Solving Inequality with Integer Constraints: A Bus Rental Problem

Renting Buses: Solving an Inequality Problem

This problem involves finding the possible values of 'm' that satisfy the given inequality and then using these values to determine the ideal combination of large and medium-sized buses to rent.

The Problem:

A school needs to rent buses for a trip. They can rent large buses, which hold a certain number of people, and medium-sized buses with a smaller capacity. The rental cost is represented by the following inequality:

1500(11-2m) + 1200(3m-11) ≤ 6000

Where:

• 'm' represents a positive integer.

Solution:

1. Solve the Inequality: First, simplify and solve the inequality: 1500(11-2m) + 1200(3m-11) ≤ 6000 16500 - 3000m + 3600m - 13200 ≤ 6000 600m + 3300 ≤ 6000 600m ≤ 2700 m ≤ 4.5

2. Identify Possible Integer Values: Since 'm' must be a positive integer, the possible values are 1, 2, 3, and 4.

3. Apply to the Bus Rental Scenario: Let's assume:

   • 'x' represents the number of large buses
   • 'y' represents the number of medium-sized buses

   We need additional information to connect 'm' to 'x' and 'y' and determine the exact combination of buses for each value of 'm'. This information might be given in the form of equations relating 'm', 'x', and 'y' or additional constraints on the number of buses needed.

   For example, if we were given the equations:

   • x = 11 - 2m
   • y = 3m - 11

   We could then substitute each possible value of 'm' into these equations to find the corresponding values of 'x' and 'y' and determine which solution(s) are feasible in the context of the problem.

Conclusion:

This problem demonstrates the process of solving an inequality with integer constraints. By carefully considering the possible values of 'm' and applying them back to the context of the problem, we can work towards finding the optimal solution for renting the appropriate number of buses.