To determine the length of line segment \( GH \), we have the following lengths based on the problem statement:
- \( GI = 3x \)
- \( HI = 2 \)
- \( GH = 5x - 6 \)
Given that point \( H \) lies on line segment \( GI \), we can set up an equation based on the relationship between these lengths. Since \( G \), \( H \), and \( I \) are collinear, we have:
\[ GH + HI = GI \]
Substituting the known values into the equation gives us:
\[ (5x - 6) + 2 = 3x \]
Now, we can simplify and solve for \( x \):
\[ 5x - 6 + 2 = 3x \]
This simplifies to:
\[ 5x - 4 = 3x \]
Next, we isolate \( x \) by subtracting \( 3x \) from both sides:
\[ 5x - 3x - 4 = 0 \] \[ 2x - 4 = 0 \]
Adding \( 4 \) to both sides gives:
\[ 2x = 4 \]
Dividing both sides by \( 2 \) results in:
\[ x = 2 \]
Now that we have \( x \), we can calculate the length of \( GH \):
\[ GH = 5x - 6 \]
Substituting \( x = 2 \):
\[ GH = 5(2) - 6 = 10 - 6 = 4 \]
Thus, the numerical length of \( GH \) is:
\[ \boxed{4} \]