Given the information:

• $FG = 5x + 1$
• $EG = 5x + 7$
• $EF = 2x - 2$

Assuming that $E$, $F$, and $G$ are collinear points and $F$ is between $E$ and $G$, the relationship between these segments can be expressed as:

$EG = EF + FG$

Substituting the given expressions:

$5x + 7 = (2x - 2) + (5x + 1)$

Simplify the equation:

$5x + 7 = 7x - 1$

Now, solve for $x$:

$5x - 7x = -1 - 7$

$-2x = -8$

$x = 4$

Now that we have the value of $x$, substitute it back into the expression for $EF$:

$EF = 2x - 2 = 2(4) - 2 = 8 - 2 = 6$

Thus, the numerical length of $\overline{EF}$ is 6.

Would you like further explanation on any of these steps?

Here are some related questions:

1. How can you verify that $E$, $F$, and $G$ are collinear?
2. What happens to the equation setup if the points are not collinear?
3. How would this method change if $F$ was not between $E$ and $G$?
4. What are the geometric interpretations of different segment lengths?
5. How does the choice of which segment to calculate affect the solution process?

Tip: Always ensure the relationships between segments are clearly understood before solving; whether they add up or subtract depends on the point's arrangement.