Absolute Value Equations Coloring Worksheet - Solve and Color Activity

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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1. $ |5x| = 15 $

We split into two cases:

- $ 5x = 15 $ → $ x = 3 $
- $ 5x = -15 $ → $ x = -3 $

✔ Solutions: $ x = 3 $ or $ x = -3 $

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2. $ |-2x| = 8 $

Note: $ |-2x| = |2x| $, so:

- $ 2x = 8 $ → $ x = 4 $
- $ 2x = -8 $ → $ x = -4 $

✔ Solutions: $ x = 4 $ or $ x = -4 $

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3. $ \left|\frac{x}{4}\right| = 24 $

Multiply both sides by 4:

- $ |x| = 96 $
- So $ x = 96 $ or $ x = -96 $

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

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4. $ |x - 7| = 10 $

Split into two cases:

- $ x - 7 = 10 $ → $ x = 17 $
- $ x - 7 = -10 $ → $ x = -3 $

✔ Solutions: $ x = 17 $ or $ x = -3 $

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5. $ |x + 2| + 3 = 7 $

Subtract 3 from both sides:

- $ |x + 2| = 4 $

Now solve:

- $ x + 2 = 4 $ → $ x = 2 $
- $ x + 2 = -4 $ → $ x = -6 $

✔ Solutions: $ x = 2 $ or $ x = -6 $

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6. $ 5\left|\frac{x}{2}\right| = 20 $

Divide both sides by 5:

- $ \left|\frac{x}{2}\right| = 4 $

Multiply both sides by 2:

- $ |x| = 8 $
- So $ x = 8 $ or $ x = -8 $

✔ Solutions: $ x = 8 $ or $ x = -8 $

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7. $ |3x| - 5 = 10 $

Add 5 to both sides:

- $ |3x| = 15 $

Then:

- $ 3x = 15 $ → $ x = 5 $
- $ 3x = -15 $ → $ x = -5 $

✔ Solutions: $ x = 5 $ or $ x = -5 $

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8. $ |-2x + 6| = 6 $

This is $ | -2x + 6 | = 6 $. Let’s solve:

Case 1: $ -2x + 6 = 6 $
→ $ -2x = 0 $ → $ x = 0 $

Case 2: $ -2x + 6 = -6 $
→ $ -2x = -12 $ → $ x = 6 $

✔ Solutions: $ x = 0 $ or $ x = 6 $

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9. $ 2|4x + 1| = 10 $

Divide both sides by 2:

- $ |4x + 1| = 5 $

Now solve:

- $ 4x + 1 = 5 $ → $ 4x = 4 $ → $ x = 1 $
- $ 4x + 1 = -5 $ → $ 4x = -6 $ → $ x = -\frac{3}{2} $

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

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10. $ 2 - |3x - 9| = 2 $

Subtract 2 from both sides:

- $ -|3x - 9| = 0 $
- So $ |3x - 9| = 0 $

Absolute value equals zero only when the inside is zero:

- $ 3x - 9 = 0 $ → $ 3x = 9 $ → $ x = 3 $

✔ Solution: $ x = 3 $ (only one solution)

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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 (not shown here) that maps each answer (or perhaps the values like 3, -3, etc.) to a specific color. Then, using the problem number and the matching color, you would color in the corresponding section of the picture.

For example:
- If the key says "3 → Red", then problem 1 (which has $ x = 3 $) would be colored red.
- If it says "both 3 and -3 → Blue", then problem 1 gets blue.

But without the key, we can't assign colors — but we have solved all the equations correctly.

Let me know if you'd like help interpreting a specific color key!