Solving Absolute Value Equations - Kuta Software - Free Printable

Absolutely! Let’s solve each absolute value equation step by step. The key idea is:

> For any expression $ |A| = B $, where $ B \geq 0 $, the solutions are:
> - $ A = B $
> - $ A = -B $

If $ B < 0 $, there is no solution, since absolute value is always non-negative.

---

1) $ |3x| = 9 $

Split into two cases:

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

✔ Solution: $ x = 3 $ or $ x = -3 $

---

2) $ |-3r| = 9 $

Note: $ |-3r| = |3r| $, so same as problem 1.

- $ -3r = 9 $ → $ r = -3 $
- $ -3r = -9 $ → $ r = 3 $

✔ Solution: $ r = 3 $ or $ r = -3 $

---

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

Split:

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

✔ Solution: $ b = 5 $ or $ b = -5 $

---

4) $ |-6m| = 30 $

Same as $ |6m| = 30 $

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

✔ Solution: $ m = 5 $ or $ m = -5 $

---

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

Split:

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

✔ Solution: $ n = 6 $ or $ n = -6 $

---

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

This is $ |5x - 4| = 16 $

Split:

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

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

---

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

This is $ |-(2r + 1)| = |2r + 1| = 11 $

So:

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

✔ Solution: $ r = 5 $ or $ r = -6 $

---

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

Split:

- $ 1 - 5s = 29 $ → $ -5s = 28 $ → $ s = -\frac{28}{5} $
- $ 1 - 5s = -29 $ → $ -5s = -30 $ → $ s = 6 $

✔ Solution: $ s = 6 $ or $ s = -\frac{28}{5} $

---

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

This is $ | -2a + 6 | = 6 $

Split:

- $ -2a + 6 = 6 $ → $ -2a = 0 $ → $ a = 0 $
- $ -2a + 6 = -6 $ → $ -2a = -12 $ → $ a = 6 $

✔ Solution: $ a = 0 $ or $ a = 6 $

---

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

First, isolate the absolute value:

→ $ |v + 8| = 2 + 5 = 7 $

Now split:

- $ v + 8 = 7 $ → $ v = -1 $
- $ v + 8 = -7 $ → $ v = -15 $

✔ Solution: $ v = -1 $ or $ v = -15 $

---

## ✔ Final Answers:

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

Let me know if you’d like to see graphs or check your work!