jmc

Since we are looking for integer solutions, we should isolate $x$ by multiplying by 5. This gives $$1\frac{1}{4}<x<3\frac{1}{3}.$$The only integers between $1\frac{1}{4}$ and $3\frac{1}{3}$ are 2 and 3. Therefore there are $\boxed{2}$ integers that solve this pair of inequalities.