$$
x < 7
$$

✔ Solution: $ x < 7 $

👉 Number line: Open circle at 7, shaded to the left.

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#### 7) $\frac{x}{7} > 4$

Multiply both sides by 7:
$$
x > 28
$$

✔ Solution: $ x > 28 $

👉 Number line: Open circle at 28, shaded to the right.

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#### 8) $\frac{x}{6} < 12$

Multiply both sides by 6:
$$
x < 72
$$

✔ Solution: $ x < 72 $

👉 Number line: Open circle at 72, shaded to the left.

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#### 9) $\frac{x}{8} + 1 \leq 7$

Subtract 1:
$$
\frac{x}{8} \leq 6
$$

Multiply by 8:
$$
x \leq 48
$$

✔ Solution: $ x \leq 48 $

👉 Number line: Closed circle at 48, shaded to the left.

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#### 10) $\frac{x}{3} - 2 \geq 9$

Add 2:
$$
\frac{x}{3} \geq 11
$$

Multiply by 3:
$$
x \geq 33
$$

✔ Solution: $ x \geq 33 $

👉 Number line: Closed circle at 33, shaded to the right.

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#### 11) $\frac{x+4}{2} < 1$

Multiply both sides by 2:
$$
x + 4 < 2
$$

Subtract 4:
$$
x < -2
$$

✔ Solution: $ x < -2 $

👉 Number line: Open circle at -2, shaded to the left.

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#### 12) $\frac{x - 9}{4} \geq -4$

Multiply both sides by 4:
$$
x - 9 \geq -16
$$

Add 9:
$$
x \geq -7
$$

✔ Solution: $ x \geq -7 $

👉 Number line: Closed circle at -7, shaded to the right.

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#### 13) $\frac{2x + 1}{7} < 3$

Multiply both sides by 7:
$$
2x + 1 < 21
$$

Subtract 1:
$$
2x < 20
$$

Divide by 2:
$$
x < 10
$$

✔ Solution: $ x < 10 $

👉 Number line: Open circle at 10, shaded to the left.

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#### 14) $\frac{7x - 5}{8} \geq -5$

Multiply both sides by 8:
$$
7x - 5 \geq -40
$$

Add 5:
$$
7x \geq -35
$$

Divide by 7:
$$
x \geq -5
$$

✔ Solution: $ x \geq -5 $

👉 Number line: Closed circle at -5, shaded to the right.

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✔ Summary of Solutions:

| Problem | Solution |
|--------|----------|
| 1 | $ x \geq 5 $ |
| 2 | $ x \leq 4 $ |
| 3 | $ x < 9 $ |
| 4 | $ x > 3 $ |
| 5 | $ x \geq 4 $ |
| 6 | $ x < 7 $ |
| 7 | $ x > 28 $ |
| 8 | $ x < 72 $ |
| 9 | $ x \leq 48 $ |
| 10 | $ x \geq 33 $ |
| 11 | $ x < -2 $ |
| 12 | $ x \geq -7 $ |
| 13 | $ x < 10 $ |
| 14 | $ x \geq -5 $ |

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📌 How to Graph on a Number Line:

- Use closed circle (●) for $ \leq $ or $ \geq $
- Use open circle (○) for $ < $ or $ > $
- Shade the direction that satisfies the inequality:
- For $ > $ or $ \geq $: shade to the right
- For $ < $ or $ \leq $: shade to the left

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🧠 Tips:

- Always perform the same operation on both sides.
- When multiplying or dividing by a negative number, reverse the inequality sign (not needed here).
- Simplify step-by-step.

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If you'd like, I can generate a visual representation of any of these number lines! Let me know which ones you’d like illustrated.