2010 AMC 10B Problem 13

We first take care of the outer absolute value. We have either $$x = 2x - |60 - 2x|$$ or $$x = -2x + |60 - 2x|.$$ These simplify to $$x = |60 - 2x|$$ and $$3x = |60 - 2x|.$$ We have $2$ cases for each equation. For the first one, we have $$x = 60 - 2x$$ and $$x = 2x - 60.$$ Solving both gives us $$3x = 60$$$$x = 20$$ and $$-x = -60$$$$x = 60.$$ For the other equation, we have $$3x = 60 - 2x$$ 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.