Let $x \in \mathbb{R}$. Prove that $x\in \mathbb{Q}\iff-x\in\mathbb{Q}$.
So I am having trouble coming up with an approach to figure out this proof. I feel as if I have the basics covered. I was going to break the $\iff$ into two if then statements. I figured simplifying the problem can potentially provide some insight.
case (1): $\quad$ If $\phantom{-}x \in\mathbb{Q} \Rightarrow -x \in \mathbb{Q}$
case (2): $\quad$ If $-x \in\mathbb{Q} \Rightarrow \phantom{-}x \in \mathbb{Q}$
And I begin with the basics:
If $x\in\mathbb{Q}$, then $x$ can be expressed as a ratio of two integers in lowest terms such that $x=\dfrac{a}{b}$ where $\gcd(a,b)=1$. Now I know that either $a$ or $b$ have to be a negative integer to show that $-x\in\mathbb{Q}$ but this is where I get stuck.
A push in the right direction would be heavily appreciated.