A community theater sells child tickets and adult tickets to a play. The group purchasing the tickets must buy at least 5 tickets in total, but they want to spend no more than $60. Child tickets cost $6 each and adult tickets cost $10 each. Which of the following systems of inequalities models this situation?
A system of inequalities puts two limits on the same purchase. A minimum ticket count needs , while a maximum budget needs . Build the count by adding tickets and the bill by multiplying each price by its ticket count. Keep tickets and dollars separate so each limit applies to the right total.
Hints
- Hint 1
A minimum includes its boundary: at least allows exactly tickets or more. Add the child and adult counts to get the total. Which inequality sign keeps and every larger count?
- Hint 2
A budget limits dollars spent, not tickets bought. Multiply each ticket count by its price, then add the two costs. Which inequality sign allows a bill of exactly $60 but rules out a higher bill?
Step-by-step
Match each limit to its units
Step 1Set the minimum ticket count
No Desmos needed. This question asks you to translate the limits, not solve for ticket counts. Adding the child tickets and adult tickets gives the total. “At least ” includes and every larger count, so write:
- Step 2
Find the total cost
Each child ticket costs $6, so child tickets cost dollars. Each adult ticket costs $10, so adult tickets cost dollars. Add them to get the total cost:
That's a dollar amount, not a ticket count.
- Step 3
Apply the spending limit
“No more than $60” allows a bill of exactly $60 but not a higher one. Cap the total cost:
Compare tickets with tickets and dollars with dollars. Together, and model both requirements. Choice D.