### Instruction: Suppose that $\alpha$ is inversely proportional to $\beta$.  If $\alpha = 4$ when $\beta = 9$, find $\alpha$ when $\beta = -72$.### Response: I recall that inverse proportionality means that the product of $\alpha$ and $\beta$ is constant, so I can write $\alpha \beta = k$ for some constant $k$.
To find the value of $k$, I can use the given values of $\alpha$ and $\beta$, so I substitute $\alpha = 4$ and $\beta = 9$ into the equation and get $4 \cdot 9 = k$, which means $k = 36$.
Now I can use the same equation to find $\alpha$ when $\beta = -72$.
I substitute $\beta = -72$ and $k = 36$ and get $\alpha \cdot (-72) = 36$.