Does |x| = |y| requires checking conditions while solving? I am trying to solve this equation $\lvert 2x \rvert = \lvert x - 2 + y \rvert$ (specifically, find set of all points $(x, y)$ satisfying equation). 
$\lvert 2x \rvert = \lvert x - 2 + y \rvert$ is equivalent to $2x = x - 2 + y \lor 2x = -(x - 2 + y)$. 
If $2x = x - 2 + y$, then I am assuming that $2x \ge 0$ and $x - 2 + y \ge 0$, but do I have to check if solutions meet those conditions? 
In general, while solving any equation with absolute values you have to check conditions, but sometimes you don't have to. 
For example,
$$\lvert\lvert x-1\rvert +5\rvert = 7$$ 
I can write it as $\lvert x - 1\rvert + 5 = 7$ or $\lvert x - 1\rvert + 5 = -7$. In this case, I don't have to check conditons.


It is obvious that $2x = x - 2 + y$ or $2x = -(x - 2 + y)$.

True, but is it obvious for you that $\lvert 2x \rvert = \lvert x - 2 + y \rvert$ is equivalent to $2x =\pm( x - 2 + y)$ ?
We have
$$\begin{align}|X|=|Y|&\iff |X|^2=|Y|^2\\&\iff X^2-Y^2=0\\&\iff (X-Y)(X+Y)=0\\&\iff X=\pm Y\end{align}$$
So, $\lvert 2x \rvert = \lvert x - 2 + y \rvert$ is equivalent to $2x = \pm (x - 2 + y)$.
Hence, we can have
$$\begin{align}\lvert 2x \rvert = \lvert x - 2 + y \rvert &\iff 2x = x - 2 + y\quad \text{or}\quad 2x = -(x - 2 + y)\\&\iff y=x+2\quad\text{or}\quad y=-3x+2\end{align}$$
You don't have to check if solutions meet the conditions.
