We are given the following information:

• $GI = 5x$
• $HI = 3x - 8$
• $GH = 3x$

Since point $H$ is on the line segment $\overline{GI}$, the total length of $GI$ is the sum of the lengths of $GH$ and $HI$, i.e.,

$GH + HI = GI$

Substitute the values for $GH$, $HI$, and $GI$ into this equation:

$3x + (3x - 8) = 5x$

Now, simplify the left-hand side:

$3x + 3x - 8 = 5x$ $6x - 8 = 5x$

Subtract $5x$ from both sides:

$6x - 5x = 8$ $x = 8$

Now that we know $x = 8$, we can find the length of $GH$:

$GH = 3x = 3(8) = 24$

Thus, the numerical length of $\overline{GH}$ is 24 units.

Would you like further clarification or details?

Here are 5 related questions to explore:

1. What is the length of $HI$ when $x = 8$?
2. How do we verify that point $H$ lies on $\overline{GI}$?
3. If $GH = 24$, what is the total length of $GI$?
4. How does the relationship between segment lengths change if $H$ is the midpoint of $\overline{GI}$?
5. Can the equation setup change if $H$ were outside $\overline{GI}$?

Tip: When dealing with line segments, always check if the sum of the smaller segments equals the total length.