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Find the number of integers $n$ that satisfy both of the inequalities $4n + 3 < 253$ and $-7n + 5 < 24$.

 Sep 10, 2023
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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.

 Sep 10, 2023

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