Is $|x| = -x$ true for $x = 0$? What are the solutions for this equation?
$|x| = -x$ 
It is clear for me that all negative numbers will fulfill this (my brother doesn't believe me, but that doesn't matter). But I'm having a discussion with my mum about whether $x=0$ fulfils it. My mum says something like "There isn't a $-0$ so this doesn't work!". I don't believe this. 
Is the equation true for $x=0$? Why or why not? What is this $-0$ thing?
 A: The set of solutions to $|x| = -x$ is $x \leq 0$. Note that $-0 = (-1) * 0 = 0$. 
A: A couple of view points.
The definition.
The definition of $\lvert x \rvert$ is
$$
\lvert x \rvert = 
\begin{cases} x & \text{ for } x\geq 0 \\
-x & \text{ for } x< 0
\end{cases}
$$
So according to the definition, $\lvert 0 \lvert = 0$ and $-0 = 0$. and so indeed $\lvert 0 \rvert = - 0$ (they are equal to the same number.
You can object to the definition and maybe in another book you have found another definition. The point is that no matter what definition you have, $\lvert 0 \rvert = -0$ will always be true.
Group theory
I don't know if the second viewpoint is helpful, but I am guessing that the confusion might be that whether indeed $-0 = 0$. To "prove" this we can turn to the definition. $0$ is the (unique) element $e$ such that $e + x = x$ for all real numbers $x$. Given a real number $x$, what is $-x$? $-x$ is the (unique) element that satisfies the equation $x + \color{blue}{(-x)} = 0$.
Now the uniqueness of these elements is a small exercise in what is called group theory. But for now just take this as facts.
So, to see that $-0 = 0$ you have to "show" that $0 + \color{blue}{0} = 0$. But this is true. 
