To solve the absolute value inequality [tex]\(3|x-1| \leq 12\)[/tex], follow these step-by-step instructions:
1. Isolate the absolute value:
Start by dividing both sides of the inequality by 3:
[tex]\[
\frac{3|x-1|}{3} \leq \frac{12}{3}
\][/tex]
This simplifies to:
[tex]\[
|x-1| \leq 4
\][/tex]
2. Solve the absolute value inequality:
Recall that [tex]\(|x - 1| \leq 4\)[/tex] means the expression inside the absolute value, [tex]\(x - 1\)[/tex], must lie between [tex]\(-4\)[/tex] and [tex]\(4\)[/tex]. This can be expressed by the compound inequality:
[tex]\[
-4 \leq x - 1 \leq 4
\][/tex]
3. Solve the compound inequality:
To clear the absolute value, separate it into two inequalities and solve for [tex]\(x\)[/tex]:
[tex]\[
-4 \leq x - 1 \quad \text{and} \quad x - 1 \leq 4
\][/tex]
Add 1 to all sides of the inequalities to solve for [tex]\(x\)[/tex]:
[tex]\[
-4 + 1 \leq x - 1 + 1 \quad \text{and} \quad x - 1 + 1 \leq 4 + 1
\][/tex]
Simplifying these gives:
[tex]\[
-3 \leq x \quad \text{and} \quad x \leq 5
\][/tex]
4. Write the final answer:
Combining the two parts of the compound inequality, we get:
[tex]\[
-3 \leq x \leq 5
\][/tex]
The correct answer is:
d. [tex]\( -3 \leq x \leq 5 \)[/tex]
