jmc

Subtract 3 and divide by 4 on both sides of the first inequality to obtain $$\begin{array}{rl}4n+3& <25\\ \Rightarrow 4n& <22\\ \Rightarrow n& <\mathrm{5.5.}\end{array}$$Similarly, the second inequality yields $$\begin{array}{rl}-7n+5& <24\\ \Rightarrow -7n& <19\\ \Rightarrow n& >-\frac{19}{7}.\end{array}$$Therefore, we are looking for all the integers between $-\frac{19}{7}$ and $5.5$. Since $-\frac{19}{7}$ is between $-3$ and $-2$ and the largest integer less than $5.5$ is 5, we need to count the number of integers between $-2$ and $5$, inclusive. There are $5$ positive integers, $2$ negative integers, and zero, so there are $\boxed{8}$ integers that satisfy both $4n+3<25$ and $-7n+5<24$.