What are the solutions of $|x+y|=|x|+|y|$? So I am having a problem in solving this type of equation. 
The problem I am dealing with is...
$$\left|(2x-1) + \frac{3x-1}x\right| = \left|2x-1\right| + \left|\frac{3x-1}x\right|$$
Please help me with this problem. 
Thank you so much. 

Another approach, assuming $x$ and $y$ are real:
$$\begin{aligned}
|x+y| = |x| + |y|
&\iff |x+y|^2 = (|x|+|y|)^2 \\
&\iff (x+y)^2 = |x|^2 + 2|x||y| + |y|^2 \\
&\iff x^2 + 2xy + y^2 = x^2 + 2|x||y| + y^2 \\
&\iff 2xy = 2|x||y| \\
&\iff xy = |x||y| \\
&\iff xy = |xy| \\
&\iff xy \geq 0 \\
\end{aligned}$$
In your case, the original problem is therefore equivalent to
$$(2x-1)\left(\frac{3x-1}{x}\right) \geq 0$$
which should be straightforward to solve.
