2010 AMC 10B Problems/Problem 13

Problem

What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$ ?

$\mathrm{(A)}\ 32 \qquad \mathrm{(B)}\ 60 \qquad \mathrm{(C)}\ 92 \qquad \mathrm{(D)}\ 120 \qquad \mathrm{(E)}\ 124$

Case 1:

$2x-|60-2x|=x, x=|60-2x|$

Case 1a:

$x=60-2x, 3x=60, x=20$

Case 1b:

$-x=60-2x, x=60$

Case 2:

$2x-|60-2x|=-x, 3x=|60-2x|$

Case 2a:

$3x=60-2x, 5x=60, x=12$

Case 2b:

$-3x=60-2x, -x=60, x=-60$

Since an absolute value cannot be negative, we exclude $x=-60$ . The answer is $20+60+12= \boxed{\mathrm {(C)}\ 92}$