SOLUTION: What is the sum of all the solutions of x = |2x-|60-2x||?

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Question 1148536: What is the sum of all the solutions of x = |2x-|60-2x||?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Find equivalent expressions for the given function

abs%282x-abs%2860-2x%29%29

for different ranges of values for x.

Case 1: x>30

When x>30, 2x>60, and (60-2x) is negative; that means

abs%2860-2x%29+=+2x-60

and then

abs%282x-abs%2860-2x%29%29+=+abs%282x-%282x-60%29%29+=+abs%2860%29+=+60

So on the interval x>30, the only solution to the given equation is x=60.

Case 2: x<30

When x<30, 2x<60, and (60-2x) is positive; that means

abs%2860-2x%29+=+60-2x

and then

abs%282x-abs%2860-2x%29%29+=+abs%282x-%2860-2x%29%29+=+abs%2860-4x%29

abs%2860-4x%29 is positive for x<15 and negative for x>15, so we need to break case 2 into two sub-cases.

Case 2a: x<15

abs%282x-abs%2860-2x%29%29+=+abs%282x-%2860-2x%29%29+=+abs%2860-4x%29+=+60-4x

So on the interval x<15, the solution to the given equation will be when

x+=+60-4x
5x+=+60
x+=+12

Case 2b: x>15 and x<30

abs%282x-abs%2860-2x%29%29+=+abs%282x-%2860-2x%29%29+=+abs%2860-4x%29+=+4x-60

So on the interval (15,30), the solution to the given equation will be when

x+=+4x-60
60+=+3x
x+=+20

ANSWER: There are three solutions to the equation: x=12, x=20, and x=60.

A graph....

graph%28400%2C200%2C-10%2C90%2C-10%2C90%2Cabs%282x-abs%2860-2x%29%29%2Cx%29