a. To represent this situation with a compound inequality, we need to consider the cost of the Halloween bags and the additional fee for filling them with candy. Let x represent the number of Halloween bags the students can purchase. Each Halloween bag costs $4.50, and there is an additional fee of $12.00 for filling the bags. Therefore, the total cost for x Halloween bags can be represented as:
Total Cost = Cost per Bag (4.50) * Number of Bags (x) + Additional Fee (12.00)
So, the compound inequality to represent the situation is:
75 ≤ 4.50x + 12.00 ≤ 180
b. Now, let's solve this compound inequality to find the possible values of x:
First, subtract 12 from all parts of the inequality:
63 ≤ 4.50x ≤ 168
To isolate x, divide all parts of the inequality by 4.50:
(63 / 4.50) ≤ (4.50x / 4.50) ≤ (168 / 4.50)
Simplify:
14 ≤ x ≤ 37.33
Now, you cannot purchase a fraction of a Halloween bag, so you need to round down to the nearest whole number since you can't buy a fraction of a bag. Therefore, the students can purchase between 14 and 37 Halloween bags to stay within their allowance. They can buy 14, 15, 16, 17, and so on, up to 37 bags.