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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.

---

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.

Learn how to calculate the length of segment GE in a geometric problem involving points H, G, F, and E. Detailed step-by-step solution provided with algebraic expressions for segment lengths and final calculation. Suitable for advanced high school students and geometry enthusiasts.

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