Sum of irrational and rational

Contradiction. Suppose $r\in\mathbb{Q}$, $x\notin\mathbb{Q}$, and $r+x=q\in\mathbb{Q}$. Then, $x=q-r\in\mathbb{Q}$ (difference of two rationals), contradicting $x\notin\mathbb{Q}$.

Conclusion. $r+x$ is irrational.

Application. $\sqrt2+1$ is irrational.