The problem shows a segment $HE$, with points $G$ and $F$ between $H$ and $E$. The lengths of different segments along this line are represented by the following algebraic expressions:

• $HG = 13x - 993$
• $GF = 3x - 129$
• $FE = 3x - 185$
• $EF = 4$

To find $GE$, we can express $GE$ as the sum of the segments $GF$ and $FE$.

So, we have: $GE = GF + FE$ Substitute the expressions for $GF$ and $FE$: $GE = (3x - 129) + (3x - 185)$ Simplifying: $GE = 3x - 129 + 3x - 185 = 6x - 314$

Thus, the length of $GE$ is $6x - 314$.

Would you like to proceed by solving for $x$, or do you want further details on the steps? Here are some follow-up questions:

1. How do we solve for $x$ if we know the total length of $HE$?
2. How can we express the length of $HE$ in terms of $x$?
3. What additional steps are needed to solve for $GE$ once $x$ is found?
4. How does knowing the value of $FE$ help solve the problem?
5. How do you verify the accuracy of the solution?

Tip: When solving problems with segments, always add smaller segments to express the larger segment.