To solve the inequality [tex]\(-9 < 3n \leq 12\)[/tex], we will break it down into two separate inequalities and solve each part individually.
1. Solve the first part of the inequality: [tex]\(-9 < 3n\)[/tex]
- Divide both sides by 3 to isolate [tex]\(n\)[/tex]:
[tex]\[
-9 < 3n \implies \frac{-9}{3} < n \implies -3 < n
\][/tex]
2. Solve the second part of the inequality: [tex]\(3n \leq 12\)[/tex]
- Divide both sides by 3 to isolate [tex]\(n\)[/tex]:
[tex]\[
3n \leq 12 \implies n \leq \frac{12}{3} \implies n \leq 4
\][/tex]
3. Combine both parts of the inequality:
- From the first part, we have [tex]\(n > -3\)[/tex]
- From the second part, we have [tex]\(n \leq 4\)[/tex]
- Combining these results gives:
[tex]\[
-3 < n \leq 4
\][/tex]
4. Identify the integer solutions:
Since [tex]\(n\)[/tex] is an integer, we need to list all integer values that satisfy [tex]\(-3 < n \leq 4\)[/tex]. The integers that are greater than [tex]\(-3\)[/tex] and less than or equal to [tex]\(4\)[/tex] are:
[tex]\[
-2, -1, 0, 1, 2, 3, 4
\][/tex]
Therefore, the integer values of [tex]\(n\)[/tex] that satisfy the inequality [tex]\(-9 < 3n \leq 12\)[/tex] are [tex]\(-2, -1, 0, 1, 2, 3, 4\)[/tex].
