Absolute Value Worksheets - Free Printable

Let's solve each of these absolute value equations step by step. The general rule for solving an equation like $ |A| = B $ is:

> $ A = B $ or $ A = -B $

We'll apply this principle to each problem.

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1) $ |a - 4| = 8 $

$$
a - 4 = 8 \quad \text{or} \quad a - 4 = -8 \\
a = 12 \quad \text{or} \quad a = -4
$$

✔ Answer: $ a = 12 $ or $ -4 $

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2) $ |6 + b| = 10 $

$$
6 + b = 10 \quad \text{or} \quad 6 + b = -10 \\
b = 4 \quad \text{or} \quad b = -16
$$

✔ Answer: $ b = 4 $ or $ -16 $

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3) $ |9 - c| = 4 $

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

✔ Answer: $ c = 5 $ or $ 13 $

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4) $ |3 + d + 5| = 7 $

Simplify inside: $ |d + 8| = 7 $

$$
d + 8 = 7 \quad \text{or} \quad d + 8 = -7 \\
d = -1 \quad \text{or} \quad d = -15
$$

✔ Answer: $ d = -1 $ or $ -15 $

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5) $ |2e| = 24 $

$$
2e = 24 \quad \text{or} \quad 2e = -24 \\
e = 12 \quad \text{or} \quad e = -12
$$

✔ Answer: $ e = 12 $ or $ -12 $

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6) $ |2f + 1| = 13 $

$$
2f + 1 = 13 \quad \text{or} \quad 2f + 1 = -13 \\
2f = 12 \quad \text{or} \quad 2f = -14 \\
f = 6 \quad \text{or} \quad f = -7
$$

✔ Answer: $ f = 6 $ or $ -7 $

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7) $ 3|g - 4| = 12 $

Divide both sides by 3:

$$
|g - 4| = 4 \\
g - 4 = 4 \quad \text{or} \quad g - 4 = -4 \\
g = 8 \quad \text{or} \quad g = 0
$$

✔ Answer: $ g = 8 $ or $ 0 $

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8) $ \frac{1}{2}|h + 2| = 3 $

Multiply both sides by 2:

$$
|h + 2| = 6 \\
h + 2 = 6 \quad \text{or} \quad h + 2 = -6 \\
h = 4 \quad \text{or} \quad h = -8
$$

✔ Answer: $ h = 4 $ or $ -8 $

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9) $ |4i| - |-7| = 1 $

Note: $ |-7| = 7 $, so:

$$
|4i| - 7 = 1 \Rightarrow |4i| = 8 \\
4i = 8 \quad \text{or} \quad 4i = -8 \\
i = 2 \quad \text{or} \quad i = -2
$$

✔ Answer: $ i = 2 $ or $ -2 $

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10) $ |2(j - 3)| = 6 $

Simplify: $ |2j - 6| = 6 $

$$
2j - 6 = 6 \quad \text{or} \quad 2j - 6 = -6 \\
2j = 12 \quad \text{or} \quad 2j = 0 \\
j = 6 \quad \text{or} \quad j = 0
$$

✔ Answer: $ j = 6 $ or $ 0 $

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11) $ |4(k + 3)| = 20 $

Simplify: $ |4k + 12| = 20 $

$$
4k + 12 = 20 \quad \text{or} \quad 4k + 12 = -20 \\
4k = 8 \quad \text{or} \quad 4k = -32 \\
k = 2 \quad \text{or} \quad k = -8
$$

✔ Answer: $ k = 2 $ or $ -8 $

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12) $ |8 - 2l| = 6 $

$$
8 - 2l = 6 \quad \text{or} \quad 8 - 2l = -6 \\
-2l = -2 \quad \text{or} \quad -2l = -14 \\
l = 1 \quad \text{or} \quad l = 7
$$

✔ Answer: $ l = 1 $ or $ 7 $

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13) $ 3|m - 5| = 21 $

Divide by 3:

$$
|m - 5| = 7 \\
m - 5 = 7 \quad \text{or} \quad m - 5 = -7 \\
m = 12 \quad \text{or} \quad m = -2
$$

✔ Answer: $ m = 12 $ or $ -2 $

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14) $ |2n - 3| = 11 $

$$
2n - 3 = 11 \quad \text{or} \quad 2n - 3 = -11 \\
2n = 14 \quad \text{or} \quad 2n = -8 \\
n = 7 \quad \text{or} \quad n = -4
$$

✔ Answer: $ n = 7 $ or $ -4 $

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15) $ |10 - 2o| = 4 $

$$
10 - 2o = 4 \quad \text{or} \quad 10 - 2o = -4 \\
-2o = -6 \quad \text{or} \quad -2o = -14 \\
o = 3 \quad \text{or} \quad o = 7
$$

✔ Answer: $ o = 3 $ or $ 7 $

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16) $ |3(p - 2)| = 24 $

Simplify: $ |3p - 6| = 24 $

$$
3p - 6 = 24 \quad \text{or} \quad 3p - 6 = -24 \\
3p = 30 \quad \text{or} \quad 3p = -18 \\
p = 10 \quad \text{or} \quad p = -6
$$

✔ Answer: $ p = 10 $ or $ -6 $

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17) $ 5|4 - q| = 30 $

Divide by 5:

$$
|4 - q| = 6 \\
4 - q = 6 \quad \text{or} \quad 4 - q = -6 \\
-q = 2 \quad \text{or} \quad -q = -10 \\
q = -2 \quad \text{or} \quad q = 10
$$

✔ Answer: $ q = -2 $ or $ 10 $

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18) $ \frac{1}{2}|r + 6| = 12 $

Multiply both sides by 2:

$$
|r + 6| = 24 \\
r + 6 = 24 \quad \text{or} \quad r + 6 = -24 \\
r = 18 \quad \text{or} \quad r = -30
$$

✔ Answer: $ r = 18 $ or $ -30 $

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19) $ |\frac{1}{2}(s - 3)| = 7 $

Multiply both sides by 2 (to eliminate fraction):

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

✔ Answer: $ s = 17 $ or $ -11 $

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20) $ |6 - \frac{1}{4}t| = 4 $

$$
6 - \frac{1}{4}t = 4 \quad \text{or} \quad 6 - \frac{1}{4}t = -4 \\
-\frac{1}{4}t = -2 \quad \text{or} \quad -\frac{1}{4}t = -10 \\
t = 8 \quad \text{or} \quad t = 40
$$

✔ Answer: $ t = 8 $ or $ 40 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1) | $ a = 12 $ or $ -4 $ |
| 2) | $ b = 4 $ or $ -16 $ |
| 3) | $ c = 5 $ or $ 13 $ |
| 4) | $ d = -1 $ or $ -15 $ |
| 5) | $ e = 12 $ or $ -12 $ |
| 6) | $ f = 6 $ or $ -7 $ |
| 7) | $ g = 8 $ or $ 0 $ |
| 8) | $ h = 4 $ or $ -8 $ |
| 9) | $ i = 2 $ or $ -2 $ |
| 10) | $ j = 6 $ or $ 0 $ |
| 11) | $ k = 2 $ or $ -8 $ |
| 12) | $ l = 1 $ or $ 7 $ |
| 13) | $ m = 12 $ or $ -2 $ |
| 14) | $ n = 7 $ or $ -4 $ |
| 15) | $ o = 3 $ or $ 7 $ |
| 16) | $ p = 10 $ or $ -6 $ |
| 17) | $ q = -2 $ or $ 10 $ |
| 18) | $ r = 18 $ or $ -30 $ |
| 19) | $ s = 17 $ or $ -11 $ |
| 20) | $ t = 8 $ or $ 40 $ |

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