Rational + irrational = always irrational?
Suppose $x$ is irrational and $x+\dfrac pq=\dfrac mn$ then, $x=\dfrac mn-\dfrac pq=\dfrac{mq-np}{nq}$ so, $x$ would then be rational. :)
Look at the contrapositive: If $x$ is rational, then $x+n$ is rational. Clearly this is a true statement.