I'm sure this is a relatively easy thing to do, I'm just not sure how to begin.
Here I wish to show that if $k \in \mathbb{Z}$ that $$ \frac{3}{2} \pm k \not \in \mathbb{Z} $$ Or, to extend, if $p,q \in \mathbb{Z}$ such that $\frac{p}{q} \not \in \mathbb{Z}$ and $q \ne 0$ that $$ \frac{p}{q} - k \not \in \mathbb{Z} $$ Not asking for a full solution, just a starting point.
Thank you!