Proof that in a field $x=0$ is equivalent to $x=-x$

Question

Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$ This means that the statement in the title is true if and only if $1\neq -1$.

But how do we know that $1\neq -1$ for an arbitary field?

Answer 1

But how do we know that $1+1\neq 0$ for an arbitrary field?

Completely valid question, and the answer is: you don't! There are fields for which $1 + 1 = 0$, called fields of characteristic $2$. The simplest example is the field with two elements, $\mathbb{F}_2 := \{ 0, 1 \}$, where $1 + 1 = 0$.

Indeed, in $\mathbb{F}_2$ (or any field of characteristic $2$) we have $1 = -1$ but $1 \neq 0$, so the result you are trying to prove is true only when the characteristic of the field is not $2$.