+622 Find the number of integers $n$ that satisfy both of the inequalities $4n + 3 < 253$ and $-7n + 5 < 24$.
Find the number of integers \(n\) that satisfy both of the inequalities:
1. \(4n + 3 < 253\)
2. \(-7n + 5 < 24\)
Solution:
1. For the first inequality, solve for \(n\):
\[4n + 3 < 253\]
Subtract 3 from both sides:
\[4n < 250\]
Divide both sides by 4:
\[n < \frac{250}{4} = 62.5\]
2. For the second inequality, solve for \(n\):
\[-7n + 5 < 24\]
Subtract 5 from both sides:
\[-7n < 19\]
Divide both sides by -7 (and reverse the inequality sign):
\[n > \frac{-19}{7}\]
3. To satisfy both inequalities, \(n\) must be an integer between \(-19/7\) and 62.5.
4. Count the integers within this range:
-3, -2, -1, 0, 1, 2, 3, ..., 61, 62
5. There are 66 integers within this range that satisfy both inequalities.
So, the number of integers \(n\) that satisfy both inequalities is 66.
