Question

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

Answer 1

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