Solving base e equation $e^x - e^{-x} = 0$

If $$x = -x$$

Then we can say that $$x = 0$$

Because

$$ x + x = -x + x $$

$$ 2x = 0 $$

$$ x = 0 $$

You are solving for $x$ when $e^x=e^{-x}$. It is not the case for all $x$ that $e^x=e^{-x}$; therefore, the statement $x=-x$ will not be true for all $x$, just the $x$ you are solving for.

Both are correct. $x=-x\Leftrightarrow 2x=0\Leftrightarrow x=0$.