Math worksheet for solving compound inequalities, including graphing solutions on number lines.

Let's solve each problem step by step from the worksheet titled "Solving Compound Inequalities".

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Part 1: Write the compound inequality shown by each graph

We are given number lines with shaded regions. We need to interpret these into compound inequalities.

#### Problem 3
Graph shows a shaded region from -4 to 2, including both endpoints (closed circles).

- This means: $ x \geq -4 $ and $ x \leq 2 $
- Combined:
Answer: $ -4 \leq x \leq 2 $

#### Problem 4
Graph shows a shaded region from -1 to 5, but only at the ends — open circles at -1 and 5, shaded in between.

- So it’s: $ x > -1 $ and $ x < 5 $
- Combined:
Answer: $ -1 < x < 5 $

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Part 2: Solve each compound inequality and graph the solutions

#### Problem 5: $ -10 \leq x - 8 < -6 $

We solve this compound inequality step by step:

Add 8 to all parts:
$$
-10 + 8 \leq x - 8 + 8 < -6 + 8 \\
-2 \leq x < 2
$$

Answer: $ -2 \leq x < 2 $

Graph: Closed circle at -2, open circle at 2, shaded in between.

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#### Problem 6: $ 12 < 4x < 28 $

Divide all parts by 4:
$$
\frac{12}{4} < x < \frac{28}{4} \\
3 < x < 7
$$

Answer: $ 3 < x < 7 $

Graph: Open circles at 3 and 7, shaded in between.

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#### Problem 7: $ -2 < 3h - 7 \leq 13 $

Add 7 to all parts:
$$
-2 + 7 < 3h \leq 13 + 7 \\
5 < 3h \leq 20
$$

Divide by 3:
$$
\frac{5}{3} < h \leq \frac{20}{3} \\
1.\overline{6} < h \leq 6.\overline{6}
$$

Answer: $ \frac{5}{3} < h \leq \frac{20}{3} $

Graph: Open circle at $ \frac{5}{3} $, closed circle at $ \frac{20}{3} $, shaded in between.

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#### Problem 8: $ x > -3 $ OR $ x < 3 $

This is an OR compound inequality.

- $ x > -3 $: everything to the right of -3
- $ x < 3 $: everything to the left of 3

Together, they cover all real numbers, because every number is either greater than -3 or less than 3 (or both).

Answer: All real numbers $ (-\infty, \infty) $

Graph: Entire number line shaded.

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#### Problem 9: $ 5k < -20 $ OR $ 2k > 8 $

Solve each part:

1. $ 5k < -20 $ → divide by 5: $ k < -4 $
2. $ 2k > 8 $ → divide by 2: $ k > 4 $

So: $ k < -4 $ OR $ k > 4 $

Answer: $ k < -4 $ or $ k > 4 $

Graph: Shade left of -4 (open circle), and right of 4 (open circle). Two separate rays.

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#### Problem 10: $ 2x + 3 < 7 $ OR $ 3x + 5 > 25 $

Solve each:

1. $ 2x + 3 < 7 $ → $ 2x < 4 $ → $ x < 2 $
2. $ 3x + 5 > 25 $ → $ 3x > 20 $ → $ x > \frac{20}{3} \approx 6.67 $

So: $ x < 2 $ OR $ x > \frac{20}{3} $

Answer: $ x < 2 $ or $ x > \frac{20}{3} $

Graph: Shade left of 2 (open), and right of $ \frac{20}{3} $ (open). Two separate rays.

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Part 3: Write a compound inequality for each problem. Graph the solutions

#### Problem 11: The human ear can distinguish sounds between 20 Hz and 20,000 Hz, inclusive.

This means the frequency $ f $ satisfies:
$$
20 \leq f \leq 20,000
$$

Answer: $ 20 \leq f \leq 20,000 $

Graph: Closed circles at 20 and 20,000, shaded between.

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#### Problem 12: For a man to lose weight, he must weigh more than 140 lbs, but at most 147 lbs.

Let $ w $ = weight.

- More than 140 lbs: $ w > 140 $
- At most 147 lbs: $ w \leq 147 $

So: $ 140 < w \leq 147 $

Answer: $ 140 < w \leq 147 $

Graph: Open circle at 140, closed circle at 147, shaded in between.

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 3 | $ -4 \leq x \leq 2 $ |
| 4 | $ -1 < x < 5 $ |
| 5 | $ -2 \leq x < 2 $ |
| 6 | $ 3 < x < 7 $ |
| 7 | $ \frac{5}{3} < h \leq \frac{20}{3} $ |
| 8 | All real numbers ($ x \in \mathbb{R} $) |
| 9 | $ k < -4 $ or $ k > 4 $ |
| 10 | $ x < 2 $ or $ x > \frac{20}{3} $ |
| 11 | $ 20 \leq f \leq 20,000 $ |
| 12 | $ 140 < w \leq 147 $ |

> Note: You should draw the graphs on the number lines provided, using open/closed circles appropriately and shading the correct intervals.

Let me know if you'd like visual representations of any of the graphs!