Absolute Value Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each absolute value inequality step by step and then graph the solution on a number line. I'll show all work for each problem.

---

1. $ |x + 4| < 7 $

Step 1: Recall that for $ |A| < B $, where $ B > 0 $, the solution is:
$$
-B < A < B
$$

So,
$$
-7 < x + 4 < 7
$$

Step 2: Subtract 4 from all parts:
$$
-7 - 4 < x < 7 - 4 \\
-11 < x < 3
$$

✔ Solution: $ (-11, 3) $

Graph:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|---->
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
(------------------------)
```
Open circles at -11 and 3, shaded in between.

---

2. $ |2x - 1| \geq 9 $

Step 1: For $ |A| \geq B $, the solution is:
$$
A \leq -B \quad \text{or} \quad A \geq B
$$

So,
$$
2x - 1 \leq -9 \quad \text{or} \quad 2x - 1 \geq 9
$$

Step 2: Solve each inequality:

- $ 2x - 1 \leq -9 $
$$
2x \leq -8 \Rightarrow x \leq -4
$$

- $ 2x - 1 \geq 9 $
$$
2x \geq 10 \Rightarrow x \geq 5
$$

✔ Solution: $ (-\infty, -4] \cup [5, \infty) $

Graph:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|---->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
[====================] [====================]
```
Closed circles at -4 and 5, shaded to the left of -4 and right of 5.

---

3. $ -|2x| - 1 > -15 $

Step 1: Add 1 to both sides:
$$
-|2x| > -14
$$

Step 2: Multiply both sides by -1 → reverse inequality:
$$
|2x| < 14
$$

Step 3: Divide both sides by 2:
$$
|x| < 7
$$

This means:
$$
-7 < x < 7
$$

✔ Solution: $ (-7, 7) $

Graph:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|---->
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
(------------------------)
```
Open circles at -7 and 7, shaded in between.

---

4. $ \left|\frac{x}{2}\right| \leq 6 $

Step 1: Multiply both sides by 2:
$$
|x| \leq 12
$$

Step 2: This gives:
$$
-12 \leq x \leq 12
$$

✔ Solution: $ [-12, 12] $

Graph:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|---->
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
[========================]
```
Closed circles at -12 and 12, shaded in between.

---

5. $ |6x| + 4 \geq 28 $

Step 1: Subtract 4 from both sides:
$$
|6x| \geq 24
$$

Step 2: Divide both sides by 6:
$$
|x| \geq 4
$$

Step 3: So:
$$
x \leq -4 \quad \text{or} \quad x \geq 4
$$

✔ Solution: $ (-\infty, -4] \cup [4, \infty) $

Graph:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|---->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
[====================] [====================]
```
Closed circles at -4 and 4, shaded to the left of -4 and right of 4.

---

6. $ 3|x| < 12 $

Step 1: Divide both sides by 3:
$$
|x| < 4
$$

Step 2: So:
$$
-4 < x < 4
$$

✔ Solution: $ (-4, 4) $

Graph:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|---->
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
(------------------------)
```
Open circles at -4 and 4, shaded in between.

---

✔ Final Answers Summary:

| Problem | Solution |
|--------|----------|
| 1 | $ (-11, 3) $ |
| 2 | $ (-\infty, -4] \cup [5, \infty) $ |
| 3 | $ (-7, 7) $ |
| 4 | $ [-12, 12] $ |
| 5 | $ (-\infty, -4] \cup [4, \infty) $ |
| 6 | $ (-4, 4) $ |

Each should be graphed accordingly with appropriate open or closed circles and shading.

Let me know if you'd like these solutions formatted into a printable worksheet!