The greatest number of people that they may invite is 78.
Explanation
Suppose, the number of people that they may invite is [tex]x[/tex]
The reception hall charges $ 37 per person, so the cost for [tex]x[/tex] number of people [tex]=37x[/tex] dollar. It also charges a $ 100 cleanup fee.
Given that, the maximum budget for their reception is $3000. So, the equation will be......
[tex]37x+100\leq 3000\\ \\ 37x\leq 2900\\ \\ x\leq \frac{2900}{37}\\ \\ x\leq 78.378...[/tex]
As the number of people will be as a whole number, so the greatest number of people that they may invite and still stay within their budget is 78.
