Question

ahmad is going to rent a truck for one day. there are two companies he can choose from, and they have the following prices. company a charges an initial fee of $55 and an additional 40 cents for every mile driven. company b has no initial fee but charges 60 cents for every mile driven. for what mileages will company a charge more than company b? use m for the number of miles driven, and solve your inequality for m.

Explanation:

Step1: Define Cost Functions

Company A's cost: \( 55 + 0.40m \) (initial fee + per - mile cost).
Company B's cost: \( 0.60m \) (per - mile cost, no initial fee).

Step2: Set Up Inequality

We want when Company A charges more than Company B, so:
\( 55 + 0.40m>0.60m \)

Step3: Solve Inequality

Subtract \( 0.40m \) from both sides:
\( 55>0.60m - 0.40m \)
\( 55 > 0.20m \)
Divide both sides by \( 0.20 \):
\( m<\frac{55}{0.20} \)
\( m < 275 \) (Wait, no, wait. Wait, we had \( 55+0.4m>0.6m \), so \( 55>0.6m - 0.4m=0.2m \), then \( m < \frac{55}{0.2}=275 \)? Wait, no, that would be when A is more than B? Wait, no, let's re - do. Wait, the inequality is \( 55 + 0.4m>0.6m \). Subtract \( 0.4m \) from both sides: \( 55>0.2m \). Then divide both sides by \( 0.2 \): \( m < 275 \)? Wait, that would mean when miles are less than 275, A is more? Wait, no, let's test with \( m = 0 \). Company A: \( 55 \), Company B: \( 0 \). So A is more. When \( m = 275 \), Company A: \( 55+0.4\times275=55 + 110 = 165 \), Company B: \( 0.6\times275 = 165 \). When \( m = 300 \), Company A: \( 55+0.4\times300=55 + 120 = 175 \), Company B: \( 0.6\times300 = 180 \). Wait, so when \( m>275 \), Company B is more? Wait, I made a mistake in the inequality direction. Wait, the question is "For what mileages will Company A charge more than Company B?". So the correct inequality is \( 55+0.4m>0.6m \). Let's solve it again.

\( 55+0.4m>0.6m \)

Subtract \( 0.4m \) from both sides:

\( 55>0.6m - 0.4m \)

\( 55>0.2m \)

Now, divide both sides by \( 0.2 \):

\( m < \frac{55}{0.2}=275 \)

Wait, but when \( m = 300 \), Company A: \( 55+0.4\times300 = 55+120 = 175 \), Company B: \( 0.6\times300 = 180 \). So A is less than B. When \( m = 200 \), Company A: \( 55 + 80=135 \), Company B: \( 120 \). So A is more. So the correct solution is \( m < 275 \). Wait, but the initial inequality setup: we want A's cost > B's cost. So \( 55 + 0.4m>0.6m \) leads to \( m < 275 \).

Wait, but let's re - express the costs. Company A: \( C_A=55 + 0.4m \), Company B: \( C_B = 0.6m \). We want \( C_A>C_B \), so \( 55+0.4m>0.6m \). Then \( 55>0.2m \), so \( m < 275 \). So for mileages less than 275 miles, Company A charges more than Company B.

Wait, but the problem says "For what mileages will Company A charge more than Company B?". So the inequality is \( 55 + 0.4m>0.6m \), which simplifies to \( m < 275 \).

Answer:

\( m < 275 \) (So the inequality is \( 55 + 0.4m>0.6m \), and the solution is \( m < 275 \))