For a school fundraiser, volunteers plan to sell raffle tickets. They must sell at least 200 tickets to cover expenses, but they printed no more than 500 tickets. Let represent the number of tickets sold.
Which inequality best represents the possible numbers of tickets that can be sold?
A compound inequality captures two limits on the same quantity. Translate “at least” into a lower bound and “no more than” into an upper bound, then put the variable between them. Both phrases allow their boundary values, so use inclusive signs. A one-sided inequality misses a limit, while strict signs exclude counts the wording allows.
Hints
- Hint 1
A lower bound is the smallest allowed value. “At least” lets the volunteers sell exactly the minimum and still cover expenses. Which inequality puts at or above that number?
- Hint 2
An upper bound is the largest allowed value. “No more than” allows exactly the maximum but nothing higher. How can you show this limit together with the lower bound?
Step-by-step
Translate both ticket limits
Step 1Set the minimum ticket count
No Desmos needed. The limit words tell you which counts are allowed.
At least means selling exactly covers expenses, and any higher count does too. So is an included lower bound:
- Step 2
Set the maximum ticket count
No more than means the volunteers can sell exactly tickets, but they can't sell more than they printed. So is an included upper bound:
- Step 3
Combine the two limits
The same must satisfy both limits, so put it between them in a compound inequality, a chain requiring both comparisons:
Both endpoints count because both limit phrases include equality. The possible whole-number ticket counts run from through , including both. Choice C.