### Instruction: Suppose $x \in [-5,-3]$ and $y \in [2,4]$. What is the smallest possible value of $\frac{x+y}{x-y}$?### Response: I want to find the minimum of a ratio of two linear expressions in $x$ and $y$.
One way to do this is to consider how the ratio changes as $x$ and $y$ vary within their given intervals.
If I fix $y$ and increase $x$, the numerator $x+y$ increases and the denominator $x-y$ decreases, so the ratio increases.
Therefore, the ratio is smaller when $x$ is smaller, and the smallest $x$ is $-5$.
If I fix $x$ and increase $y$, the numerator $x+y$ increases and the denominator $x-y$ increases, so the ratio decreases.
Therefore, the ratio is smaller when $y$ is larger, and the largest $y$ is $4$.
So the smallest possible value of the ratio occurs when $x=-5$ and $y=4$.
Plugging these values into the expression, I get $\frac{-5+4}{-5-4} = \frac{-1}{-9} = \frac{1}{9}$.