Rationals plus Irrational are Everywhere Dense in Irrationals

Theorem
Let $\left({\R \setminus \Q, \tau_d}\right)$ be the irrational number space under the Euclidean topology $\tau_d$.

Let $x \in \R \setminus \Q$ be an arbitrary irrational number.

Let $S_x$ be the set defined as:
 * $S_x := \left\{{x + q: q \in \Q}\right\}$

Then $S_x$ is everywhere dense in $\left({\R \setminus \Q, \tau_d}\right)$.