To solve the inequality [tex]\( |2n + 5| > 1 \)[/tex], we will consider the definition of the absolute value function. The inequality [tex]\(|x| > a\)[/tex] implies [tex]\(x < -a\)[/tex] or [tex]\(x > a\)[/tex].
Let's break it down step-by-step:
1. Start with the given inequality:
[tex]\[
|2n + 5| > 1
\][/tex]
2. Since the absolute value inequality [tex]\(|x| > a\)[/tex] translates to [tex]\(x < -a\)[/tex] or [tex]\(x > a\)[/tex], we can write:
[tex]\[
2n + 5 < -1 \quad \text{or} \quad 2n + 5 > 1
\][/tex]
3. Solve each part of the inequality separately.
First part: [tex]\(2n + 5 < -1\)[/tex]
Subtract 5 from both sides:
[tex]\[
2n < -1 - 5
\][/tex]
[tex]\[
2n < -6
\][/tex]
Divide both sides by 2:
[tex]\[
n < -3
\][/tex]
Second part: [tex]\(2n + 5 > 1\)[/tex]
Subtract 5 from both sides:
[tex]\[
2n > 1 - 5
\][/tex]
[tex]\[
2n > -4
\][/tex]
Divide both sides by 2:
[tex]\[
n > -2
\][/tex]
4. Combining the two parts, we get:
[tex]\[
n < -3 \quad \text{or} \quad n > -2
\][/tex]
Thus, the solution to the inequality [tex]\( |2n + 5| > 1 \)[/tex] is:
[tex]\[
n < -3 \quad \text{or} \quad n > -2
\][/tex]
The correct answer choice is:
[tex]\[
n < -3 \quad \text{or} \quad n > -2
\][/tex]
