Question 1 of 10

Which of the following is the solution to [tex]3|x-1| \geq 12[/tex]?

A. [tex]x \leq -3[/tex] and [tex]x \geq 5[/tex]

B. [tex]x \geq 5[/tex]

C. [tex]x \geq -3[/tex] or [tex]x \geq 5[/tex]

D. [tex]x \leq -3[/tex] or [tex]x \geq 5[/tex]



Answer :

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]