1. **State the problem:** A car and a motorcycle start from the same place at noon, traveling in the same direction. 2. **Define variables:** Let $m$ be the speed of the motorcycle in mph. 3. **Express the car's speed:** The car's speed is 30 mph slower than twice the motorcycle's speed, so car speed $c = 2m - 30$. 4. **Use the distance formula:** Distance = speed $\times$ time. 5. **Set up the equation for the distance difference after 2 hours:** The car is 20 miles ahead, so $$2c - 2m = 20$$ 6. **Substitute $c$:** $$2(2m - 30) - 2m = 20$$ 7. **Simplify:** $$4m - 60 - 2m = 20$$ $$2m - 60 = 20$$ 8. **Add 60 to both sides:** $$2m - \cancel{60} + 60 = 20 + 60$$ $$2m = 80$$ 9. **Divide both sides by 2:** $$\frac{\cancel{2}m}{\cancel{2}} = \frac{80}{2}$$ $$m = 40$$ 10. **Find car speed:** $$c = 2(40) - 30 = 80 - 30 = 50$$ **Final answer:** - Motorcycle speed = 40 mph - Car speed = 50 mph