We are given the inequality:

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

This inequality involves an absolute value expression. To solve it, we'll need to break it down into two cases because the absolute value function measures the distance from zero and can result in both positive and negative expressions.

### Step 1: Set up two inequalities (without the absolute value)
The general rule is:

\[
\left| A \right| < B \quad \text{implies} \quad -B < A < B
\]

In our case, \( A = \frac{x}{3} - \frac{2}{3} \) and \( B = 2 \). So, the inequality becomes:

\[
-2 < \frac{x}{3} - \frac{2}{3} < 2
\]

### Step 2: Solve the compound inequality

#### 1. Left Inequality:
\[
-2 < \frac{x}{3} - \frac{2}{3}
\]

Add \(\frac{2}{3}\) to both sides:
\[
-2 + \frac{2}{3} < \frac{x}{3}
\]

Convert \(-2\) to a fraction with denominator 3:
\[
\frac{-6}{3} + \frac{2}{3} < \frac{x}{3}
\]

Simplify:
\[
\frac{-4}{3} < \frac{x}{3}
\]

Multiply both sides by 3:
\[
-4 < x
\]

#### 2. Right Inequality:
\[
\frac{x}{3} - \frac{2}{3} < 2
\]

Add \(\frac{2}{3}\) to both sides:
\[
\frac{x}{3} < 2 + \frac{2}{3}
\]

Convert 2 to a fraction with denominator 3:
\[
\frac{x}{3} < \frac{6}{3} + \frac{2}{3}
\]

Simplify:
\[
\frac{x}{3} < \frac{8}{3}
\]

Multiply both sides by 3:
\[
x < 8
\]

### Step 3: Combine the two results

The solution to the compound inequality is:
\[
-4 < x < 8
\]

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

Do you want more details or have any questions?

Here are 5 related questions:
1. How do you solve absolute value inequalities with "greater than" instead of "less than"?
2. Can absolute value inequalities result in no solution?
3. How do you graph the solution of this inequality on a number line?
4. What is the difference between strict inequalities and non-strict inequalities?
5. How can you solve an absolute value inequality involving a quadratic expression?

**Tip**: Always check for any extraneous solutions, especially in absolute value equations or inequalities.

Learn how to solve the absolute value inequality |x/3 - 2/3| < 2 using step-by-step methods. This solution covers key concepts in algebra, such as breaking down absolute value expressions and solving compound inequalities. Ideal for students in Grades 8-10.

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