If $x = - x$, then adding $x$ to each side, we have \begin{equation} 2 x = x + x = - x + x = 0 \end{equation} Dividing each side by $2$, we then get $x = 0 / 2 = 0$. Conversely, if $x = 0$, then $x = 0 = - 0 = - x$. $\square$