1. **State the problem:** We want to find the average speed driving to Trigville and the average speed driving home given the distances and speeds. 2. **Set up the rational equation:** - Speed to Trigville: $x$ km/h - Distance to Trigville: 330 km - Time to Trigville: $\frac{330}{x}$ hours - Speed returning: $x - 35$ km/h - Distance returning: 330 km - Time returning: $\frac{330}{x - 35}$ hours 3. **Use the formula for average speed:** Average speed $= \frac{\text{total distance}}{\text{total time}}$ 4. **Write the equation for average speed:** $$\text{Average speed} = \frac{330 + 330}{\frac{330}{x} + \frac{330}{x - 35}} = \frac{660}{\frac{330}{x} + \frac{330}{x - 35}}$$ 5. **Simplify the denominator:** $$\frac{330}{x} + \frac{330}{x - 35} = 330 \left(\frac{1}{x} + \frac{1}{x - 35}\right) = 330 \frac{(x - 35) + x}{x(x - 35)} = 330 \frac{2x - 35}{x(x - 35)}$$ 6. **Substitute back:** $$\text{Average speed} = \frac{660}{330 \frac{2x - 35}{x(x - 35)}} = \frac{660}{330} \cdot \frac{x(x - 35)}{2x - 35} = 2 \cdot \frac{x(x - 35)}{2x - 35}$$ 7. **Given the average speed is the same as the speed to Trigville $x$, set up the equation:** $$x = 2 \cdot \frac{x(x - 35)}{2x - 35}$$ 8. **Solve for $x$:** Multiply both sides by $2x - 35$: $$x(2x - 35) = 2x(x - 35)$$ Expand both sides: $$2x^2 - 35x = 2x^2 - 70x$$ Subtract $2x^2$ from both sides: $$-35x = -70x$$ Add $70x$ to both sides: $$35x = 0$$ Divide both sides by 35: $$x = 0$$ 9. **Check for extraneous solutions:** $x=0$ is not valid for speed. So, check the original equation again. 10. **Rearrange the original equation:** $$x = 2 \cdot \frac{x(x - 35)}{2x - 35}$$ Multiply both sides by $2x - 35$: $$x(2x - 35) = 2x(x - 35)$$ Expand: $$2x^2 - 35x = 2x^2 - 70x$$ Subtract $2x^2$ from both sides: $$-35x = -70x$$ Add $70x$ to both sides: $$35x = 0$$ Divide by 35: $$x = 0$$ No valid solution from this approach, so try another method. 11. **Alternative approach: The average speed for the round trip is given by the harmonic mean:** $$\text{Average speed} = \frac{2 \cdot \text{speed}_1 \cdot \text{speed}_2}{\text{speed}_1 + \text{speed}_2} = \frac{2x(x - 35)}{x + (x - 35)} = \frac{2x(x - 35)}{2x - 35}$$ 12. **Set the average speed equal to 60 km/h (example) or solve for $x$ if given average speed. Since no average speed is given, the problem is to find $x$ such that the total time equals a certain value or to find $x$ from the equation:** Given the problem, the rational equation to solve is: $$\frac{330}{x} + \frac{330}{x - 35} = \text{total time}$$ Without total time, we cannot solve for $x$ numerically. 13. **Summary:** The rational equation is: $$\frac{330}{x} + \frac{330}{x - 35} = T$$ where $T$ is total time. The average speed is: $$\frac{660}{T} = \frac{2x(x - 35)}{2x - 35}$$ Without additional data, $x$ cannot be numerically determined. **Final answer:** The rational equation to solve for $x$ is: $$\frac{330}{x} + \frac{330}{x - 35} = T$$ The average speed for the round trip is: $$\frac{2x(x - 35)}{2x - 35}$$