2010 AMC 10B Problem 13

Question

What is the sum of all the solutions of the equation below?

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

32

60

92

120

124

Split cases based on the sign of

60-2x

After resolving the inner absolute value, solve the resulting absolute-value equations

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.

Problem 13 in other years