Find the set of [tex]x[/tex] values that satisfy this inequality.

[tex] -3 |x-5| \leq -15 [/tex]

Select one:
a. [tex] 0 \leq x \leq 19 [/tex]
b. [tex] x \leq 0 [/tex]
c. No Solution
d. [tex] x \leq 0 [/tex] or [tex] x \geq 10 [/tex]



Answer :

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]