$x$ is rational, $\frac{x}{2}$ is rational, and $3x-1$ is rational are equivalent How do we prove that the three statements below about the real number $x$ are equivalent?
(i) $\displaystyle x$ is rational
(ii) $\displaystyle \frac{x}{2}$ is rational
(iii) $\displaystyle 3x-1$ is rational

If $x$ is rational, then $x=\frac{p}{q}$ for some integer $p$ and some nonzero integer $q$. Then $\frac{x}{2}=\frac{p}{2q}$ with $p$ an integer and $2q$ a nonzero integer.
If $\frac{x}{2}$ is rational, then $\frac{x}{2}=\frac{m}{n}$ for some integer $m$ and some nonzero integer $n$. Then $3x-1=\frac{6m-n}{n}$ with $6m-n$ an integer and $n$ a nonzero integer.
If $3x-1$ is rational, then $3x-1=\frac{a}{b}$ for some integer $a$ and some nonzero integer $b$. Then $x=\frac{a+b}{3b}$ with $a+b$ an integer and $3b$ a nonzero integer.
