Demand function for good x

In your case there could be multiple solutions to the problem not just $y=0$. Any combination of $x$ and $y$ that would satisfy the budget with equality would be optimal in this case.

Hence I think it would probably be best just express demand for $x$ as:

$x=frac{I}{P_x} - 4 y$

In fact this also gives you all potential optimum solutions to the problem since you can verify that for some parameters for example $I=100$ and $P_x=1$, when you consume 0 units of $y$ (implying that $x=100$) you get exactly the same utility as when you consume $x=96$ and $y=1$, or $x=92$ and $y=2$ and so on. So the $x= frac{I}{P_x}-4y$ gives the combination of all possible optimum solutions. I personally don't see an reason to just assume that $y$ should be zero in such situation (unless this is for a classroom setting and that was recommended by instructor in these situations).