Everything You Need to Know about the Compound Inequalities ... - Free Printable

Let's solve each compound inequality and graph it on the number line as shown in the worksheet. I'll go through each problem step by step.

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1) $ 30 > x \geq -20 $

This is a compound inequality:
$ -20 \leq x < 30 $

- Step 1: Rewrite clearly: $ x \geq -20 $ AND $ x < 30 $
- Graphing:
- Closed circle at $-20$ (since it’s $\geq$)
- Open circle at $30$ (since it’s $<$)
- Shade between $-20$ and $30$

✔ Graph: Solid dot at $-20$, open circle at $30$, shaded line in between.

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2) $ x > 3 $ or $ x < -2 $

This is an OR compound inequality.

- Two separate parts:
- $ x > 3 $: open circle at 3, shade to the right
- $ x < -2 $: open circle at -2, shade to the left

- Since it's "or", we shade both regions.

✔ Graph: Open circle at $-2$, shaded to the left; open circle at $3$, shaded to the right.

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3) $ -12 \leq x \leq 12 $

- This is a closed interval.
- $ x \geq -12 $ AND $ x \leq 12 $
- Closed circles at both $-12$ and $12$
- Shade between them

✔ Graph: Solid dots at $-12$ and $12$, shaded line between.

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4) $ x \leq 15 $ and $ x \geq 25 $

- $ x \leq 15 $: values less than or equal to 15
- $ x \geq 25 $: values greater than or equal to 25
- There is no overlap — no number can be both ≤15 and ≥25

✘ No solution

✔ Graph: No shading — just write "No solution" or leave blank.

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5) $ x \leq 5 $ or $ x > 10 $

- $ x \leq 5 $: closed circle at 5, shade to the left
- $ x > 10 $: open circle at 10, shade to the right
- Since it's "or", combine both regions

✔ Graph: Closed dot at 5, shaded left; open dot at 10, shaded right.

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6) $ 2 < x < 3 $

- Open circle at 2 (not included), open circle at 3 (not included)
- Shade between 2 and 3

✔ Graph: Open dots at 2 and 3, shaded line in between.

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7) $ 40 \geq x \geq -50 $

Rewriting: $ -50 \leq x \leq 40 $

- Closed circles at $-50$ and $40$
- Shade between

✔ Graph: Solid dots at $-50$ and $40$, shaded line between.

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8) $ x \leq -7 $ and $ x > -28 $

- $ x > -28 $: open circle at $-28$, shade right
- $ x \leq -7 $: closed circle at $-7$, shade left
- Intersection: values that satisfy both → $ -28 < x \leq -7 $

✔ Graph: Open dot at $-28$, closed dot at $-7$, shaded between.

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9) $ x < -5 $ or $ x > 9 $

- $ x < -5 $: open circle at $-5$, shade left
- $ x > 9 $: open circle at $9$, shade right
- “Or” → both regions

✔ Graph: Open dot at $-5$, shaded left; open dot at $9$, shaded right.

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10) $ 1 < x < 4 $

- Open circle at 1, open circle at 4
- Shade between

✔ Graph: Open dots at 1 and 4, shaded line between.

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✔ Summary of Graphs:

| Problem | Inequality | Graph Description |
|--------|------------|-------------------|
| 1 | $ -20 \leq x < 30 $ | Closed at -20, open at 30, shaded in between |
| 2 | $ x > 3 $ or $ x < -2 $ | Shaded left of -2 and right of 3 |
| 3 | $ -12 \leq x \leq 12 $ | Closed at both ends, shaded between |
| 4 | $ x \leq 15 $ and $ x \geq 25 $ | No solution |
| 5 | $ x \leq 5 $ or $ x > 10 $ | Shaded left of 5 (closed) and right of 10 (open) |
| 6 | $ 2 < x < 3 $ | Open at 2 and 3, shaded between |
| 7 | $ -50 \leq x \leq 40 $ | Closed at both ends, shaded between |
| 8 | $ -28 < x \leq -7 $ | Open at -28, closed at -7, shaded between |
| 9 | $ x < -5 $ or $ x > 9 $ | Shaded left of -5 (open) and right of 9 (open) |
| 10 | $ 1 < x < 4 $ | Open at 1 and 4, shaded between |

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📌 Tips for Graphing:
- Use closed circles (●) for $ \leq $ or $ \geq $
- Use open circles (○) for $ < $ or $ > $
- For AND: find overlapping region
- For OR: combine both regions

You can now fill in the graphs on the worksheet using these instructions! Let me know if you'd like a visual version.