To find the possible values of $x$, I need to isolate it in the inequality by multiplying both sides by $5$. This gives me $5 \cdot \frac14 < x < 5 \cdot \frac23$, or $1.25 < x < 3.33$. Now I need to count how many integers are between $1.25$ and $3.33$. I can see that $x = 2$ and $x = 3$ are the only two integers that satisfy the inequality, since any smaller or larger integer would be outside the range. Therefore, the answer is $2$, since there are only two integers $x$ that make the inequality true. 
For how many integers $x$ do we have $\frac14 < \frac{x}{5} < \frac23$?