Absolute Value Equations Coloring Activity - Free Printable

Let's solve each of the absolute value equations step by step. Since this is a coloring worksheet, we need to find the solution(s) for each equation, and then use those answers to determine the correct color (based on a key that would be on a separate page — which isn't provided here). But since you're asking for the solutions, I'll solve all 10 problems.

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🔷 Problem 1: $ |5x| = 15 $

The absolute value equation means:
$$
5x = 15 \quad \text{or} \quad 5x = -15
$$

Solve both:
- $ x = 3 $
- $ x = -3 $

✔ Solutions: $ x = 3 $ or $ x = -3 $

---

🔷 Problem 2: $ |-2x| = 8 $

Absolute value: $ |-2x| = |2x| = 8 $

So:
$$
2x = 8 \quad \text{or} \quad 2x = -8
$$

Solve:
- $ x = 4 $
- $ x = -4 $

✔ Solutions: $ x = 4 $ or $ x = -4 $

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🔷 Problem 3: $ \left| \frac{x}{4} \right| = 24 $

This means:
$$
\frac{x}{4} = 24 \quad \text{or} \quad \frac{x}{4} = -24
$$

Multiply both sides by 4:
- $ x = 96 $
- $ x = -96 $

✔ Solutions: $ x = 96 $ or $ x = -96 $

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🔷 Problem 4: $ |x - 7| = 10 $

Break into two cases:
$$
x - 7 = 10 \quad \text{or} \quad x - 7 = -10
$$

Solve:
- $ x = 17 $
- $ x = -3 $

✔ Solutions: $ x = 17 $ or $ x = -3 $

---

🔷 Problem 5: $ |x + 2| + 3 = 7 $

First subtract 3 from both sides:
$$
|x + 2| = 4
$$

Now:
$$
x + 2 = 4 \quad \text{or} \quad x + 2 = -4
$$

Solve:
- $ x = 2 $
- $ x = -6 $

✔ Solutions: $ x = 2 $ or $ x = -6 $

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🔷 Problem 6: $ 5 \left| \frac{x}{2} \right| = 20 $

Divide both sides by 5:
$$
\left| \frac{x}{2} \right| = 4
$$

Then:
$$
\frac{x}{2} = 4 \quad \text{or} \quad \frac{x}{2} = -4
$$

Multiply by 2:
- $ x = 8 $
- $ x = -8 $

✔ Solutions: $ x = 8 $ or $ x = -8 $

---

🔷 Problem 7: $ |3x| - 5 = 10 $

Add 5 to both sides:
$$
|3x| = 15
$$

So:
$$
3x = 15 \quad \text{or} \quad 3x = -15
$$

Solve:
- $ x = 5 $
- $ x = -5 $

✔ Solutions: $ x = 5 $ or $ x = -5 $

---

🔷 Problem 8: $ |-2x + 6| = 6 $

Set up two equations:
$$
-2x + 6 = 6 \quad \text{or} \quad -2x + 6 = -6
$$

Solve first:
- $ -2x = 0 $ → $ x = 0 $

Second:
- $ -2x = -12 $ → $ x = 6 $

✔ Solutions: $ x = 0 $ or $ x = 6 $

---

🔷 Problem 9: $ 2|4x + 1| = 10 $

Divide both sides by 2:
$$
|4x + 1| = 5
$$

Now:
$$
4x + 1 = 5 \quad \text{or} \quad 4x + 1 = -5
$$

First:
- $ 4x = 4 $ → $ x = 1 $

Second:
- $ 4x = -6 $ → $ x = -\frac{3}{2} $

✔ Solutions: $ x = 1 $ or $ x = -\frac{3}{2} $

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🔷 Problem 10: $ 2 - |3x - 9| = 2 $

Subtract 2 from both sides:
$$
-|3x - 9| = 0
$$

Multiply both sides by -1:
$$
|3x - 9| = 0
$$

Absolute value is zero only when inside is zero:
$$
3x - 9 = 0 \Rightarrow x = 3
$$

✔ Solution: $ x = 3 $

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✔ Final Answers Summary:

| Problem | Solution(s) |
|--------|-------------|
| 1 | $ x = 3 $ or $ x = -3 $ |
| 2 | $ x = 4 $ or $ x = -4 $ |
| 3 | $ x = 96 $ or $ x = -96 $ |
| 4 | $ x = 17 $ or $ x = -3 $ |
| 5 | $ x = 2 $ or $ x = -6 $ |
| 6 | $ x = 8 $ or $ x = -8 $ |
| 7 | $ x = 5 $ or $ x = -5 $ |
| 8 | $ x = 0 $ or $ x = 6 $ |
| 9 | $ x = 1 $ or $ x = -\frac{3}{2} $ |
| 10 | $ x = 3 $ |

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🎨 How to Use This in the Worksheet:

You would now look at the coloring key (which is likely on a separate page) and match each solution to a color. For example:

- If the key says “$ x = 3 $” → red, then color problem #1 and #10 red.
- If “$ x = -3 $” → blue, then color part of #1 blue.
- And so on.

Since the key isn’t included, you’ll need to refer to it to assign colors.

But your job is done: you’ve solved all the equations!

Let me know if you’d like help interpreting a specific color key or checking your work.