Find an Ideal of $\mathbb{Z}+x \mathbb{Q}[ x ]$ that is NOT principal 
The ring $\mathbb{Z}+x \mathbb{Q}[ x ]$ cannot be a principal ideal domain since it is not a unique factorization domain. Find an ideal of $\mathbb{Z}+x \mathbb{Q}[ x ]$ that is not principal. 

My book gives no examples of how to show an ideal is not principal. I'm pretty sure if I let $I=(2,1/2 x)$ then I can show it's not principal. 
I'm pretty confident that the best route to go is to do a proof by contradiction. But how do I start? What is my initial assumption? 
Thanks!

Call $R= \Bbb{Z}+ x \Bbb{Q}[x]$. I highly suspect that $R$ is a Bezout domain, (i.e. every finitely generated ideal is principal), so I give you a non finitely generated ideal. Consider the ideal
$$I=(x, x/2 , x/4 , x/8 , \dots) = \bigcup_{k \ge 1} \left( \frac{1}{2^k}x \right)$$
Clearly, for all $k \ge 1$ we have $$\left( \frac{1}{2^k}x \right) \subsetneq \left( \frac{1}{2^{k-1}}x \right)$$
because $$\frac{1}{2^{k-1}}x = 2 \frac{1}{2^k}x$$
and $2$ is not a unit of $R$
: this shows that $R$ is not Noetherian, and that the union of this chain of ideals is not finitely generated.
