Maya is deciding between two truck rental companies. Company A charges an initial fee of $40 for the rental plus $2 per mile driven. Company B charges an initial fee of $70 for the rental plus $1 per mile driven. Let AA represent the amount Company A would charge if Maya drives xx miles, and let BB represent the amount Company B would charge if Maya drives xx miles. Write an equation for each situation, in terms of x, commax, and determine the interval of miles driven, x, commax, for which Company A is cheaper than Company B.

Question

GPT-4o mini · Answer

Let $ A(x) $ represent the cost from Company A and $ B(x) $ represent the cost from Company B. The equation for Company A can be expressed as: $$ A(x) = 40 + 2x $$ where $ 40 $ is the initial fee and $ 2x $ represents the cost per mile driven. The equation for Company B can be expressed as: $$ B(x) = 70 + 1x $$ where $ 70 $ is the initial fee and $ x $ represents the cost per mile driven. To find the interval of miles driven $ x $ for which Company A is cheaper than Company B, we need to set up the inequality: $$ A(x) < B(x) $$ Substituting the equations gives: $$ 40 + 2x < 70 + 1x $$ Now, let's solve this inequality step-by-step: 1. Subtract $ 1x $ from both sides: $$ 40 + 2x - 1x < 70 $$ $$ 40 + x < 70 $$ 2. Subtract $ 40 $ from both sides: $$ x < 70 - 40 $$ $$ x < 30 $$ Thus, the interval of miles driven $ x $ for which Company A is cheaper than Company B is: $$ x < 30 $$ In conclusion, Company A is cheaper than Company B when Maya drives fewer than 30 miles.