How many integers $n$ satisfy both of the inequalities $4n + 3 < 25$ and $-7n + 5 < 24$?
Subtract 3 and divide by 4 on both sides of the first inequality to obtain \begin{align*}
4n + 3 &< 25 \\
\Rightarrow\qquad 4n &< 22 \\
\Rightarrow\qquad n &< 5.5.
\end{align*}Similarly, the second inequality yields \begin{align*}
-7n + 5 &< 24 \\
\Rightarrow\qquad -7n &< 19 \\
\Rightarrow\qquad n &> -\frac{19}{7}.
\end{align*}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 $-7n + 5 < 24$.