1. **State the problem:** Solve the equation $$\frac{x - 4}{2022} + \frac{x - 3}{2023} + \frac{x - 2}{2024} = 3.$$\n\n2. **Formula and approach:** To solve this linear equation, we combine like terms by finding a common denominator or by multiplying through by the product of denominators to clear fractions.\n\n3. **Multiply both sides by the product of denominators:** The denominators are 2022, 2023, and 2024. Multiply both sides by $$2022 \times 2023 \times 2024$$ to eliminate fractions:\n$$ (x - 4) \times 2023 \times 2024 + (x - 3) \times 2022 \times 2024 + (x - 2) \times 2022 \times 2023 = 3 \times 2022 \times 2023 \times 2024.$$\n\n4. **Expand each term:**\n$$ x \times 2023 \times 2024 - 4 \times 2023 \times 2024 + x \times 2022 \times 2024 - 3 \times 2022 \times 2024 + x \times 2022 \times 2023 - 2 \times 2022 \times 2023 = 3 \times 2022 \times 2023 \times 2024.$$\n\n5. **Group the x terms and constants:**\n$$ x(2023 \times 2024 + 2022 \times 2024 + 2022 \times 2023) = 3 \times 2022 \times 2023 \times 2024 + 4 \times 2023 \times 2024 + 3 \times 2022 \times 2024 + 2 \times 2022 \times 2023.$$\n\n6. **Calculate the sums:**\n- Calculate $$S_x = 2023 \times 2024 + 2022 \times 2024 + 2022 \times 2023.$$\n- Calculate $$S_c = 3 \times 2022 \times 2023 \times 2024 + 4 \times 2023 \times 2024 + 3 \times 2022 \times 2024 + 2 \times 2022 \times 2023.$$\n\n7. **Evaluate each product:**\n- $$2023 \times 2024 = 4,091,552$$\n- $$2022 \times 2024 = 4,089,328$$\n- $$2022 \times 2023 = 4,085,506$$\n\nSum $$S_x = 4,091,552 + 4,089,328 + 4,085,506 = 12,266,386.$$\n\n8. **Calculate constants:**\n- $$3 \times 2022 \times 2023 \times 2024 = 3 \times 8,273,113,552 = 24,819,340,656$$\n- $$4 \times 2023 \times 2024 = 4 \times 4,091,552 = 16,366,208$$\n- $$3 \times 2022 \times 2024 = 3 \times 4,089,328 = 12,267,984$$\n- $$2 \times 2022 \times 2023 = 2 \times 4,085,506 = 8,171,012$$\n\nSum $$S_c = 24,819,340,656 + 16,366,208 + 12,267,984 + 8,171,012 = 24,856,145,860.$$\n\n9. **Solve for x:**\n$$ x = \frac{S_c}{S_x} = \frac{24,856,145,860}{12,266,386} \approx 2025.5.$$\n\n**Final answer:** $$x \approx 2025.5.$$