jmc

Subtracting $3$ from both sides of the inequality gives $5n>-13$, and dividing both sides by $5$ gives $n>-\frac{13}{5}$. Since $-\frac{13}{5}$ is between $-\frac{15}{5}=-3$ and $-\frac{10}{5}=-2$, the smallest integer $n$ satisfying our inequality is $\boxed{-2}$.