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|$ (Error compiling LaTeX. Unknown error_msg)

Case 1a:

$x=60-2x

3x=60

x=20$ (Error compiling LaTeX. Unknown error_msg)

Case 1b:

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

Case 2:

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

3x=|60-2x|$ (Error compiling LaTeX. Unknown error_msg)

Case 2a:

$3x=60-2x

5x=60

x=12$ (Error compiling LaTeX. Unknown error_msg)

Case 2b:

$-3x=60-2x

-x=60

x=-60$ (Error compiling LaTeX. Unknown error_msg)

Since an absolute value cannot be negative, we exclude $x=-60$ . The answer is $20+60+12=92$