1. **State the problem:** Solve the equation $\frac{x-2}{4} - 3 = \frac{x-4}{5}$ for $x$. 2. **Formula and rules:** To solve equations with fractions, first eliminate the denominators by multiplying both sides by the least common denominator (LCD). Here, the denominators are 4 and 5, so the LCD is 20. 3. **Multiply both sides by 20:** $$20 \times \left(\frac{x-2}{4} - 3\right) = 20 \times \frac{x-4}{5}$$ 4. **Distribute multiplication:** $$20 \times \frac{x-2}{4} - 20 \times 3 = 20 \times \frac{x-4}{5}$$ Simplify each term: $$5(x-2) - 60 = 4(x-4)$$ 5. **Expand both sides:** $$5x - 10 - 60 = 4x - 16$$ Simplify: $$5x - 70 = 4x - 16$$ 6. **Isolate $x$ terms on one side:** $$5x - 4x = -16 + 70$$ Simplify: $$x = 54$$ 7. **Final answer:** $$\boxed{54}$$