The problem in the image asks to find the length of segment $GE$ given the points $H$, $G$, $F$, and $E$ on a line segment with the following information:

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

Since $G$ and $F$ are points between $H$ and $E$, the length $HE$ can be expressed as the sum of the lengths of the individual segments:

$HE = HG + GF + FE$

Substituting the given expressions:

$13x - 993 = (3x - 129) + (3x - 185) + 4$

Now, let's solve for $x$.

Step 1: Simplify the equation

$13x - 993 = 3x - 129 + 3x - 185 + 4$

Combine like terms on the right-hand side:

$13x - 993 = 6x - 310$

Step 2: Isolate $x$

Subtract $6x$ from both sides:

$7x - 993 = -310$

Add 993 to both sides:

$7x = 683$

Finally, divide by 7:

$x = 97.57 \quad (\text{approx.})$

Step 3: Calculate $GE$

To find $GE$, we subtract $HG$ from $HE$:

$GE = HE - HG = (13x - 993) - (3x - 129)$

Substituting $x = 97.57$:

$GE = (13(97.57) - 993) - (3(97.57) - 129)$

Calculate the values:

$HE = 13(97.57) - 993 = 1268.41 - 993 = 275.41$

$HG = 3(97.57) - 129 = 292.71 - 129 = 163.71$

So:

$GE = 275.41 - 163.71 = 111.7$

Therefore, the length of $GE$ is approximately $111.7$ units.

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

1. How would the solution change if the segments were in a different order?
2. Can you verify the length of segment $GF$ with the given $x$ value?
3. What is the significance of verifying the individual segment lengths before final calculation?
4. How can you cross-check the solution using another method?
5. How does rounding affect the final answer in this context?

Tip: Always double-check your arithmetic calculations, especially when dealing with decimals and large equations.