To solve the inequality $|3x + 2| \geq 3$, we need to consider two cases because the absolute value expression can either be positive or negative:

Case 1: $3x + 2 \geq 3$

1. Solve for $x$: $3x + 2 \geq 3$ Subtract 2 from both sides: $3x \geq 1$ Divide by 3: $x \geq \frac{1}{3}$

Case 2: $3x + 2 \leq -3$

1. Solve for $x$: $3x + 2 \leq -3$ Subtract 2 from both sides: $3x \leq -5$ Divide by 3: $x \leq -\frac{5}{3}$

Final Solution

Combining both cases, the solution is: $x \leq -\frac{5}{3} \quad \text{or} \quad x \geq \frac{1}{3}$

Interval Notation

The solution in interval notation is: $(-\infty, -\frac{5}{3}] \cup [\frac{1}{3}, \infty)$

Would you like more details or have any questions about this solution? Here are some related questions:

1. What is the difference between solving absolute value inequalities and absolute value equations?
2. How do you solve compound inequalities involving "and" versus "or"?
3. How do you graph the solution set of an absolute value inequality on a number line?
4. Can absolute value inequalities have no solution? If so, under what conditions?
5. How would the solution change if the inequality was $|3x + 2| < 3$ instead?

Tip: When solving absolute value inequalities, always remember to split the inequality into two cases to account for both the positive and negative scenarios of the absolute value expression.