1. **State the problem:** Solve the equation $$\frac{2}{3} \left( \frac{1}{4} x - 2 \right) = \frac{1}{5} \left( \frac{4}{3} x - 1 \right)$$. 2. **Write the formula and rules:** We will use the distributive property to remove parentheses and then solve for $x$ by isolating it on one side. 3. **Distribute both sides:** $$\frac{2}{3} \times \frac{1}{4} x - \frac{2}{3} \times 2 = \frac{1}{5} \times \frac{4}{3} x - \frac{1}{5} \times 1$$ 4. **Calculate each term:** $$\frac{2}{3} \times \frac{1}{4} x = \frac{2}{12} x = \frac{1}{6} x$$ $$\frac{2}{3} \times 2 = \frac{4}{3}$$ $$\frac{1}{5} \times \frac{4}{3} x = \frac{4}{15} x$$ $$\frac{1}{5} \times 1 = \frac{1}{5}$$ 5. **Rewrite the equation:** $$\frac{1}{6} x - \frac{4}{3} = \frac{4}{15} x - \frac{1}{5}$$ 6. **Bring all $x$ terms to one side and constants to the other:** $$\frac{1}{6} x - \frac{4}{15} x = - \frac{1}{5} + \frac{4}{3}$$ 7. **Find common denominators and subtract:** $$\frac{1}{6} x - \frac{4}{15} x = \frac{5}{30} x - \frac{8}{30} x = -\frac{3}{30} x = -\frac{1}{10} x$$ $$- \frac{1}{5} + \frac{4}{3} = -\frac{3}{15} + \frac{20}{15} = \frac{17}{15}$$ 8. **Equation becomes:** $$-\frac{1}{10} x = \frac{17}{15}$$ 9. **Solve for $x$ by dividing both sides:** $$x = \frac{17}{15} \div -\frac{1}{10} = \frac{17}{15} \times -10 = -\frac{170}{15}$$ 10. **Simplify the fraction:** $$-\frac{170}{15} = -\frac{34}{3}$$ **Final answer:** $$x = -\frac{34}{3}$$