The inequality given is:

\[
\left| \frac{x}{3} - \frac{2}{3} \right| < 2
\]

### Step-by-step solution:

1. The absolute value inequality \( |A| < B \) implies that \( -B < A < B \). Here, \( A = \frac{x}{3} - \frac{2}{3} \) and \( B = 2 \).
   
   So, we can rewrite the inequality without the absolute value as:
   \[
   -2 < \frac{x}{3} - \frac{2}{3} < 2
   \]

2. Now, add \(\frac{2}{3}\) to all three sides of the inequality:
   \[
   -2 + \frac{2}{3} < \frac{x}{3} < 2 + \frac{2}{3}
   \]
   Simplifying the left and right sides:
   \[
   -\frac{6}{3} + \frac{2}{3} < \frac{x}{3} < \frac{6}{3} + \frac{2}{3}
   \]
   \[
   -\frac{4}{3} < \frac{x}{3} < \frac{8}{3}
   \]

3. Now, multiply all parts of the inequality by 3 to eliminate the denominator:
   \[
   3 \times \left( -\frac{4}{3} \right) < x < 3 \times \left( \frac{8}{3} \right)
   \]
   \[
   -4 < x < 8
   \]

### Final solution:

The solution to the inequality is:
\[
x \in (-4, 8)
\]

### Would you like more details on any step or have any questions?

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Here are some related questions to consider:
1. How do we handle absolute value inequalities when the inequality is \(\geq\)?
2. What is the solution for \( | \frac{x}{5} - 1 | < 3 \)?
3. How do we solve \( |x^2 - 4| < 5 \)?
4. What are the rules for solving inequalities with negative coefficients?
5. How do we graph inequalities on a number line?

**Tip:** When solving absolute value inequalities, always consider both the positive and negative cases to avoid missing solutions.

Learn how to solve the absolute value inequality |x/3 - 2/3| < 2 with a clear step-by-step solution. This problem covers important algebraic concepts like absolute value inequalities and the application of inequalities properties. Suitable for students in Grades 8-10.

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