Useful (and often tested) fraction property: \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\)

So we can rewrite and simplify our target expression as follows: \(\frac{x + y}{xy}=\frac{x}{xy}+\frac{y}{xy}=\frac{1}{y}+\frac{1}{x}\)

From here we must recognize that we can minimize the value of \(\frac{1}{y}+\frac{1}{x}\) by maximizing the values of \(x\) and \(y\)

Since \(x\) and \(y\) are integers and \(3 < x < y < 9\), the maximum values \(x\) and \(y\) are \(x=7\) and \(y=8\)

This means the minimum value of \(\frac{1}{y}+\frac{1}{x}=\frac{1}{8}+\frac{1}{7}=\frac{7}{56}+\frac{8}{56}=\frac{15}{56}\)

Answer: \(\frac{15}{56}\)