Absolute Value Inequalities Worksheets with Answer Key - Free Printable

Let's solve each absolute value inequality step by step and then graph the solution on the number line. I'll go through all 8 problems one at a time.

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1. $ |y - 2| - 5 < -2 $

Step 1: Isolate the absolute value expression.

$$
|y - 2| - 5 < -2
$$

Add 5 to both sides:

$$
|y - 2| < 3
$$

Step 2: Solve the compound inequality.

$$
-3 < y - 2 < 3
$$

Add 2 to all parts:

$$
-1 < y < 5
$$

✔ Solution: $ y \in (-1, 5) $

Graph: Draw an open circle at -1 and 5, and shade between them.

---

2. $ |-6b| \leq 60 $

Step 1: Simplify the absolute value.

Note: $ |-6b| = |6b| = 6|b| $, so:

$$
6|b| \leq 60
$$

Divide both sides by 6:

$$
|b| \leq 10
$$

Step 2: Solve.

$$
-10 \leq b \leq 10
$$

✔ Solution: $ b \in [-10, 10] $

Graph: Closed circles at -10 and 10, shade between.

---

3. $ |7x + 4| \geq 74 $

Step 1: Split into two cases.

Case 1: $ 7x + 4 \geq 74 $

$$
7x \geq 70 \Rightarrow x \geq 10
$$

Case 2: $ 7x + 4 \leq -74 $

$$
7x \leq -78 \Rightarrow x \leq -\frac{78}{7} \approx -11.14
$$

✔ Solution: $ x \leq -\frac{78}{7} $ or $ x \geq 10 $

Graph: Closed circles at $ x = -\frac{78}{7} $ (≈ -11.14) and $ x = 10 $, shade left and right.

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4. $ |-4 - 3x| \leq -16 $

Step 1: Analyze the inequality.

The absolute value is always ≥ 0, so:

$$
|-4 - 3x| \geq 0
$$

But the inequality says it’s ≤ -16, which is impossible because a non-negative number cannot be less than or equal to a negative number.

✔ Solution: No solution

Graph: Blank number line (no shading)

---

5. $ 9 - 2|4x + 1| > 3 $

Step 1: Isolate the absolute value.

$$
-2|4x + 1| > 3 - 9 = -6
$$

Divide both sides by -2 → reverse inequality sign:

$$
|4x + 1| < 3
$$

Step 2: Solve.

$$
-3 < 4x + 1 < 3
$$

Subtract 1:

$$
-4 < 4x < 2
$$

Divide by 4:

$$
-1 < x < \frac{1}{2}
$$

✔ Solution: $ x \in (-1, 0.5) $

Graph: Open circles at -1 and 0.5, shade between.

---

6. $ |x| + 7 > 16 $

Step 1: Subtract 7 from both sides.

$$
|x| > 9
$$

Step 2: Solve.

$$
x < -9 \quad \text{or} \quad x > 9
$$

✔ Solution: $ x \in (-\infty, -9) \cup (9, \infty) $

Graph: Open circles at -9 and 9, shade left of -9 and right of 9.

---

7. $ \frac{|y - 4|}{5} \leq 2 $

Step 1: Multiply both sides by 5:

$$
|y - 4| \leq 10
$$

Step 2: Solve.

$$
-10 \leq y - 4 \leq 10
$$

Add 4:

$$
-6 \leq y \leq 14
$$

✔ Solution: $ y \in [-6, 14] $

Graph: Closed circles at -6 and 14, shade between.

---

8. $ \frac{|2 + 3x|}{2} \geq 5 $

Step 1: Multiply both sides by 2:

$$
|2 + 3x| \geq 10
$$

Step 2: Solve.

Case 1: $ 2 + 3x \geq 10 $

$$
3x \geq 8 \Rightarrow x \geq \frac{8}{3} \approx 2.67
$$

Case 2: $ 2 + 3x \leq -10 $

$$
3x \leq -12 \Rightarrow x \leq -4
$$

✔ Solution: $ x \leq -4 $ or $ x \geq \frac{8}{3} $

Graph: Closed circles at -4 and $ \frac{8}{3} $, shade left of -4 and right of $ \frac{8}{3} $.

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## ✔ Final Answers Summary:

| Problem | Solution |
|--------|---------|
| 1 | $ -1 < y < 5 $ |
| 2 | $ -10 \leq b \leq 10 $ |
| 3 | $ x \leq -\frac{78}{7} $ or $ x \geq 10 $ |
| 4 | No solution |
| 5 | $ -1 < x < \frac{1}{2} $ |
| 6 | $ x < -9 $ or $ x > 9 $ |
| 7 | $ -6 \leq y \leq 14 $ |
| 8 | $ x \leq -4 $ or $ x \geq \frac{8}{3} $ |

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Now, for the graphs, here's how you'd draw each on the provided number lines:

1. Problem 1: Open circles at -1 and 5, shaded in between.
2. Problem 2: Closed circles at -10 and 10, shaded in between.
3. Problem 3: Closed circles at ≈-11.14 and 10, shaded left and right.
4. Problem 4: No shading (empty).
5. Problem 5: Open circles at -1 and 0.5, shaded in between.
6. Problem 6: Open circles at -9 and 9, shaded outside.
7. Problem 7: Closed circles at -6 and 14, shaded in between.
8. Problem 8: Closed circles at -4 and ≈2.67, shaded outside.

Let me know if you'd like these graphs drawn visually!