https://file.aiquickdraw.com/mathbot/file/1725476549962by5d5xl5.jpg

To solve the problem depicted in the image, we need to identify and use the relevant mathematical relationships between the given segments.

### Given:
- Segment \( \overline{FP} \) is 8 units long.
- Segment \( \overline{PD} \) is \( x - 9 \) units long.
- The entire segment \( \overline{FD} \) is \( x + 8 \) units long.

### Objective:
To find the value of \( x \) using the equation derived from the sum of the segments.

### Approach:
Since \( \overline{FD} \) is the sum of segments \( \overline{FP} \), \( \overline{PB} \), and \( \overline{BD} \):

\[
\overline{FP} + \overline{PB} + \overline{BD} = \overline{FD}
\]

Given that \( \overline{PB} \) is equal to the unknown segment we need to find, assume it's the segment we need to solve for.

We have:
\[
8 + \overline{PB} + (x - 9) = x + 8
\]

Now, solve for \( \overline{PB} \):

1. Combine like terms:
\[
8 + \overline{PB} + x - 9 = x + 8
\]

2. Simplify the equation:
\[
\overline{PB} - 1 = 8
\]

3. Solve for \( \overline{PB} \):
\[
\overline{PB} = 9
\]

Since \( \overline{PB} \) corresponds to the unknown segment, substitute into the equation:

\[
x = 17
\]

Thus, \( x = 17 \).

Would you like more details or further clarification? 

### Related Questions:
1. What if the length of \( \overline{PD} \) was instead \( x + 9 \)? How would that change the result?
2. Can you set up and solve a similar problem where \( \overline{FP} \) is \( x + 3 \) and \( \overline{BD} \) is \( x - 2 \)?
3. How would the solution change if the entire segment \( \overline{FD} \) was \( 2x + 7 \)?
4. What geometric principles are involved when dealing with line segments and algebraic expressions?
5. How can the problem be generalized for any given lengths?

### Tip:
Always double-check the setup of your equations to ensure all segments are accounted for properly in geometric problems involving algebra.

Using the answer from the previous box in the blank box, find the value of x. Show or explain your work and provide the answer in the space below.

This problem involves finding the value of x using the Segment Addition Postulate. The problem is suitable for students in Grades 6-8, covering key algebraic concepts and basic linear equations.

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