To solve the compound inequality \( 5 \geq \frac{7n + 1}{3} > -2 \), we can break it into two separate inequalities and solve each one.
1. Solve the left inequality: \[ 5 \geq \frac{7n + 1}{3} \]
First, multiply both sides by 3 to eliminate the fraction: \[ 15 \geq 7n + 1 \]
Next, subtract 1 from both sides: \[ 14 \geq 7n \]
Now, divide both sides by 7: \[ 2 \geq n \quad \text{or} \quad n \leq 2 \]
2. Solve the right inequality: \[ \frac{7n + 1}{3} > -2 \]
Again, multiply both sides by 3: \[ 7n + 1 > -6 \]
Subtract 1 from both sides: \[ 7n > -7 \]
Now, divide both sides by 7: \[ n > -1 \]
Combining the results: We have: \[ -1 < n \leq 2 \]
The compound inequality can be expressed as: \[ -1 < n \leq 2 \]
In terms of integers, the possible values for \( n \) are: \[ n = 0, 1, 2 \]
Thus, the final answer for \( n \) as a compound inequality with integers is: \[ n \in {0, 1, 2} \]