To determine the minimum number of miles that Joseph must drive for Company A to be a better buy than Company B, we can set up the cost equations for both companies based on the number of miles driven, \( x \).
The cost for Company A can be expressed as: \[ \text{Cost}_A = 0.25x + 50 \]
The cost for Company B can be expressed as: \[ \text{Cost}_B = 0.45x + 20 \]
We need to find the smallest integer \( x \) such that: \[ \text{Cost}_A < \text{Cost}_B \] This gives us the inequality: \[ 0.25x + 50 < 0.45x + 20 \]
To solve for \( x \), we first isolate the terms involving \( x \): \[ 50 - 20 < 0.45x - 0.25x \] \[ 30 < 0.20x \]
Now, we divide both sides by 0.20 to solve for \( x \): \[ x > \frac{30}{0.20} \] \[ x > 150 \]
This means \( x \) must be greater than 150. Since we are looking for the minimum number of miles, the smallest whole number that satisfies this inequality is: \[ x = 151 \]
To confirm, we should compare the costs at \( x = 151 \):
- For Company A: \[ \text{Cost}_A = 0.25(151) + 50 = 37.75 + 50 = 87.75 \]
- For Company B: \[ \text{Cost}_B = 0.45(151) + 20 = 67.95 + 20 = 87.95 \]
Since \( 87.75 < 87.95 \), Company A is indeed the better option for 151 miles.
Thus, the minimum number of miles Joseph must drive for Company A to be a better buy is: \[ \boxed{151} \]
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