SOLUTION: What is the number of distinct solutions of the equation |x-|2x+1||=3?

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Question 1148537: What is the number of distinct solutions of the equation |x-|2x+1||=3?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


2x%2B1=0 when x = -1/2. So consider two cases, one with x < -1/2 and the other with x > -1/2.

Case 1: x < -1/2; 2x+1 < 0

abs%282x%2B1%29+=+-2x-1
abs%28x-abs%282x%2B1%29%29+=+abs%28x-%28-2x-1%29%29+=+abs%283x%2B1%29+=+-3x-1

The solution on the interval x < -1/2 is then

-3x-1+=+3
-3x+=+4
x+=+-4%2F3

Case 2: x > -1/2; 2x+1 > 0

abs%282x%2B1%29+=+2x%2B1
abs%28x-abs%282x%2B1%29%29+=+abs%28x-%282x%2B1%29%29+=+abs%28-x-1%29+=+x%2B1

The solution on the interval x > -1/2 is then

x%2B1+=+3
x+=+2

ANSWER: There are two solutions to the equation: -4/3 and 2.

A graph, showing the intersection of the two graphs at x=-4/3 and x=2....

graph%28400%2C200%2C-3%2C3%2C-1%2C5%2Cabs%28x-abs%282x%2B1%29%29%2C3%29