Let $X$ be a quasi-projective variety and two distinct points $P,Q\in X$. Prove there is an affine open subset $\mathcal{U}\subset X$ containing both $P$ and $Q$.
Assuming $X\subset \mathbb{A}^n$ is an affine variety, we can take any linear form $L$ with $L(P)=L(Q)=0$. Therefore $P,Q\notin V(L+1)$, so that $P,Q\in X\setminus V(L+1)$, which is an affine open subset of $X$.
I'm trying to argue that we can assume $X$ is affine without loss of generality, but I don't know if this is possible. If not, how could I solve this?