Given matrices $A, B \in \Bbb R^{3 \times 3}$ whose ranks satisfy $\mbox{rank} (A), \mbox{rank} (B) \geq 2$, I would like to prove that for large (or small) enough scalar $\alpha \in \mathbb{R} \setminus \{0\}$ the following does hold.

$$\mbox{rank} (A+\alpha B) \geq 2$$

This seems to be true by hand waving argument, but I would like to find some short proof or reference that formally proves it. Thanks.