We are given that the line segment is made up of three points, F, G, and H. The segments between these points are labeled as:

• $FG = 6x + 5$
• $GH = 7x$
• $FH = 18$

Since $F$, $G$, and $H$ are collinear, the total distance $FH$ must equal the sum of the distances $FG + GH$:

$FH = FG + GH$

Substituting the given expressions:

$18 = (6x + 5) + 7x$

Now, let's solve for $x$:

$18 = 6x + 5 + 7x$ $18 = 13x + 5$ $18 - 5 = 13x$ $13 = 13x$ $x = 1$

Now that we know $x = 1$, we can substitute it into the expression for $GH$:

$GH = 7x = 7(1) = 7$

Thus, $GH = 7$.

Would you like more details or have any further questions?

Related questions:

1. What is the length of $FG$ when $x = 1$?
2. How do we verify the total length $FH$ after finding $FG$ and $GH$?
3. What method can be used to solve for $x$ in similar problems?
4. What happens if $x$ results in a negative distance? Is that physically meaningful?
5. Can this process be extended to cases where there are more segments along a line?

Tip: When solving linear equations involving segments, always check that the total length agrees with the sum of individual parts!