To solve the compound inequality \( 50 > 7n + 8 > 22 \), we can break it into two separate inequalities:
- \( 50 > 7n + 8 \)
- \( 7n + 8 > 22 \)
Solving the first inequality:
\[ 50 > 7n + 8 \] Subtract 8 from both sides: \[ 50 - 8 > 7n \] \[ 42 > 7n \] Now, divide both sides by 7: \[ \frac{42}{7} > n \] \[ 6 > n \] or \[ n < 6 \]
Solving the second inequality:
\[ 7n + 8 > 22 \] Subtract 8 from both sides: \[ 7n > 22 - 8 \] \[ 7n > 14 \] Now, divide both sides by 7: \[ n > \frac{14}{7} \] \[ n > 2 \]
Combined results:
Now we can combine the two results: \[ 2 < n < 6 \]
Thus, the solution to the inequality \( 50 > 7n + 8 > 22 \) is: \[ \boxed{(2, 6)} \]