This question came in one of class-room discussion.
Given finitely many distinct real numbers $x_1,x_2,\cdots, x_n$, does there exists a real number $y$ such that $y+x_1, y+x_2, \cdots, y+x_n$ are irrational?
(Similar question can be posed with expectation that $y+x_i$ are rational for all $i$.)
HINT (if you are familiar with uncountable sets): For any $x$, how many $y$ are there such that $x+y$ is rational? How does this compare to the whole set of reals? Do you see how to bring multiple different $x$ into it?
Note, by the way, that this works even if we have infinitely many (but countably many) $x$.
You may also be interested in the Baire category theorem.
For the second question, the answer is no even for two values of $x$: do you see why $y+0$ and $y+\pi$ can never both be rational?