Since $\alpha$ is inversely proportional to $\beta$, by definition $\alpha\beta = k$ for some constant $k$.  Plugging in, we see that $4\cdot 9 = k$, so $k = 36$.  So when $\beta = -72$, we have that $-72\alpha = 36$, or $\alpha = -\frac{1}{2}$.
Suppose that $\alpha$ is inversely proportional to $\beta$.  If $\alpha = 4$ when $\beta = 9$, find $\alpha$ when $\beta = -72$.