Solving the first inequality, we have $4n+3<253$. Subtracting three from both sides gives us $4n<250$. Dividing both sides by $4$ gives $n<62.5$, so the possible values of $n$ satisfying the first inequality are 61, 60, 59, ..., 1. There are 61 such values. Solving the second inequality, we have $-7n+5<24.$ Subtracting 5 from both sides gives $-7n<19$, and dividing both sides by $-7$ gives $n>-19/7$, or $n\geq -3$. The possible values of $n$ satisfying the second inequality are $-3$, $-2$, $-1$, 0, 1, 2,.... The smallest such value is $n= -3$, the next is $n= -2$, and so on - so in total, there are $3$ such values. In order to satisfy both inequalities, we must have $-3
Therefore, there are $\boxed{59}$ possible values of $n$.
Find the number of integers $n$ that satisfy both of the inequalities $4n + 3 < 253$ and $-7n + 5 < 24$.
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