Final answer:
The inequality to determine how many people are needed for Joe's Cooking to be a better deal than Party Time Caterers is 12p + 150 < 16p.
Step-by-step explanation:
To determine how many people, p, must be at the event for Joe's Cooking to be the better deal compared to Party Time Caterers, we must set up an inequality. Party Time Caterers charges a flat rate of $16 per person, while Joe's Cooking charges $12 per person plus a $150 set up fee. We need to find the number of people for which Joe's Cooking total cost will be less than Party Time Caterers' cost.
The inequality representing this scenario will be:
12p + 150 < 16p
Cancelling p on both sides and isolating p will give us the number of people needed for Joe's Cooking to cost less than Party Time Caterers, making it the more economical choice for Marnie's fundraising event.