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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.

Which of the following intervals could be part of the solution of the inequality |3x - 3| < 3?

Learn how to solve the absolute value inequality |3x - 3| < 3 and identify the correct interval that satisfies the inequality. This problem covers absolute value properties and interval notation, suitable for Grades 9-11.

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