We need to solve the inequality:

\[
\frac{x}{2x - 1} < \frac{2}{3}
\]

### Step-by-step solution:

1. **Cross-multiply** (since both sides are positive as \(2x - 1 \neq 0\), we can safely cross-multiply):
   \[
   3x < 2(2x - 1)
   \]

2. **Expand and simplify** the right-hand side:
   \[
   3x < 4x - 2
   \]

3. **Move terms involving \(x\)** to one side:
   \[
   3x - 4x < -2
   \]
   \[
   -x < -2
   \]

4. **Multiply both sides by \(-1\)** (and remember to reverse the inequality sign):
   \[
   x > 2
   \]

### Conclusion:
The solution is \(x > 2\).

However, we need to check for any restrictions:

- The denominator \(2x - 1\) must not be zero, which gives \(x \neq \frac{1}{2}\).
- Additionally, the expression for \(x\) must be valid in all intervals.

Thus, the solution is:

\[
x > 2
\]

Would you like a detailed check of intervals or have any other questions?

---

### Related questions:
1. How do you solve rational inequalities in general?
2. Can you explain the importance of critical points in solving inequalities?
3. What are the common mistakes to avoid when solving inequalities?
4. How does multiplying by a negative number affect an inequality?
5. What does cross-multiplying do in the context of rational inequalities?

**Tip:** Always check for critical points where the denominator may cause the inequality to be undefined.

Learn how to solve the rational inequality x/(2x-1) < 2/3 with a detailed step-by-step solution. This guide covers algebraic concepts, including cross-multiplication and handling inequalities when multiplying by negative numbers. Suitable for students in Grades 9-11.

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