Solving Absolute Value Equations - COLORING ACTIVITY - Amped Up ... - Free Printable

We are given 12 absolute value equations to solve. Let’s go through each one step-by-step, isolating the absolute value expression and then solving for the variable.

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1.
Equation:
$$
\left| \frac{n}{10} \right| + 1 = 2
$$

Step 1: Subtract 1 from both sides:
$$
\left| \frac{n}{10} \right| = 1
$$

Step 2: Use definition of absolute value:
$$
\frac{n}{10} = 1 \quad \text{or} \quad \frac{n}{10} = -1
$$

Step 3: Multiply both sides by 10:
$$
n = 10 \quad \text{or} \quad n = -10
$$

✔ Answer: $ n = \pm 10 $

---

2.
Equation:
$$
-7|-5x| = 0
$$

Step 1: Divide both sides by -7:
$$
|-5x| = 0
$$

Step 2: Absolute value equals zero only if the inside is zero:
$$
-5x = 0 \Rightarrow x = 0
$$

✔ Answer: $ x = 0 $

---

3.
Equation:
$$
|x - 1| - 1 = 0
$$

Step 1: Add 1 to both sides:
$$
|x - 1| = 1
$$

Step 2:
$$
x - 1 = 1 \quad \text{or} \quad x - 1 = -1
$$

Step 3: Solve:
$$
x = 2 \quad \text{or} \quad x = 0
$$

✔ Answer: $ x = 0, 2 $

---

4.
Equation:
$$
2|-6n| - 8 = 100
$$

Step 1: Add 8 to both sides:
$$
2|-6n| = 108
$$

Step 2: Divide by 2:
$$
|-6n| = 54
$$

Step 3:
$$
-6n = 54 \quad \text{or} \quad -6n = -54
$$

Step 4: Solve:
$$
n = -9 \quad \text{or} \quad n = 9
$$

✔ Answer: $ n = \pm 9 $

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5.
Equation:
$$
5 + 9|b - 2| = 95
$$

Step 1: Subtract 5:
$$
9|b - 2| = 90
$$

Step 2: Divide by 9:
$$
|b - 2| = 10
$$

Step 3:
$$
b - 2 = 10 \quad \text{or} \quad b - 2 = -10
$$

Step 4: Solve:
$$
b = 12 \quad \text{or} \quad b = -8
$$

✔ Answer: $ b = -8, 12 $

---

6.
Equation:
$$
-3|m - 1| - 6 = -12
$$

Step 1: Add 6 to both sides:
$$
-3|m - 1| = -6
$$

Step 2: Divide by -3:
$$
|m - 1| = 2
$$

Step 3:
$$
m - 1 = 2 \quad \text{or} \quad m - 1 = -2
$$

Step 4: Solve:
$$
m = 3 \quad \text{or} \quad m = -1
$$

✔ Answer: $ m = -1, 3 $

---

7.
Equation:
$$
3 + |v + 3| = -2
$$

Step 1: Subtract 3:
$$
|v + 3| = -5
$$

BUT — absolute value cannot be negative. So this equation has no solution.

✔ Answer: No solution

---

8.
Equation:
$$
|5x + 5| - 3 = 22
$$

Step 1: Add 3:
$$
|5x + 5| = 25
$$

Step 2:
$$
5x + 5 = 25 \quad \text{or} \quad 5x + 5 = -25
$$

Step 3: Solve each:

- First: $ 5x = 20 \Rightarrow x = 4 $
- Second: $ 5x = -30 \Rightarrow x = -6 $

✔ Answer: $ x = -6, 4 $

---

9.
Equation:
$$
-6|b - 2| = -42
$$

Step 1: Divide both sides by -6:
$$
|b - 2| = 7
$$

Step 2:
$$
b - 2 = 7 \quad \text{or} \quad b - 2 = -7
$$

Step 3: Solve:
$$
b = 9 \quad \text{or} \quad b = -5
$$

✔ Answer: $ b = -5, 9 $

---

10.
Equation:
$$
5|r - 4| - 10 = 10
$$

Step 1: Add 10:
$$
5|r - 4| = 20
$$

Step 2: Divide by 5:
$$
|r - 4| = 4
$$

Step 3:
$$
r - 4 = 4 \quad \text{or} \quad r - 4 = -4
$$

Step 4: Solve:
$$
r = 8 \quad \text{or} \quad r = 0
$$

✔ Answer: $ r = 0, 8 $

---

11.
Equation:
$$
7 + |-4m - 8| = 19
$$

Step 1: Subtract 7:
$$
|-4m - 8| = 12
$$

Note: $ |-4m - 8| = |-(4m + 8)| = |4m + 8| $, so we can write:

$$
|4m + 8| = 12
$$

Step 2:
$$
4m + 8 = 12 \quad \text{or} \quad 4m + 8 = -12
$$

Step 3: Solve:

- First: $ 4m = 4 \Rightarrow m = 1 $
- Second: $ 4m = -20 \Rightarrow m = -5 $

✔ Answer: $ m = -5, 1 $

---

12.
Equation:
$$
6 - |2m + 3| = -5
$$

Step 1: Subtract 6:
$$
-|2m + 3| = -11
$$

Step 2: Multiply both sides by -1:
$$
|2m + 3| = 11
$$

Step 3:
$$
2m + 3 = 11 \quad \text{or} \quad 2m + 3 = -11
$$

Step 4: Solve:

- First: $ 2m = 8 \Rightarrow m = 4 $
- Second: $ 2m = -14 \Rightarrow m = -7 $

✔ Answer: $ m = -7, 4 $

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## ✔ Final Answers Summary:

1. $ n = \pm 10 $
2. $ x = 0 $
3. $ x = 0, 2 $
4. $ n = \pm 9 $
5. $ b = -8, 12 $
6. $ m = -1, 3 $
7. No solution
8. $ x = -6, 4 $
9. $ b = -5, 9 $
10. $ r = 0, 8 $
11. $ m = -5, 1 $
12. $ m = -7, 4 $

Let me know if you’d like a graph or explanation for any specific problem!