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, which will be used to match a color key (not shown here). I'll solve each problem and provide the answer(s) so you can use them with the color key.

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1. $ |5x| = 15 $

Break 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 this becomes $ |2x| = 8 $

- $ 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 $
- $ x = -96 $

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

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4. $ |x - 7| = 10 $

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 $
- $ x = -8 $

✔ Solutions: $ x = 8 $ or $ x = -8 $

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

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

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

✔ Solutions: $ x = 5 $ or $ x = -5 $

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

This is equivalent to $ | -2x + 6 | = 6 $

Set up two cases:
- $ -2x + 6 = 6 $ → $ -2x = 0 $ → $ x = 0 $
- $ -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:
- $ 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 $

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

Absolute value is zero only when the inside is zero:
- $ 3x - 9 = 0 $ → $ 3x = 9 $ → $ x = 3 $

✔ Solution: $ x = 3 $

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✔ Final Answers Summary:

| Problem | Answer(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 Worksheet

Once you have these answers, go to the coloring page (not shown here), and use the color key that matches answers to colors. For example:

- If the key says "3" → red, then color problem #1 in red.
- If it says "-3" → blue, then you might need to check if the key uses individual values or sets.

Since multiple problems may have the same solution (like ±3, ±4), look carefully at how the key is structured.

If you share the color key, I can help you assign the correct colors!

Let me know if you want this formatted as a printable answer sheet or need help interpreting the color key!