A local theater seats 160 people. Adult tickets cost $15 each and student tickets cost $9 each. For an upcoming performance, the theater must collect at least $1,800 in ticket revenue. If is the number of adult tickets and is the number of student tickets sold, which system of inequalities represents these constraints?
A system of inequalities describes limits that must hold at the same time. For ticket counts, write one expression for seats used and another for money collected, then match each to its limit. “At most” means no greater than; “at least” means no less than. Don’t mistake a seating capacity for a required sellout.
Hints
- Hint 1
A capacity is an upper limit, not a promise that every seat is filled. Since each ticket uses one seat, what inequality keeps the total number of tickets within the available seats?
- Hint 2
For revenue, multiply the number of each kind of ticket by its price, then add. “At least” includes the exact target; which inequality symbol includes that amount and every amount above it?
Step-by-step
Translate each limit
Step 1Limit the number of tickets
No Desmos needed. You’re translating the theater’s limits, not solving a given inequality. Each adult or student ticket uses one seat, so is the number of seats used. The theater can sell 160 or fewer tickets; it doesn’t have to sell out. So the capacity gives .
- Step 2
Set the revenue minimum
Adult tickets bring in dollars, and student tickets bring in dollars, so total revenue is . The theater needs at least $1,800, including exactly $1,800, so . Choose the system that stays within capacity while meeting that revenue minimum. Choice A.