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.