Final answer:
The theatre club needs to sell tickets such that their total revenue is at least $400, with inequality $4a + $5d ≥ $400. If 40 tickets are sold in advance, at least 48 tickets must be sold at the door to meet the goal.
Step-by-step explanation:
To write an inequality for the number of tickets the theater club needs to sell to raise at least $400, let's denote the number of advance tickets as a and the number of tickets sold at the door as d. Since tickets cost $4 in advance and $5 at the door, the inequality representing the total revenue R would be $4a + $5d ≥ $400.
If the club sells 40 tickets in advance, we can substitute a = 40 into the inequality to find out the least number of tickets they need to sell at the door. The revenue from the advance tickets is $4 × 40 = $160. Subtracting this from the $400 goal, we have $400 - $160 = $240 needed from door sales. Thus, the inequality becomes $5d ≥ $240, and solving for d yields d ≥ $240/$5, so d ≥ 48.
Therefore, the theater club needs to sell at least 48 tickets at the door to meet their goal, assuming they sell 40 tickets in advance.