Solving Absolute Value Equations - Kuta Software - Free Printable

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

> If $ |A| = B $, then $ A = B $ or $ A = -B $, provided $ B \geq 0 $.
> (If $ B < 0 $, there is no solution because absolute values are always non-negative.)

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1) $ |3x| = 9 $

We split into two cases:
- $ 3x = 9 $ → $ x = 3 $
- $ 3x = -9 $ → $ x = -3 $

✔ Solutions: $ x = 3 $ or $ x = -3 $

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2) $ |-3r| = 9 $

Note: $ |-3r| = |3r| $, so this is equivalent to $ |3r| = 9 $

Split:
- $ 3r = 9 $ → $ r = 3 $
- $ 3r = -9 $ → $ r = -3 $

✔ Solutions: $ r = 3 $ or $ r = -3 $

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3) $ \left|\frac{b}{5}\right| = 1 $

Split:
- $ \frac{b}{5} = 1 $ → $ b = 5 $
- $ \frac{b}{5} = -1 $ → $ b = -5 $

✔ Solutions: $ b = 5 $ or $ b = -5 $

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4) $ |-6m| = 30 $

$ |-6m| = |6m| $, so $ |6m| = 30 $

Split:
- $ 6m = 30 $ → $ m = 5 $
- $ 6m = -30 $ → $ m = -5 $

✔ Solutions: $ m = 5 $ or $ m = -5 $

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5) $ \left|\frac{n}{3}\right| = 2 $

Split:
- $ \frac{n}{3} = 2 $ → $ n = 6 $
- $ \frac{n}{3} = -2 $ → $ n = -6 $

✔ Solutions: $ n = 6 $ or $ n = -6 $

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6) $ |-4 + 5x| = 16 $

Let’s rewrite: $ |5x - 4| = 16 $

Split:
- $ 5x - 4 = 16 $ → $ 5x = 20 $ → $ x = 4 $
- $ 5x - 4 = -16 $ → $ 5x = -12 $ → $ x = -\frac{12}{5} $

✔ Solutions: $ x = 4 $ or $ x = -\frac{12}{5} $

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7) $ |-2r - 1| = 11 $

Split:
- $ -2r - 1 = 11 $ → $ -2r = 12 $ → $ r = -6 $
- $ -2r - 1 = -11 $ → $ -2r = -10 $ → $ r = 5 $

✔ Solutions: $ r = -6 $ or $ r = 5 $

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8) $ |1 - 5a| = 29 $

Split:
- $ 1 - 5a = 29 $ → $ -5a = 28 $ → $ a = -\frac{28}{5} $
- $ 1 - 5a = -29 $ → $ -5a = -30 $ → $ a = 6 $

✔ Solutions: $ a = -\frac{28}{5} $ or $ a = 6 $

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9) $ |-2n + 6| = 6 $

Split:
- $ -2n + 6 = 6 $ → $ -2n = 0 $ → $ n = 0 $
- $ -2n + 6 = -6 $ → $ -2n = -12 $ → $ n = 6 $

✔ Solutions: $ n = 0 $ or $ n = 6 $

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10) $ |v + 8| - 5 = 2 $

First, isolate the absolute value:
$$
|v + 8| = 2 + 5 = 7
$$

Now solve:
- $ v + 8 = 7 $ → $ v = -1 $
- $ v + 8 = -7 $ → $ v = -15 $

✔ Solutions: $ v = -1 $ or $ v = -15 $

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✔ Final Answers:

1) $ x = 3 $ or $ x = -3 $
2) $ r = 3 $ or $ r = -3 $
3) $ b = 5 $ or $ b = -5 $
4) $ m = 5 $ or $ m = -5 $
5) $ n = 6 $ or $ n = -6 $
6) $ x = 4 $ or $ x = -\frac{12}{5} $
7) $ r = -6 $ or $ r = 5 $
8) $ a = -\frac{28}{5} $ or $ a = 6 $
9) $ n = 0 $ or $ n = 6 $
10) $ v = -1 $ or $ v = -15 $

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