To solve the inequality [tex]\(-3 \cdot |x - 5| \leq -15\)[/tex], let's first isolate the absolute value expression. Follow these steps:
1. Start with the given inequality:
[tex]\[
-3 \cdot |x - 5| \leq -15
\][/tex]
2. Divide both sides by [tex]\(-3\)[/tex]. Remember that when you divide or multiply an inequality by a negative number, you must reverse the inequality sign:
[tex]\[
\left| x - 5 \right| \geq \frac{15}{3}
\][/tex]
Simplifying the right-hand side gives:
[tex]\[
\left| x - 5 \right| \geq 5
\][/tex]
3. The absolute value inequality [tex]\(\left| x - 5 \right| \geq 5\)[/tex] means that the expression inside the absolute value, [tex]\(x - 5\)[/tex], is either greater than or equal to 5, or less than or equal to -5:
[tex]\[
x - 5 \geq 5 \quad \text{or} \quad x - 5 \leq -5
\][/tex]
Solve each part separately:
- For [tex]\(x - 5 \geq 5\)[/tex]:
[tex]\[
x \geq 10
\][/tex]
- For [tex]\(x - 5 \leq -5\)[/tex]:
[tex]\[
x \leq 0
\][/tex]
4. Combining these results, the solution set is:
[tex]\[
x \leq 0 \quad \text{or} \quad x \geq 10
\][/tex]
Therefore, the correct answer is:
d. [tex]\(x \leq 0 \text{ or } x \geq 10\)[/tex]
