To solve the inequality [tex]\(3|x-1| \geq 12\)[/tex], let’s go through the steps methodically.
1. Isolate the absolute value expression:
[tex]\[
3|x-1| \geq 12
\][/tex]
Divide both sides by 3 to simplify:
[tex]\[
|x-1| \geq 4
\][/tex]
2. Break the absolute value inequality into two separate inequalities:
The absolute value expression [tex]\( |x-1| \geq 4 \)[/tex] implies two cases:
[tex]\[
x-1 \geq 4 \quad \text{or} \quad x-1 \leq -4
\][/tex]
3. Solve each inequality separately:
- For the first inequality [tex]\( x-1 \geq 4 \)[/tex]:
[tex]\[
x \geq 4 + 1
\][/tex]
[tex]\[
x \geq 5
\][/tex]
- For the second inequality [tex]\( x-1 \leq -4 \)[/tex]:
[tex]\[
x \leq -4 + 1
\][/tex]
[tex]\[
x \leq -3
\][/tex]
4. Combine the solutions:
The solutions from the two inequalities are [tex]\( x \geq 5 \)[/tex] or [tex]\( x \leq -3 \)[/tex].
Therefore, the solution to the inequality [tex]\(3|x-1| \geq 12\)[/tex] is:
[tex]\[
x \leq -3 \quad \text{or} \quad x \geq 5
\][/tex]
This corresponds to option B in the given choices.
Final Answer: B. [tex]\( x \leq -3 \)[/tex] or [tex]\( x \geq 5 \)[/tex].
