The problem is to solve the inequality $3 - 2|3x - 1| \ge -7$.

1. Problem Description

The problem is to solve the inequality $3 - 2|3x - 1| \ge -7$.

2. Solution Steps

First, we isolate the absolute value term.

Subtract 3 from both sides of the inequality:

$3 - 2|3x - 1| - 3 \ge -7 - 3$

$-2|3x - 1| \ge -10$

Next, divide both sides by -

2. Remember that dividing by a negative number reverses the inequality sign:

$\frac{-2|3x - 1|}{-2} \le \frac{-10}{-2}$

$|3x - 1| \le 5$

Now we consider the definition of absolute value, which implies that

$-5 \le 3x - 1 \le 5$

We solve this compound inequality by adding 1 to all parts:

$-5 + 1 \le 3x - 1 + 1 \le 5 + 1$

$-4 \le 3x \le 6$

Finally, divide all parts by 3:

$\frac{-4}{3} \le \frac{3x}{3} \le \frac{6}{3}$

$-\frac{4}{3} \le x \le 2$

3. Final Answer

The solution to the inequality is $-\frac{4}{3} \le x \le 2$.