To solve the absolute value inequality [tex]\( |3x - 1| > 11 \)[/tex], we can break it down into two separate inequalities due to the nature of the absolute value function.
The absolute value inequality [tex]\( |A| > B \)[/tex] is equivalent to the two inequalities:
[tex]\[ A > B \text{ or } A < -B \][/tex]
Applying this to our inequality [tex]\( |3x - 1| > 11 \)[/tex], we get:
[tex]\[ 3x - 1 > 11 \text{ or } 3x - 1 < -11 \][/tex]
Let's solve each inequality separately:
1. [tex]\( 3x - 1 > 11 \)[/tex]
Add 1 to both sides:
[tex]\[ 3x > 12 \][/tex]
Divide both sides by 3:
[tex]\[ x > 4 \][/tex]
2. [tex]\( 3x - 1 < -11 \)[/tex]
Add 1 to both sides:
[tex]\[ 3x < -10 \][/tex]
Divide both sides by 3:
[tex]\[ x < -\frac{10}{3} \][/tex]
Thus, the solutions to the inequalities [tex]\( |3x - 1| > 11 \)[/tex] are:
[tex]\[ x > 4 \text{ or } x < -\frac{10}{3} \][/tex]
So the correct solution for the absolute value inequality is:
[tex]\[ x > 4 \text{ or } x < -\frac{10}{3} \][/tex]
This corresponds to option (b):
[tex]\[ \text{b. } x > 4 \text{ or } x < -\frac{10}{3} \][/tex]
