To determine the integer values of [tex]\( n \)[/tex] that satisfy the inequality [tex]\( -15 < 3n \leq 6 \)[/tex], we will follow a step-by-step approach.
1. Break down the inequality into two parts:
[tex]\[-15 < 3n\][/tex]
and
[tex]\[3n \leq 6\][/tex]
2. Solve the first part of the inequality:
[tex]\[-15 < 3n\][/tex]
To isolate [tex]\( n \)[/tex], divide both sides by [tex]\( 3 \)[/tex]:
[tex]\[
\frac{-15}{3} < \frac{3n}{3} \implies -5 < n
\][/tex]
This simplifies to:
[tex]\[
n > -5
\][/tex]
3. Solve the second part of the inequality:
[tex]\[3n \leq 6\][/tex]
Again, isolate [tex]\( n \)[/tex] by dividing both sides by [tex]\( 3 \)[/tex]:
[tex]\[
\frac{3n}{3} \leq \frac{6}{3} \implies n \leq 2
\][/tex]
4. Combine the results of both inequalities:
We now have:
[tex]\[
-5 < n \leq 2
\][/tex]
5. Interpret [tex]\( n \)[/tex] as an integer:
Since [tex]\( n \)[/tex] must be an integer, we need to list all the integer values that fall within the range:
[tex]\[
-5 < n \leq 2
\][/tex]
This means [tex]\( n \)[/tex] can be any integer greater than [tex]\(-5\)[/tex] and less than or equal to [tex]\( 2 \)[/tex].
The integer values of [tex]\( n \)[/tex] that satisfy the inequality are:
[tex]\[
n = -4, -3, -2, -1, 0, 1, 2
\][/tex]
So, the values of [tex]\( n \)[/tex] that satisfy [tex]\( -15 < 3n \leq 6 \)[/tex] are [tex]\( -4, -3, -2, -1, 0, 1, 2 \)[/tex].
