To solve the absolute value inequality [tex]\(|2n - 3| > 11\)[/tex], let's follow these steps:
1. Understand the absolute value inequality: The expression [tex]\(|A| > B\)[/tex] (where [tex]\(A\)[/tex] is an expression and [tex]\(B\)[/tex] is a positive number) can be interpreted as two separate inequalities: [tex]\(A > B\)[/tex] or [tex]\(A < -B\)[/tex].
2. Apply this to our specific inequality:
[tex]\[
|2n - 3| > 11
\][/tex]
This translates into two separate inequalities:
[tex]\[
2n - 3 > 11 \quad \text{or} \quad 2n - 3 < -11
\][/tex]
3. Solve the first inequality [tex]\(2n - 3 > 11\)[/tex]:
[tex]\[
2n - 3 > 11
\][/tex]
Add 3 to both sides to isolate terms involving [tex]\(n\)[/tex]:
[tex]\[
2n > 14
\][/tex]
Divide both sides by 2:
[tex]\[
n > 7
\][/tex]
4. Solve the second inequality [tex]\(2n - 3 < -11\)[/tex]:
[tex]\[
2n - 3 < -11
\][/tex]
Add 3 to both sides to isolate terms involving [tex]\(n\)[/tex]:
[tex]\[
2n < -8
\][/tex]
Divide both sides by 2:
[tex]\[
n < -4
\][/tex]
5. Combine the solutions: The solutions to the two inequalities are:
[tex]\[
n > 7 \quad \text{or} \quad n < -4
\][/tex]
Therefore, the solution to the absolute value inequality [tex]\(|2n - 3| > 11\)[/tex] is:
[tex]\[
n > 7 \quad \text{or} \quad n < -4
\][/tex]
This means that [tex]\(n\)[/tex] must be either greater than 7 or less than -4 to satisfy the inequality.
