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In this problem, we are given a line segment \( FH \), where the total length is \( FH = 11 \). The segment is divided into two parts: 

- \( FG = 7 \)
- \( GH = x + 1 \)

We need to solve for \( x \) and find the length of \( GH \).

### Step 1: Set up the equation
Since the total length of \( FH \) is 11, the sum of \( FG \) and \( GH \) should equal 11:
\[
FG + GH = FH
\]
Substitute the known values:
\[
7 + (x + 1) = 11
\]

### Step 2: Solve for \( x \)
Simplify the equation:
\[
7 + x + 1 = 11
\]
\[
x + 8 = 11
\]
Subtract 8 from both sides:
\[
x = 3
\]

### Step 3: Find \( GH \)
Now that we know \( x = 3 \), substitute it back into the expression for \( GH \):
\[
GH = x + 1 = 3 + 1 = 4
\]

### Final Answer:
- \( x = 3 \)
- \( GH = 4 \)

Would you like more details on any step? Here are some related questions:

1. How would the equation change if \( FH \) had a different length?
2. What is the general formula for finding \( GH \) if \( FG \) and \( FH \) were unknown?
3. How would this problem change if \( G \) were not exactly between \( F \) and \( H \)?
4. What would happen if the segments included variables for both \( FG \) and \( GH \)?
5. How can we verify the solution using different methods?

**Tip:** Always double-check your work by adding up the parts of the segment to ensure they match the total length!

FH = 11, FG = 7, GH = x + 1. Solve for x and GH.

Solve for x and the length of GH in the line segment FH = 11, where FG = 7 and GH = x + 1. This problem covers algebraic concepts, including setting up linear equations and solving for unknown variables. Suitable for students in Grades 6-8.

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