The error in Rick's equation is that he incorrectly subtracted $100 from $200, which does not reflect the total cost of the rental including the mileage fee. The correct equation should represent the total cost as the base fee of $100 plus $0.25 per mile (25 cents times the number of miles driven), and that total cost should not exceed his budget of $200.
The correct equation is:
\[ 100 + 0.25x \leq 200 \]
To solve for \( x \) (the number of miles he can drive), Rick would then subtract 100 from both sides:
\[ 0.25x \leq 100 \]
And then divide both sides by 0.25:
\[ x \leq 400 \]
So, Rick can drive up to 400 miles without exceeding his budget.