To solve the inequality $|3x - 2| \leq 4$, we need to remove the absolute value by breaking it down into two separate inequalities.

The inequality $|3x - 2| \leq 4$ means: $-4 \leq 3x - 2 \leq 4$

Now, we will solve for $x$ by isolating it in both inequalities.

Step 1: Solve the Left Inequality

$-4 \leq 3x - 2$ Add 2 to both sides: $-2 \leq 3x$ Now, divide by 3: $-\frac{2}{3} \leq x$

Step 2: Solve the Right Inequality

$3x - 2 \leq 4$ Add 2 to both sides: $3x \leq 6$ Now, divide by 3: $x \leq 2$

Step 3: Combine the Results

The solution to the inequality is: $-\frac{2}{3} \leq x \leq 2$

Final Answer in Interval Notation

In interval notation, the solution is: $\left[ -\frac{2}{3}, 2 \right]$

So, the correct choice is A: The solution set is $\left[ -\frac{2}{3}, 2 \right]$.

Would you like further details or have any questions?

Here are some related questions for further practice:

1. How would you solve $|4x + 5| \leq 7$ in interval notation?
2. What are the steps for solving $|2x - 3| > 5$?
3. How do you interpret absolute value inequalities geometrically?
4. What are the differences between strict inequalities $<, >$ and inclusive inequalities $\leq, \geq$ in absolute value equations?
5. How would you convert the interval notation $(-\infty, -1] \cup [3, \infty)$ into inequality form?

Tip: Always remember that absolute value inequalities $|ax + b| \leq c$ are solved by creating two separate inequalities: $-c \leq ax + b \leq c$.