Solve the absolute value inequality. Write the solution in interval notation.
[tex]\[ 3|x-5|+12 \geq 15 \][/tex]
Select one: a. [tex]\[[-4, 14]\][/tex] b. [tex]\[[4, 6]\][/tex] c. [tex]\[(-\infty, 4] \cup [6, \infty)\][/tex] d. [tex]\[(-\infty, -4] \cup [14, \infty)\][/tex]
To solve the inequality [tex]\(3|x-5|+12 \geq 15\)[/tex], let's go through the steps one by one.
1. Isolate the absolute value expression: [tex]\[3|x - 5| + 12 \geq 15\][/tex]
2. Subtract 12 from both sides to simplify the inequality: [tex]\[3|x - 5| \geq 3\][/tex]
3. Divide both sides by 3 to isolate the absolute value term: [tex]\[|x - 5| \geq 1\][/tex]
4. Interpret the absolute value inequality: The inequality [tex]\(|x - 5| \geq 1\)[/tex] means that the expression inside the absolute value term [tex]\(x - 5\)[/tex] is either greater than or equal to 1, or less than or equal to -1.
This results in two separate inequalities: [tex]\[
x - 5 \geq 1 \quad \text{or} \quad x - 5 \leq -1
\][/tex]
5. Solve each inequality separately: [tex]\[
x - 5 \geq 1
\][/tex] Adding 5 to both sides, we get: [tex]\[
x \geq 6
\][/tex]
For the second inequality: [tex]\[
x - 5 \leq -1
\][/tex] Adding 5 to both sides, we get: [tex]\[
x \leq 4
\][/tex]
6. Combine the solutions: The solution to the inequality [tex]\(|x - 5| \geq 1\)[/tex] is [tex]\(x \geq 6\)[/tex] or [tex]\(x \leq 4\)[/tex].
7. Express the solution in interval notation: The solution can be written as: [tex]\[
(-\infty, 4] \cup [6, \infty)
\][/tex]
Thus, the correct option is: [tex]\[
\boxed{(-\infty, 4] \cup [6, \infty)}
\][/tex]
