The drama club is selling tickets to their play to raise money for the show's expenses. Each student ticket sells for $8 and each adult ticket sells for $12. The auditorium can hold at most 131 people. The drama club must make at least $1400 from ticket sales to cover the show's costs. If 82 adult tickets were sold, determine all possible values for the number of student tickets that the drama club must sell in order to meet the show's expenses.
Let the number of student tickets sold be represented by the variable $x$.
Since each student ticket sells for $8, the total revenue from student tickets is $8x.
Since each adult ticket sells for $12, the total revenue from adult tickets is $12(82) = $984.
Therefore, the total revenue from all tickets is $8x + $984.
We know that the drama club must make at least $1400 from ticket sales. This can be represented as the inequality:
$8x + $984 ≥ $1400
Solving for $x, we get:
$8x ≥ $416
x ≥ 52
So, the drama club must sell at least 52 student tickets in order to meet the show's expenses.