Given:
Total budget = $3500
The hall charges a $50 clean up fee plus $38 per person
Let the number of persons = x
So, the total cost = 38x + 50
The greatest number of people will follow the inequality:
![38x+50\le3500](https://img.qammunity.org/2023/formulas/mathematics/college/l0gbg6ve2qbjhp5a727jjzp3qrlp2xmer1.png)
solve for x:
![\begin{gathered} 38x\le3500-50 \\ 38x\le3450 \\ x\le(3450)/(38) \\ \\ x\le90.789 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/p0w1dn7gzw9tzukwtjzj42r3ll141vzsfk.png)
So, the answer will be:
The greatest number of people they can invite = 90 persons