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Understanding the cost structure of each plan is crucial to determine the most economical choice. In Plan A, the cost consists of a fixed daily rate combined with a variable cost that depends on the number of miles driven. Specifically, you pay \(30 per day plus \)0.18 per mile, meaning the total cost, \( C_A \), can be expressed as \( C_A = 30 + 0.18m \), where \( m \) represents the miles driven. On the other hand, Plan B offers a simple cost structure with a flat daily rate of $50. This plan includes unlimited mileage without any additional charges per mile. The key to solving this exercise is comparing these two cost structures to find out when Plan B becomes more financially favorable than Plan A.

In solving any problem involving linear inequalities, defining your variables is a fundamental step. In this exercise, the variable \( m \) represents the number of miles driven. Clear variable definition is crucial, as it allows us to establish a mathematical equation or inequality that corresponds to the problem scenario. The cost equations \( C_A = 30 + 0.18m \) and \( C_B = 50 \) are derived based on this variable, illustrating how costs change with miles driven under each plan. Without this step, you would be unable to frame the problem accurately or reach a valid solution.

Solving the inequality \( 50 < 30 + 0.18m \) is the goal to determine when Plan B is the cheaper option. This inequality shows us how \( Plan \ B \) becomes less expensive as the mileage increases beyond a certain point. We start by subtracting 30 from both sides of the inequality to get \( 20 < 0.18m \), streamlining the equation. Next, to isolate \( m \), divide both sides by 0.18, resulting in \( \frac{20}{0.18} < m \), equivalent to \( m > 111.11 \). Since you cannot drive a fractional mile, the conclusion is straightforward: Plan B is cost-effective only if you drive more than 111 miles, rounding up to at least 112 miles. This process of inequality solving is integral to finding the breakeven point in cost comparison scenarios like this one.