Kevin Wilda - Free Printable

Let's solve each absolute value equation step by step. The general rule for solving absolute value equations is:

> If $ |A| = B $, then $ A = B $ or $ A = -B $, provided $ B \geq 0 $.
> If $ B < 0 $, there is no solution.

We'll go through each equation one by one.

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Row 1

1. $ |x - 2| = 6 $

$$
x - 2 = 6 \quad \text{or} \quad x - 2 = -6 \\
x = 8 \quad \text{or} \quad x = -4
$$

✔ Solutions: $ x = 8, -4 $

2. $ |x + 4| = 9 $

$$
x + 4 = 9 \quad \text{or} \quad x + 4 = -9 \\
x = 5 \quad \text{or} \quad x = -13
$$

✔ Solutions: $ x = 5, -13 $

3. $ |x - 6| = 14 $

$$
x - 6 = 14 \quad \text{or} \quad x - 6 = -14 \\
x = 20 \quad \text{or} \quad x = -8
$$

✔ Solutions: $ x = 20, -8 $

4. $ |x + 3| = 20 $

$$
x + 3 = 20 \quad \text{or} \quad x + 3 = -20 \\
x = 17 \quad \text{or} \quad x = -23
$$

✔ Solutions: $ x = 17, -23 $

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Row 2

5. $ |2x| = 14 $

$$
2x = 14 \quad \text{or} \quad 2x = -14 \\
x = 7 \quad \text{or} \quad x = -7
$$

✔ Solutions: $ x = 7, -7 $

6. $ \left| \frac{x}{3} \right| = 12 $

$$
\frac{x}{3} = 12 \quad \text{or} \quad \frac{x}{3} = -12 \\
x = 36 \quad \text{or} \quad x = -36
$$

✔ Solutions: $ x = 36, -36 $

7. $ |-4x| = 20 $

Note: $ |-4x| = |4x| $, so this becomes:

$$
|4x| = 20 \Rightarrow 4x = 20 \quad \text{or} \quad 4x = -20 \\
x = 5 \quad \text{or} \quad x = -5
$$

✔ Solutions: $ x = 5, -5 $

8. $ \left| \frac{x}{7} \right| = 14 $

$$
\frac{x}{7} = 14 \quad \text{or} \quad \frac{x}{7} = -14 \\
x = 98 \quad \text{or} \quad x = -98
$$

✔ Solutions: $ x = 98, -98 $

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Row 3

9. $ |x| + 2 = 6 $

Subtract 2 from both sides:

$$
|x| = 4 \Rightarrow x = 4 \quad \text{or} \quad x = -4
$$

✔ Solutions: $ x = 4, -4 $

10. $ 4|x| = 24 $

Divide both sides by 4:

$$
|x| = 6 \Rightarrow x = 6 \quad \text{or} \quad x = -6
$$

✔ Solutions: $ x = 6, -6 $

11. $ \frac{|x|}{4} = 12 $

Multiply both sides by 4:

$$
|x| = 48 \Rightarrow x = 48 \quad \text{or} \quad x = -48
$$

✔ Solutions: $ x = 48, -48 $

12. $ 9 + |x| = 27 $

Subtract 9:

$$
|x| = 18 \Rightarrow x = 18 \quad \text{or} \quad x = -18
$$

✔ Solutions: $ x = 18, -18 $

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Row 4

13. $ 5 + |x - 3| = 19 $

Subtract 5:

$$
|x - 3| = 14 \Rightarrow x - 3 = 14 \quad \text{or} \quad x - 3 = -14 \\
x = 17 \quad \text{or} \quad x = -11
$$

✔ Solutions: $ x = 17, -11 $

14. $ |7 - 2x| + 7 = 34 $

Subtract 7:

$$
|7 - 2x| = 27 \Rightarrow 7 - 2x = 27 \quad \text{or} \quad 7 - 2x = -27 \\
-2x = 20 \Rightarrow x = -10 \\
-2x = -34 \Rightarrow x = 17
$$

✔ Solutions: $ x = -10, 17 $

15. $ 3|6 - 4x| = 54 $

Divide by 3:

$$
|6 - 4x| = 18 \Rightarrow 6 - 4x = 18 \quad \text{or} \quad 6 - 4x = -18 \\
-4x = 12 \Rightarrow x = -3 \\
-4x = -24 \Rightarrow x = 6
$$

✔ Solutions: $ x = -3, 6 $

16. $ \frac{|6 - 3x|}{5} = 3 $

Multiply both sides by 5:

$$
|6 - 3x| = 15 \Rightarrow 6 - 3x = 15 \quad \text{or} \quad 6 - 3x = -15 \\
-3x = 9 \Rightarrow x = -3 \\
-3x = -21 \Rightarrow x = 7
$$

✔ Solutions: $ x = -3, 7 $

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Row 5

17. $ |x + 5| = 29 $

$$
x + 5 = 29 \quad \text{or} \quad x + 5 = -29 \\
x = 24 \quad \text{or} \quad x = -34
$$

✔ Solutions: $ x = 24, -34 $

18. $ 16 + |x| = 44 $

Subtract 16:

$$
|x| = 28 \Rightarrow x = 28 \quad \text{or} \quad x = -28
$$

✔ Solutions: $ x = 28, -28 $

19. $ |7x| = 35 $

$$
7x = 35 \quad \text{or} \quad 7x = -35 \\
x = 5 \quad \text{or} \quad x = -5
$$

✔ Solutions: $ x = 5, -5 $

20. $ 9 + |x - 4| = 57 $

Subtract 9:

$$
|x - 4| = 48 \Rightarrow x - 4 = 48 \quad \text{or} \quad x - 4 = -48 \\
x = 52 \quad \text{or} \quad x = -44
$$

✔ Solutions: $ x = 52, -44 $

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✔ Final Answers (Summary):

| Equation | Solution(s) |
|--------|-------------|
| 1. $ |x - 2| = 6 $ | $ x = 8, -4 $ |
| 2. $ |x + 4| = 9 $ | $ x = 5, -13 $ |
| 3. $ |x - 6| = 14 $ | $ x = 20, -8 $ |
| 4. $ |x + 3| = 20 $ | $ x = 17, -23 $ |
| 5. $ |2x| = 14 $ | $ x = 7, -7 $ |
| 6. $ \left| \frac{x}{3} \right| = 12 $ | $ x = 36, -36 $ |
| 7. $ |-4x| = 20 $ | $ x = 5, -5 $ |
| 8. $ \left| \frac{x}{7} \right| = 14 $ | $ x = 98, -98 $ |
| 9. $ |x| + 2 = 6 $ | $ x = 4, -4 $ |
| 10. $ 4|x| = 24 $ | $ x = 6, -6 $ |
| 11. $ \frac{|x|}{4} = 12 $ | $ x = 48, -48 $ |
| 12. $ 9 + |x| = 27 $ | $ x = 18, -18 $ |
| 13. $ 5 + |x - 3| = 19 $ | $ x = 17, -11 $ |
| 14. $ |7 - 2x| + 7 = 34 $ | $ x = -10, 17 $ |
| 15. $ 3|6 - 4x| = 54 $ | $ x = -3, 6 $ |
| 16. $ \frac{|6 - 3x|}{5} = 3 $ | $ x = -3, 7 $ |
| 17. $ |x + 5| = 29 $ | $ x = 24, -34 $ |
| 18. $ 16 + |x| = 44 $ | $ x = 28, -28 $ |
| 19. $ |7x| = 35 $ | $ x = 5, -5 $ |
| 20. $ 9 + |x - 4| = 57 $ | $ x = 52, -44 $ |

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Let me know if you'd like these written in a formatted table or want explanations for any specific ones!