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What is the solution to the inequality [tex]|2n+5| \ \textgreater \ 1[/tex]?

A. [tex]-3 \ \textgreater \ n \ \textgreater \ -2[/tex]
B. [tex]2 \ \textless \ n \ \textless \ 3[/tex]
C. [tex]n \ \textless \ -3 \text{ or } n \ \textgreater \ -2[/tex]
D. [tex]n \ \textless \ 2 \text{ or } n \ \textgreater \ 3[/tex]


Sagot :

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]
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