Show that $\Bbb Q^n \cup (\Bbb R-\Bbb Q)^n$ is connected

Hint: Let $E = \mathbb Q^n \cup (\mathbb R\setminus \mathbb Q)^n.$ For $x\in \mathbb R^n,$ let $L(x) = \{tx: t\in \mathbb R\}.$ ($L(x)$ is the line in $\mathbb R^n$ through the origin and $x.$) Verify that if $q = (q_1,\dots ,q_n)\in \mathbb Q^n$ and each $q_k \ne 0,$ then $L(q)\subset E.$ Consider the union of all of such lines.