Check if intersection of several 2D half-planes is empty
Your problem can be solved in linear time.

This paper describes a method to solve a system of $n$ linear inequalities with at most two variables per inequality and $m$ distinct variables in total in $O(n m^3 \log m)$ time. (I am swapping the meaning of $n$ and $m$ with respect to the paper in order to keep the name $n$ consistent with your question).

In your case $m=2$ therefore the resulting time complexity is $O(n)$.