How do you solve |x + 3| = abs(x-2)|x+3|=|x−2|?

1 Answer
sente
Apr 28, 2016

x=-1/2x=−12

Explanation:

|x+3| = |x-2||x+3|=|x−2|

=> x+3 = |x-2|⇒x+3=|x−2|
or
-(x+3) = |x-2|−(x+3)=|x−2|

=> x+3 = x-2⇒x+3=x−2
or
x+3 = -(x-2)x+3=−(x−2)
or
-(x+3)=x-2−(x+3)=x−2
or
-(x+3)=-(x-2)−(x+3)=−(x−2)

The first and fourth equations have no solution as you can cancel xx (or -x−x) from both sides to obtain 2=-32=−3, which is a contradiction.

The second and third equations have the same solution, as one can be obtained from the other by multiplying both sides by -1−1. Then, we may simply solve

x+3 = -(x-2)x+3=−(x−2)

=> x+3 = -x+2⇒x+3=−x+2

=>2x=-1⇒2x=−1

:.x = -1/2

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