Certainly! Let's solve the compound inequality step-by-step:
[tex]\[ 20 > 3n - 7 \geq -28 \][/tex]
This compound inequality actually consists of two separate inequalities:
1. [tex]\( 20 > 3n - 7 \)[/tex]
2. [tex]\( 3n - 7 \geq -28 \)[/tex]
We will solve each inequality separately and then determine the combined solution set.
### Solving [tex]\( 20 > 3n - 7 \)[/tex]
1. Add 7 to both sides:
[tex]\[ 20 + 7 > 3n - 7 + 7 \][/tex]
[tex]\[ 27 > 3n \][/tex]
2. Divide both sides by 3:
[tex]\[ \frac{27}{3} > n \][/tex]
[tex]\[ 9 > n \][/tex]
Which can be written as:
[tex]\[ n < 9 \][/tex]
### Solving [tex]\( 3n - 7 \geq -28 \)[/tex]
1. Add 7 to both sides:
[tex]\[ 3n - 7 + 7 \geq -28 + 7 \][/tex]
[tex]\[ 3n \geq -21 \][/tex]
2. Divide both sides by 3:
[tex]\[ \frac{3n}{3} \geq \frac{-21}{3} \][/tex]
[tex]\[ n \geq -7 \][/tex]
### Combining the results
Now we have the two inequalities:
1. [tex]\( n < 9 \)[/tex]
2. [tex]\( n \geq -7 \)[/tex]
Putting them together, we get the combined solution set:
[tex]\[ -7 \leq n < 9 \][/tex]
Therefore, the solution to the compound inequality [tex]\( 20 > 3n - 7 \geq -28 \)[/tex] is:
[tex]\[ -7 \leq n < 9 \][/tex]
