Solve the inequality $-2 + 3|5 - x| \le 4$.

1. Problem Description

Solve the inequality $-2 + 3|5 - x| \le 4$.

2. Solution Steps

First, isolate the absolute value term by adding 2 to both sides of the inequality:

$-2 + 3|5 - x| + 2 \le 4 + 2$

$3|5 - x| \le 6$

Next, divide both sides by 3:

$\frac{3|5 - x|}{3} \le \frac{6}{3}$

$|5 - x| \le 2$

Now, we can rewrite the absolute value inequality as a compound inequality:

$-2 \le 5 - x \le 2$

We can split this into two separate inequalities:

$5 - x \le 2$ and $5 - x \ge -2$

Solve the first inequality:

$5 - x \le 2$

$-x \le 2 - 5$

$-x \le -3$

$x \ge 3$

Solve the second inequality:

$5 - x \ge -2$

$-x \ge -2 - 5$

$-x \ge -7$

$x \le 7$

Combining these two inequalities, we have:

$3 \le x \le 7$

3. Final Answer

$3 \le x \le 7$