Graphing compound inequalities worksheet with eight problems and number lines for visual representation.

Let's solve each compound inequality and graph it on the number line. We'll go through each one step by step.

---

1. $ x < 9 $ and $ x \geq 3 $

- This is a "and" compound inequality → both conditions must be true.
- So, $ x $ must be greater than or equal to 3 and less than 9.
- In interval notation: $ [3, 9) $
- On the number line:
- Closed circle at 3 (because $ x \geq 3 $)
- Open circle at 9 (because $ x < 9 $)
- Shade between them.

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
●===================○
```

---

2. $ x > -1 $ and $ x < -3 $

- $ x > -1 $ means values greater than -1
- $ x < -3 $ means values less than -3
- But no number can be both greater than -1 and less than -3.
- These two sets do not overlap.
- So, no solution.

✔ Graph:
No shading — empty set.

```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
```
→ No points shaded

---

3. $ -4 < x \leq 4 $

- This is a compound inequality: $ x $ is greater than -4 and less than or equal to 4.
- Interval notation: $ (-4, 4] $
- On the number line:
- Open circle at -4 (since $ x > -4 $)
- Closed circle at 4 (since $ x \leq 4 $)
- Shade between.

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
○===================●
```

---

4. $ -2 < x \leq 2 $

- $ x $ is greater than -2 and less than or equal to 2.
- Interval: $ (-2, 2] $
- Open circle at -2, closed circle at 2, shade in between.

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
○===================●
```

---

5. $ x \geq 12 $ or $ x \leq -16 $

- This is an "or" compound inequality → either condition is acceptable.
- So, all numbers ≤ -16 OR ≥ 12
- Two separate rays:
- One from $ -\infty $ to -16 (including -16)
- One from 12 to $ +\infty $ (including 12)

Interval: $ (-\infty, -16] \cup [12, \infty) $

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
●=================== ===================●
```
(Shade left of -16 including -16, and right of 12 including 12)

---

6. $ x > 8 $ and $ x < 16 $

- Both conditions: $ x $ must be greater than 8 and less than 16.
- So: $ 8 < x < 16 $
- Interval: $ (8, 16) $
- Open circles at 8 and 16, shade between.

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-20 -16 -12 -8 -4 0 4 8 12 16 20
○===================○
```

---

7. $ x \leq -2 $ and $ x \geq -5 $

- $ x \geq -5 $ and $ x \leq -2 $
- So: $ -5 \leq x \leq -2 $
- Interval: $ [-5, -2] $
- Closed circles at both ends, shade between.

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
●===================●
```

---

8. $ -5 \leq x < 1 $

- $ x $ is greater than or equal to -5 and less than 1.
- Interval: $ [-5, 1) $
- Closed circle at -5, open circle at 1, shade between.

✔ Graph:
```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
●===================○
```

---

✔ Summary Table:

| Problem | Inequality | Solution | Graph Description |
|--------|------------|----------|-------------------|
| 1 | $ x < 9 $ and $ x \geq 3 $ | $ [3, 9) $ | Closed at 3, open at 9, shaded between |
| 2 | $ x > -1 $ and $ x < -3 $ | No solution | Empty set |
| 3 | $ -4 < x \leq 4 $ | $ (-4, 4] $ | Open at -4, closed at 4, shaded between |
| 4 | $ -2 < x \leq 2 $ | $ (-2, 2] $ | Open at -2, closed at 2, shaded between |
| 5 | $ x \geq 12 $ or $ x \leq -16 $ | $ (-\infty, -16] \cup [12, \infty) $ | Two rays: left to -16 (closed), right from 12 (closed) |
| 6 | $ x > 8 $ and $ x < 16 $ | $ (8, 16) $ | Open at 8 and 16, shaded between |
| 7 | $ x \leq -2 $ and $ x \geq -5 $ | $ [-5, -2] $ | Closed at both ends, shaded between |
| 8 | $ -5 \leq x < 1 $ | $ [-5, 1) $ | Closed at -5, open at 1, shaded between |

---

You can now draw these graphs on the provided number lines using the descriptions above.

Let me know if you'd like a printable version or visual diagrams!