Solve for n.

5≥ 7n+1/3 >–2
Write your answer as a compound inequality with integers.

1 answer

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: \[ n \in {0, 1, 2} \]