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A doctor’s office schedules 15-minute appointments and half-hour appointments for weekdays. The doctor limits these appointments to, at most, 30 hours per week. Write an inequality to represent the number of 15-minute appointments, x, and the number of half-hour appointments, y, the doctor may have in a week.
To write an inequality for the number of appointments the doctor can have in a week, we need to consider the time each appointment takes.
Let's first assign a value to each type of appointment:
- A 15-minute appointment takes 0.25 hours (15 minutes ÷ 60 minutes/hour).
- A half-hour appointment takes 0.5 hours.
Let x be the number of 15-minute appointments and y be the number of half-hour appointments.
The total time the doctor spends on 15-minute appointments would be 0.25x hours, and the total time spent on half-hour appointments would be 0.5y hours.
The total time spent on all appointments should not exceed 30 hours in a week, so the inequality would be:
0.25x + 0.5y ≤ 30
This inequality represents the number of 15-minute and half-hour appointments the doctor may have in a week while adhering to the 30-hour limit.