Coordinates

The first step is to consider some triangles.

Once we find x, we can see that there are two congruent triangles with hypotenuse length $10\sqrt{3}$.

One of the legs of this triangle will be y1 - y2, which is 14.

Using the pythagorean theorem, we can find the distance between the two x coordinates. 

$\sqrt{(10\sqrt{3})^2 - 14^2}$

This simplifies to:

$\sqrt{104}$

This simplifies to:

$2\sqrt{26}$

This means the x coordinate is $2\sqrt{26}$ units away from the x coordinate of (4,7), and then we have these 2 possible values of x:

$4 + 2\sqrt{26}$ and $4 - 2\sqrt{26}$

Since the question is looking for the sum of the possible values of x, then we just add the two possible values together:

$4 + 2\sqrt{26} + 4 - 2\sqrt{26}$

This simplifies to 8.

Therefore, the sum of all possible values of x is 8.