Thread: absolute value on both side of equation

1. absolute value on both side of equation

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.

solving the second part we come to know that the right part of this is also true for all real numbers.

2. Please state the original problem without using an image.
As is, it appears as if you have jumped in the middle of a solution.

3. the original problem that i am trying to solve is |3x^2 + 2x|= x|3x+2|

4. First I would notice that $\displaystyle \left| {3x^2 + 2x} \right| = |x|\left| {3x + 2} \right|$.
Then we have to solve $\displaystyle |x|\left| {3x + 2} \right|=x|3x+2|$

Therefore we have three problems to consider.

$\displaystyle x < \frac{{ - 2}}{3}\, \Rightarrow \,( - x)( - 3x - 2) = x( - 3x - 2)$.

$\displaystyle \frac{{ - 2}}{3} \leqslant x < 0\, \Rightarrow \,( - x)(3x + 2) = x(3x + 2)$.

$\displaystyle 0 \leqslant x \Rightarrow \,(x)(3x + 2) = x(3x + 2)$.