For what integer $x$ is $\frac 35 < \frac x7 < \frac 79$?
Multiplying by 7, we have $\frac{21}{5}<x<\frac{49}{9}$. Note that $\frac{21}{5}$ is between $\frac{20}{5}=4$ and $\frac{25}{5}=5$, while $\frac{49}{9}$ is between $\frac{45}{9}=5$ and $\frac{54}{9}=6$.  Thus, the only integer between $\frac{21}{5}$ and $\frac{49}{9}$ is $\boxed{5}$.