Write a system of linear inequalities that describe the different numbers of tickets that can be sold

Question

For a concert event, there are $100 reserved seat tickets and $90 general admission tickets. There are 3000 reserved seats available, and fire regulations limit the number of paid ticket holders to 4000. The promoter must take in at least $367,500 in ticket sales. Write a system of linear inequalities that describe the different numbers of tickets that can be sold. (Let x to represent the number of reserved seat tickets and y to represent the number of general admission tickets.)

Answer 1

Firstly we are bound in that we can only have up to 3000 reserved seats and therefore 3000 reserved tickets:

x ≤ 3000.

If we can have at most 4000 attendees then we can only sell at most 4000 total tickets. Since x represents the number of reserved tickets and y represents the number of general admission tickets, the total amount of tickets to be sold is x + y. Since this has to be less than or equal to 4000:

x + y ≤ 4000.

This gives us two inequalities, but we can write one more. We know the promoter must take in at least $367,500. This means the total revenue of the ticket sales must be greater than or equal to 367500 dollars. If each reserved ticket is sold at $100 and we sell x amount of them, the revenue for the reserved tickets would be 100 times x; similarly for the general admission tickets their revenue would be 90 times y. Thus,

100x + 90y ≥ 367,500.

Answer 2

$367,500 in ticket sales means 100x + 90y ≥ 367500

3000 reserved seats are available means x ≤ 3000

Fire regulations limit the number of paid ticket holders to 4000 means x + y ≤ 4000