To determine the solution to the inequality [tex]\(3|x-1| \geq 12\)[/tex], let's break down the problem step by step.
1. Isolate the absolute value expression:
The original inequality is:
[tex]\[
3|x-1| \geq 12
\][/tex]
Divide both sides by 3 to simplify:
[tex]\[
|x-1| \geq 4
\][/tex]
2. Interpret the absolute value inequality:
The absolute value inequality [tex]\( |x-1| \geq 4 \)[/tex] means that the expression inside the absolute value, [tex]\( x-1 \)[/tex], can be either greater than or equal to 4 or less than or equal to -4. This translates into two separate inequalities:
[tex]\[
x-1 \geq 4
\][/tex]
and
[tex]\[
x-1 \leq -4
\][/tex]
3. Solve these inequalities separately:
For the first inequality:
[tex]\[
x-1 \geq 4
\][/tex]
Add 1 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq 5
\][/tex]
For the second inequality:
[tex]\[
x-1 \leq -4
\][/tex]
Add 1 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x \leq -3
\][/tex]
4. Combine the solutions:
The overall solution to the inequality [tex]\( 3|x-1| \geq 12 \)[/tex] is that [tex]\( x \)[/tex] must be either greater than or equal to 5, or less than or equal to -3. Thus, we write the combined solution as:
[tex]\[
x \leq -3 \quad \text{or} \quad x \geq 5
\][/tex]
This matches option D in the provided choices:
D. [tex]\( x \leq -3 \)[/tex] or [tex]\( x \geq 5 \)[/tex]
