Compound Inequalities Worksheet #1: Students write compound inequalities and graph solutions on number lines.

Let's solve each problem step by step from the Compound Inequalities Worksheet #1. The goal is to:

1. Write a compound inequality without using "and" or "or" (i.e., combine into one inequality when possible).
2. Graph the solution set on the number line.

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Problem 1:

Given: $ x < 5 $ and $ 0 \leq x $

- This means $ x $ must be greater than or equal to 0 and less than 5.
- Combine: $ 0 \leq x < 5 $
- Graph: Closed circle at 0, open circle at 5, shaded between them.

✔ Answer: $ 0 \leq x < 5 $

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Problem 2:

Given: $ x > -2 $ and $ x < 3 $

- So $ x $ is greater than -2 and less than 3.
- Combine: $ -2 < x < 3 $
- Graph: Open circles at -2 and 3, shaded in between.

✔ Answer: $ -2 < x < 3 $

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Problem 3:

Given: $ m \geq -5 $ and $ m < 3 $

- $ m $ is greater than or equal to -5 and less than 3.
- Combine: $ -5 \leq m < 3 $
- Graph: Closed circle at -5, open circle at 3, shaded between.

✔ Answer: $ -5 \leq m < 3 $

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Problem 4:

Given: $ p < -5 $ and $ p > 4 $

- $ p $ must be less than -5 AND greater than 4 → impossible!
- No number can be both less than -5 and greater than 4.
- No solution
- Graph: No shading (empty)

✔ Answer: No solution (or ∅)

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Problem 5:

Given: $ s < 3 $ or $ s \geq 1 $

- "Or" means either condition is acceptable.
- $ s < 3 $ covers everything less than 3.
- $ s \geq 1 $ covers from 1 onward.
- Together: All real numbers, because:
- If $ s < 1 $, it satisfies $ s < 3 $
- If $ s \geq 1 $, it satisfies $ s \geq 1 $
- So every number satisfies at least one condition.

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

Graph: Entire number line shaded.

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Problem 6:

Given: $ n \leq -5 $ or $ n \geq -1 $

- Two separate regions:
- $ n \leq -5 $: all numbers ≤ -5
- $ n \geq -1 $: all numbers ≥ -1
- There’s a gap between -5 and -1 (not including -5 to -1)
- Graph: Shade left of -5 (closed), and right of -1 (closed). Leave space between.

✔ Answer: $ n \leq -5 $ or $ n \geq -1 $

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Problem 7:

Given: $ w > -3 $ and $ w < 1 $

- $ w $ is greater than -3 and less than 1.
- Combine: $ -3 < w < 1 $
- Graph: Open circles at -3 and 1, shaded between.

✔ Answer: $ -3 < w < 1 $

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Problem 8:

Given: $ x < -5 $ or $ x \geq 0 $

- Two parts:
- $ x < -5 $: all numbers less than -5
- $ x \geq 0 $: all numbers ≥ 0
- Gap: From -5 to 0 (not including -5 to 0)
- Graph: Shade left of -5 (open), and right of 0 (closed).

✔ Answer: $ x < -5 $ or $ x \geq 0 $

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## ✔ Final Answers Summary:

| Problem | Compound Inequality | Graph Description |
|--------|---------------------|-------------------|
| 1 | $ 0 \leq x < 5 $ | Closed at 0, open at 5, shaded between |
| 2 | $ -2 < x < 3 $ | Open at -2 and 3, shaded between |
| 3 | $ -5 \leq m < 3 $ | Closed at -5, open at 3, shaded between |
| 4 | No solution | Empty (no shading) |
| 5 | All real numbers | Entire number line shaded |
| 6 | $ n \leq -5 $ or $ n \geq -1 $ | Shaded left of -5 (closed), right of -1 (closed) |
| 7 | $ -3 < w < 1 $ | Open at -3 and 1, shaded between |
| 8 | $ x < -5 $ or $ x \geq 0 $ | Shaded left of -5 (open), right of 0 (closed) |

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🔍 Tips for Graphing:

- Use closed circle (●) for $ \leq $ or $ \geq $
- Use open circle (○) for $ < $ or $ > $
- For "and": intersection (overlap)
- For "or": union (combine both parts)

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