https://file.aiquickdraw.com/mathbot/file/1725505258604mfc0osz8.jpg

The image contains multiple absolute value equations to solve and then graph their solution sets. I'll help solve these step by step and explain how to interpret and graph the solutions.

### 1. |y| = 3
- **Solution**: The absolute value of \( y \) equals 3, so \( y = 3 \) or \( y = -3 \).
- **Graph**: Place points at \( y = 3 \) and \( y = -3 \) on the number line.

### 2. |x - 4| = 4
- **Solution**: Break this down into two cases:
  - \( x - 4 = 4 \) gives \( x = 8 \).
  - \( x - 4 = -4 \) gives \( x = 0 \).
- **Graph**: Place points at \( x = 8 \) and \( x = 0 \) on the number line.

### 3. |y + 3| = 2
- **Solution**: Two cases:
  - \( y + 3 = 2 \) gives \( y = -1 \).
  - \( y + 3 = -2 \) gives \( y = -5 \).
- **Graph**: Place points at \( y = -1 \) and \( y = -5 \) on the number line.

### 4. |2x + 1| = 5
- **Solution**:
  - \( 2x + 1 = 5 \) gives \( 2x = 4 \), so \( x = 2 \).
  - \( 2x + 1 = -5 \) gives \( 2x = -6 \), so \( x = -3 \).
- **Graph**: Place points at \( x = 2 \) and \( x = -3 \) on the number line.

### 5. |3y - 2| = 1
- **Solution**:
  - \( 3y - 2 = 1 \) gives \( 3y = 3 \), so \( y = 1 \).
  - \( 3y - 2 = -1 \) gives \( 3y = 1 \), so \( y = \frac{1}{3} \).
- **Graph**: Place points at \( y = 1 \) and \( y = \frac{1}{3} \) on the number line.

### 6. |x + 2| = 4
- **Solution**:
  - \( x + 2 = 4 \) gives \( x = 2 \).
  - \( x + 2 = -4 \) gives \( x = -6 \).
- **Graph**: Place points at \( x = 2 \) and \( x = -6 \) on the number line.

### 7. |x| = 0
- **Solution**: \( x = 0 \).
- **Graph**: Place a point at \( x = 0 \) on the number line.

### 8. |y + 0.2| = 0.8
- **Solution**:
  - \( y + 0.2 = 0.8 \) gives \( y = 0.6 \).
  - \( y + 0.2 = -0.8 \) gives \( y = -1 \).
- **Graph**: Place points at \( y = 0.6 \) and \( y = -1 \) on the number line.

### 9. |d - 100| = 60
- **Solution**:
  - \( d - 100 = 60 \) gives \( d = 160 \).
  - \( d - 100 = -60 \) gives \( d = 40 \).
- **Graph**: Place points at \( d = 160 \) and \( d = 40 \) on the number line.

### 10. |11 - y| = 11
- **Solution**:
  - \( 11 - y = 11 \) gives \( y = 0 \).
  - \( 11 - y = -11 \) gives \( y = 22 \).
- **Graph**: Place points at \( y = 0 \) and \( y = 22 \) on the number line.

### 11. |y| = 11
- **Solution**: \( y = 11 \) or \( y = -11 \).
- **Graph**: Place points at \( y = 11 \) and \( y = -11 \) on the number line.

### 12. |2x + 4| = 6
- **Solution**:
  - \( 2x + 4 = 6 \) gives \( 2x = 2 \), so \( x = 1 \).
  - \( 2x + 4 = -6 \) gives \( 2x = -10 \), so \( x = -5 \).
- **Graph**: Place points at \( x = 1 \) and \( x = -5 \) on the number line.

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Would you like a more detailed explanation for any of these solutions?

Here are five related questions for further exploration:
1. What happens when the absolute value is equal to zero?
2. How do you graph inequalities involving absolute values?
3. Can you solve an absolute value equation with a variable on both sides?
4. How does the solution change when you have nested absolute values?
5. What is the geometric interpretation of absolute value on the number line?

**Tip**: When solving absolute value equations, always break the problem into two cases—positive and negative—and solve both separately!

Solve each absolute value equation and graph the solution set.

Learn how to solve and graph absolute value equations step by step. This lesson focuses on key algebraic concepts involving absolute value equations, including solutions represented on the number line. Suitable for students in Grades 6-8.

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