You can rent a car for the day from company A for 29.00...

You can rent a car for the day from company A for $\$ 29.00$ plus $\$ 0.12$ a mile. Company B charges $\$ 22.00$ plus $\$ 0.21$ a mile. Find the number of miles $m$ (to the nearest mile) per day for which it is cheaper to rent from company A.

Key Concepts

Linear Functions

Linear functions are mathematical models that represent relationships with a constant rate of change. They are expressed in the form y = mx + b, where m is the rate of change and b is the starting value or fixed cost. In cost analysis, they are used to represent both fixed fees and variable charges that depend on a unit such as miles driven.

Linear Inequalities

Linear inequalities involve comparing two linear functions to determine the range of values for which one function is greater or smaller than the other. In cost comparison problems, setting up an inequality helps determine at what point one pricing structure becomes more economical than another by comparing total costs as functions of a variable.

Break-even Analysis

Break-even analysis is the process of finding the point where two cost functions are equal. By solving for when the costs from different pricing models intersect, one can determine the threshold value beyond which one option is cheaper than the other. This concept is essential in decision-making and cost comparisons.

Rounding

Rounding is the process of adjusting numerical results to a specified degree of precision, often to simplify practical interpretations. In many real-world applications, such as determining the number of miles or items, values are rounded to the nearest whole number to provide a usable answer.

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