Solution:

We first take care of the outer absolute value. We have either $$x = 2x - |60 - 2x|$$ or $$x = -2x + |60 - 2x|.$$

These simplify to $$x = |60 - 2x| \text{ and } 3x = |60 - 2x|.$$

We have $2$ cases for each equation. For the first one, we have $$x = 60 - 2x \text{ and } x = 2x - 60.$$

Solving both gives us $$3x = 60$$$$x = 20$$ and $$-x = -60$$$$x = 60.$$

For the other equation, we have $$3x = 60 - 2x \text{ and } 3x = 2x - 60.$$

Again solving both, we have $$5x = 60$$$$x = 12$$ and $x = -60.$ Note that $x$ cannot be negative since the right side of the original equation is nonnegative.

Adding up all the solutions gives us $$20 + 60 + 12 = 92.$$

Thus, C is the correct answer.