2010 AMC 10B Problems/Problem 13

Problem

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

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

Solution

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=92$

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2010 AMC 10B Problems/Problem 13

Problem

Solution