!<Answer>!
To solve the system of inequalities graphically, let's first define our variables:
a: number of student tickets sold
y: number of adult tickets sold
Given the information provided, we can write the following inequalities:
1. The total number of tickets sold cannot exceed the capacity of the auditorium:
a + y ≤ 110
2. The drama club must make a minimum of $790 from ticket sales:
6a + 10y ≥ 790
3. The number of student tickets sold cannot exceed 20:
a ≤ 20
Now, let's graph these inequalities on a coordinate plane:
For the first inequality, a + y ≤ 110, we draw the line a + y = 110. This line represents the boundary where the total number of tickets sold is equal to the maximum capacity of the auditorium. Since the inequality is inclusive, we shade the region below or on the line.
For the second inequality, 6a + 10y ≥ 790, we draw the line 6a + 10y = 790. This line represents the boundary where the total revenue from ticket sales is equal to the minimum required amount. Since the inequality is greater than or equal to, we shade the region above or on the line.
For the third inequality, a ≤ 20, we draw the vertical line a = 20. This line represents the boundary where the number of student tickets sold cannot exceed 20. Since the inequality is inclusive, we shade the region to the left or on the line.
The solution to the system of inequalities is the region where all the shaded areas overlap. By examining the graph, we can find one possible solution within the overlapping region, which corresponds to a specific combination of student and adult tickets sold that satisfies all the given conditions.
