We need to solve the inequality $3 - 2|x - 5| \le -15$.

1. Problem Description

We need to solve the inequality $3 - 2|x - 5| \le -15$.

2. Solution Steps

First, subtract 3 from both sides of the inequality:

$3 - 2|x - 5| - 3 \le -15 - 3$

$-2|x - 5| \le -18$

Next, divide both sides by -

2. Remember to flip the inequality sign since we are dividing by a negative number:

$\frac{-2|x - 5|}{-2} \ge \frac{-18}{-2}$

$|x - 5| \ge 9$

This inequality means that the distance between $x$ and 5 is greater than or equal to

9. This translates into two separate inequalities:

$x - 5 \ge 9$ or $x - 5 \le -9$

For the first inequality, $x - 5 \ge 9$, add 5 to both sides:

$x - 5 + 5 \ge 9 + 5$

$x \ge 14$

For the second inequality, $x - 5 \le -9$, add 5 to both sides:

$x - 5 + 5 \le -9 + 5$

$x \le -4$

Therefore, the solution is $x \ge 14$ or $x \le -4$.

3. Final Answer

$x \le -4$ or $x \ge 14$