We are given the inequality:

$|3x - 3| < 3$

Step 1: Solve the inequality

To solve this absolute value inequality, we split it into two inequalities by removing the absolute value:

$-3 < 3x - 3 < 3$

Step 2: Solve both sides of the inequality

Now, solve for $x$ step by step.

1. Add 3 to all parts of the inequality:

$0 < 3x < 6$

2. Divide everything by 3:

$0 < x < 2$

Step 3: Interpret the solution

This inequality tells us that the solution to $|3x - 3| < 3$ is $x$ between 0 and 2 (but not including 0 or 2, since the inequality is strict).

Step 4: Check the provided intervals

Now, let’s look at the options:

• A. $\left[\frac{1}{2}, \frac{5}{2}\right)$: This is not part of the solution, because $\frac{5}{2} = 2.5$ is outside the valid range $0 < x < 2$.
• B. $\left[\frac{1}{4}, \frac{3}{2}\right]$: This interval is entirely within $0 < x < 2$, so it could be part of the solution.
• C. $\left(-\frac{1}{2}, \frac{1}{2}\right]$: This includes $x = -\frac{1}{2}$, which is outside the valid range, so it's not part of the solution.
• D. $\left[0, \frac{1}{2}\right)$: This interval includes 0, which violates the strict inequality $x > 0$.

Conclusion:

The correct answer is B. $\left[\frac{1}{4}, \frac{3}{2}\right]$, as this interval lies entirely within $0 < x < 2$.

Would you like further clarification or additional details?

Related Questions:

1. How would the solution change if the inequality were $|3x - 3| \leq 3$?
2. What if the inequality was $|3x - 3| > 3$? How would that affect the intervals?
3. How can we graph the solution to $|3x - 3| < 3$ on a number line?
4. What method can we use to solve absolute value inequalities in general?
5. How do strict inequalities affect the endpoints of intervals in solutions?

Tip:

Always remember to handle absolute value inequalities by splitting them into two separate inequalities—one for the positive case and one for the negative case.