inequality problem

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.