To solve the absolute value inequality [tex]\( 4|5x - 1| - 2 > 14 \)[/tex], we will proceed step-by-step to isolate [tex]\( x \)[/tex] and find its range.
1. Isolate the absolute value expression:
Start by adding 2 to both sides of the inequality:
[tex]\[
4|5x - 1| - 2 + 2 > 14 + 2
\][/tex]
Simplifies to:
[tex]\[
4|5x - 1| > 16
\][/tex]
2. Divide both sides by 4:
[tex]\[
|5x - 1| > 4
\][/tex]
3. Break this absolute value inequality into two separate inequalities:
[tex]\[
5x - 1 > 4 \quad \text{or} \quad 5x - 1 < -4
\][/tex]
4. Solve each of these inequalities separately:
- For the first inequality [tex]\( 5x - 1 > 4 \)[/tex]:
[tex]\[
5x - 1 > 4
\][/tex]
Add 1 to both sides:
[tex]\[
5x > 5
\][/tex]
Divide both sides by 5:
[tex]\[
x > 1
\][/tex]
- For the second inequality [tex]\( 5x - 1 < -4 \)[/tex]:
[tex]\[
5x - 1 < -4
\][/tex]
Add 1 to both sides:
[tex]\[
5x < -3
\][/tex]
Divide both sides by 5:
[tex]\[
x < -\frac{3}{5}
\][/tex]
5. Combine the solutions:
The solution to the inequality [tex]\( |5x - 1| > 4 \)[/tex] is the union of the solutions from the two inequalities:
[tex]\[
x < -\frac{3}{5} \quad \text{or} \quad x > 1
\][/tex]
Thus, the correct answer is:
b. [tex]\( x < -\frac{3}{5} \)[/tex] or [tex]\( x > 1 \)[/tex]
