1. **State the problem:** Solve for $x$ in the equation $$\frac{x-1}{x} - \frac{2x+1}{3x} = \frac{2}{3}.$$\n\n2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to combine terms and clear fractions by multiplying both sides. Here, the denominators are $x$ and $3x$, so the common denominator is $3x$.\n\n3. **Multiply both sides by $3x$ to clear denominators:**\n$$3x \times \left(\frac{x-1}{x} - \frac{2x+1}{3x}\right) = 3x \times \frac{2}{3}$$\nSimplify each term:\n$$3x \times \frac{x-1}{x} = 3 \cancel{x} \times \frac{x-1}{\cancel{x}} = 3(x-1)$$\n$$3x \times \frac{2x+1}{3x} = \cancel{3} \cancel{x} \times \frac{2x+1}{\cancel{3} \cancel{x}} = 2x+1$$\n$$3x \times \frac{2}{3} = \cancel{3} x \times \frac{2}{\cancel{3}} = 2x$$\nSo the equation becomes:\n$$3(x-1) - (2x+1) = 2x$$\n\n4. **Expand and simplify:**\n$$3x - 3 - 2x - 1 = 2x$$\n$$ (3x - 2x) - 4 = 2x$$\n$$x - 4 = 2x$$\n\n5. **Isolate $x$:**\n$$x - 4 = 2x$$\nSubtract $x$ from both sides:\n$$\cancel{x} - 4 = 2x - \cancel{x}$$\n$$-4 = x$$\n\n6. **Final answer:**\n$$\boxed{x = -4}$$