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Let's analyze each of the given options to find the correct answer:

1. **Option 1:**  
   - Midpoint: 2  
   - Distance from midpoint: 8  
   - Equation: \(|x + 2| = 8\)  
   - Solving this equation gives \( x + 2 = 8 \) or \( x + 2 = -8 \).  
   - This simplifies to \( x = 6 \) or \( x = -10 \).  
   - However, the midpoint formula \( \frac{x_1 + x_2}{2} = 2 \) should hold for \( x_1 = 6 \) and \( x_2 = -10 \), which satisfies the midpoint. This is incorrect because it represents the distance improperly.

2. **Option 2:**  
   - Midpoint: -8  
   - Distance from midpoint: 2  
   - Equation: \(|x + 8| = 2\)  
   - Solving gives \( x + 8 = 2 \) or \( x + 8 = -2 \).  
   - This simplifies to \( x = -6 \) or \( x = -10 \).  
   - The midpoint formula for \( x_1 = -6 \) and \( x_2 = -10 \) gives the correct midpoint of -8. This is the correct distance and midpoint.

3. **Option 3:**  
   - Midpoint: 8  
   - Distance from midpoint: 8  
   - Equation: \(|x - 8| = 2\)  
   - Solving gives \( x - 8 = 2 \) or \( x - 8 = -2 \).  
   - This simplifies to \( x = 10 \) or \( x = 6 \).  
   - Midpoint calculation for \( x_1 = 10 \) and \( x_2 = 6 \) results in 8, but the distance is incorrect.

4. **Option 4:**  
   - Midpoint: 2  
   - Distance from midpoint: 8  
   - Equation: \(|x - 2| = 8\)  
   - Solving gives \( x - 2 = 8 \) or \( x - 2 = -8 \).  
   - This simplifies to \( x = 10 \) or \( x = -6 \).  
   - Midpoint calculation for \( x_1 = 10 \) and \( x_2 = -6 \) results in 2, and the distance is 8.  

**Conclusion:** The correct answer is **Option 4**, where the midpoint is 2, and the distance to either point is 8, satisfying the given equation \(|x - 2| = 8\).

Let me know if you'd like more details or have any further questions!

Here are 5 questions related to the problem above:
1. How do you derive the formula for finding the midpoint between two points?
2. What does the absolute value equation represent in terms of distance?
3. How can you verify the midpoint and distance for a given pair of points?
4. What are the properties of absolute value equations when used to represent distances?
5. Can the distance formula and midpoint formula be used for points in 3D space?

**Tip:** Always check your solution by verifying the midpoint and the absolute distance to ensure accuracy.

Find the correct midpoint and distance from the given options.

Learn how to solve problems involving midpoints and distances using absolute value equations. This detailed solution explores key mathematical concepts such as midpoint calculation and distance determination, suitable for students in Grades 7-10.

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