To solve the inequality [tex]\( |2n + 5| > 1 \)[/tex], we need to consider the definition of the absolute value function. The absolute value [tex]\( |x| \)[/tex] of a number [tex]\( x \)[/tex] is defined to be the distance of [tex]\( x \)[/tex] from 0 on the number line. Therefore, [tex]\( |2n + 5| > 1 \)[/tex] means that [tex]\( 2n + 5 \)[/tex] is more than 1 unit away from 0. This can be translated into two separate inequalities:
1. [tex]\( 2n + 5 > 1 \)[/tex]
2. [tex]\( 2n + 5 < -1 \)[/tex]
Let's solve these inequalities one by one.
### Solving the first inequality: [tex]\( 2n + 5 > 1 \)[/tex]
1. Subtract 5 from both sides:
[tex]\[
2n + 5 - 5 > 1 - 5
\][/tex]
[tex]\[
2n > -4
\][/tex]
2. Divide both sides by 2:
[tex]\[
n > -2
\][/tex]
### Solving the second inequality: [tex]\( 2n + 5 < -1 \)[/tex]
1. Subtract 5 from both sides:
[tex]\[
2n + 5 - 5 < -1 - 5
\][/tex]
[tex]\[
2n < -6
\][/tex]
2. Divide both sides by 2:
[tex]\[
n < -3
\][/tex]
### Combining the solutions
The solutions to the original inequality are those values of [tex]\( n \)[/tex] that satisfy either of the two inequalities:
- [tex]\( n > -2 \)[/tex]
- [tex]\( n < -3 \)[/tex]
Therefore, the solution to the inequality [tex]\( |2n + 5| > 1 \)[/tex] is:
[tex]\[
n < -3 \quad \text{or} \quad n > -2
\][/tex]
In interval notation, this is:
[tex]\[
(-\infty, -3) \cup (-2, \infty)
\][/tex]
Thus, the correct answer is:
[tex]\[ n < -3 \quad \text{or} \quad n > -2 \][/tex]
