Exercise
If $a$ is rational and if $x$ is irrational, prove that $a + x$ is irrational.
Attempt
Since $a$ is rational, we can express it as $\frac{m}{n}$.
Let us assume that $a + x$ is rational. Then, we can express it as $\frac{q}{p}$.
We then get $\frac{m}{n} + x = \frac{q}{p}$.
I would say that $x$ cannot be put over the same denominator as $m$, therefore the LHS cannot be expressed in a fraction and is hence irrational; but, that would be a lie because indeed $\frac{m}{n} + x = \frac{m + nx}{n}$.
Request
I'm not sure on how to proceed. Hints, next steps, or full solutions are welcome.
If $a+x$ is rational, then $a+x - a = x$ is also rational, since the rationals are closed under subtraction:
If $\frac{r}{s}, \frac{p}{q} \in \mathbb{Q}$, then $\frac{r}{s} - \frac{p}{q} = \frac{rq-sp}{qs} \in \mathbb{Q}$ as well.