To determine the integer values of [tex]\( n \)[/tex] that satisfy the inequality [tex]\( -15 < 3n \leq 6 \)[/tex], we need to follow these steps:
1. Divide the entire inequality by 3 to isolate [tex]\( n \)[/tex]:
[tex]\[
-15 < 3n \leq 6
\][/tex]
Dividing everything by 3, we get:
[tex]\[
\frac{-15}{3} < \frac{3n}{3} \leq \frac{6}{3}
\][/tex]
Simplifying this, we have:
[tex]\[
-5 < n \leq 2
\][/tex]
2. Find the integer values of [tex]\( n \)[/tex] within this range:
We need to find all integers [tex]\( n \)[/tex] that are greater than [tex]\(-5\)[/tex] and less than or equal to [tex]\( 2 \)[/tex]. These are the values of [tex]\( n \)[/tex] that satisfy the inequality:
[tex]\[
-5 < n \leq 2
\][/tex]
Listing the integers within this range, we start from the smallest integer greater than [tex]\(-5\)[/tex] and go up to [tex]\(2\)[/tex]:
[tex]\[
-4, -3, -2, -1, 0, 1, 2
\][/tex]
Therefore, the integer values of [tex]\( n \)[/tex] that satisfy the inequality [tex]\( -15 < 3n \leq 6 \)[/tex] are:
[tex]\[
n \in \{-4, -3, -2, -1, 0, 1, 2\}
\][/tex]
