Question

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.

Answer 1

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.