To solve the inequality [tex]\( |4x - 6| > 14 \)[/tex], let's break it down step-by-step:
1. Understanding the Absolute Value Inequality:
- The inequality [tex]\( |A| > B \)[/tex] means that [tex]\( A \)[/tex] is greater than [tex]\( B \)[/tex] or less than [tex]\(-B\)[/tex]. Therefore, [tex]\( |4x - 6| > 14 \)[/tex] can be split into two separate inequalities:
[tex]\[
4x - 6 > 14 \quad \text{or} \quad 4x - 6 < -14
\][/tex]
2. Solving the First Inequality [tex]\( 4x - 6 > 14 \)[/tex]:
- Add 6 to both sides to isolate the [tex]\( 4x \)[/tex]:
[tex]\[
4x - 6 + 6 > 14 + 6
\][/tex]
[tex]\[
4x > 20
\][/tex]
- Divide both sides by 4:
[tex]\[
x > 5
\][/tex]
3. Solving the Second Inequality [tex]\( 4x - 6 < -14 \)[/tex]:
- Add 6 to both sides to isolate the [tex]\( 4x \)[/tex]:
[tex]\[
4x - 6 + 6 < -14 + 6
\][/tex]
[tex]\[
4x < -8
\][/tex]
- Divide both sides by 4:
[tex]\[
x < -2
\][/tex]
4. Combining the Results:
- The solution to [tex]\( |4x - 6| > 14 \)[/tex] is the combination of the two separate inequalities:
[tex]\[
x < -2 \quad \text{or} \quad x > 5
\][/tex]
So the solution of the inequality [tex]\( |4x - 6| > 14 \)[/tex] is:
[tex]\[ x < -2 \text{ or } x > 5 \][/tex]
Among the provided options:
- [tex]\( x < -2 \)[/tex] or [tex]\( x > 5 \)[/tex] is the correct answer.
