Absolute value on both side of equation

[ziaharipur]

I solved this absolute value equation and I am confuse about the answer given in the book for this question, the answer in the book is x > = 0 or x = -2/3 please tell me how can I get this answer from the following solution. Please tell the method which should be true for all such type of questions.
look at the image below
[PLAIN]http://img261.imageshack.us/img261/1577/imgsb.jpg

 

[aq1q]

ok well... one of the mistakes I see is how you reached the conclusion "3x +2 =3x +2 for all real nobs"

well, this isn't true... is it? ex: x= -1. |3(-1)+2(-1)| does not equal (-1)|3(-1)+2|.

Break it up like this:
{|3x^2 + 2x|= |x||3x+2| = x |3x + 2| } \Rightarrow { |x|=x , for (3x+2) \neq 0 }

Hence, this is true for all x \geq 0.

Now, separately consider the case (3x+2)=0. What do you get? x= -2/3

Note: LaTeX is being crazy!