Jina is going to rent a truck for one day. There are two companies she can choose from, and they have the following prices.
Company A charges an initial fee of $\$ 77.50$ and an additional 30 cents for every mile driven.
Company $B$ charges an initial fee of $\$ 30$ and an additional 80 cents for every mile driven.
For what mileages will Company A charge no more than Company B?
Write your. answer as an inequality, using $m$ for the number of miles driven.
Answer
Final Answer: The number of miles for which Company A will charge no more than Company B is represented by the inequality \(\boxed{m \leq 95}\).
Step 1 ：Jina is considering renting a truck for one day from either Company A or Company B. Company A charges an initial fee of $77.50 and an additional 30 cents for every mile driven. Company B charges an initial fee of $30 and an additional 80 cents for every mile driven.
Step 2 ：We need to find the number of miles (m) for which the cost of renting from Company A is less than or equal to the cost of renting from Company B. This can be represented as an inequality where the cost of renting from Company A is less than or equal to the cost of renting from Company B.
Step 3 ：The cost of renting from Company A is given by the equation \(77.50 + 0.30m\) and the cost of renting from Company B is given by the equation \(30 + 0.80m\).
Step 4 ：We can set these two equations equal to each other and solve for m to find the number of miles where the cost of renting from both companies is the same. Then, any number of miles less than this will result in Company A being cheaper.
Step 5 ：Solving the equation gives us \(m = 95\). This means that for 95 miles, the cost of renting from both companies is the same.
Step 6 ：Therefore, for any number of miles less than 95, the cost of renting from Company A will be less than the cost of renting from Company B.
Step 7 ：Final Answer: The number of miles for which Company A will charge no more than Company B is represented by the inequality \(\boxed{m \leq 95}\).
