To solve the inequality [tex]\(-3|n-2| < -9\)[/tex], let's break it down step by step.
1. Isolate the absolute value term:
[tex]\[
-3|n-2| < -9
\][/tex]
We start by dividing both sides of the inequality by [tex]\(-3\)[/tex], keeping in mind that dividing by a negative number reverses the inequality sign:
[tex]\[
|n-2| > 3
\][/tex]
2. Understand the inequality involving the absolute value:
The absolute value inequality [tex]\(|n-2| > 3\)[/tex] can be broken into two inequalities:
[tex]\[
n - 2 > 3 \quad \text{or} \quad n - 2 < -3
\][/tex]
3. Solve each inequality separately:
a. [tex]\( n - 2 > 3 \)[/tex]
[tex]\[
n > 5
\][/tex]
b. [tex]\( n - 2 < -3 \)[/tex]
[tex]\[
n < -1
\][/tex]
4. Combine the solutions:
[tex]\[
n > 5 \quad \text{or} \quad n < -1
\][/tex]
5. Evaluate the provided options:
- Option A: [tex]\( n = -5 \)[/tex]
[tex]\[
-5 < -1 \quad \text{(This satisfies \( n < -1 \))}
\][/tex]
- Option B: [tex]\( n = -1 \)[/tex]
[tex]\[
-1 \not< -1 \quad \text{(This does not satisfy \( n < -1 \))}
\][/tex]
- Option C: [tex]\( n = 1 \)[/tex]
[tex]\[
1 \not> 5 \quad \text{and} \quad 1 \not< -1 \quad \text{(This does not satisfy either inequality)}
\][/tex]
- Option D: [tex]\( n = 5 \)[/tex]
[tex]\[
5 \not> 5 \quad \text{(This does not satisfy \( n > 5 \))}
\][/tex]
From these evaluations, only option [tex]\(A\)[/tex] ([tex]\(n = -5\)[/tex]) satisfies one of the inequalities ([tex]\( n < -1 \)[/tex]).
Therefore, the possible value of [tex]\( n \)[/tex] is [tex]\(\boxed{-5}\)[/tex].
