Answer
a) $2.1x-325>0$
b) $x>154.76$; profit is made when at least $155$ boxes of popcorn are sold
Work Step by Step
a) Each package of the $75$ empty boxes costs $\$0.25$ per box. With four boxes bought, then the cost for the empty boxes is
$$
4(75)(0.25)=75\text{ dollars}
.$$
The fee charged by the company is $\$250$ per game plus $\$0.15$ per box of popcorn sold. Therefore, the cost due to the company is
$$
250+0.15x
.$$
Adding the cost in buying the four packages with the cost due to the company, then the total cost, $TC$, of the athletic boosters is
$$\begin{aligned}
TC&=250+0.15x+75
\\&=
325+0.15x
.\end{aligned}
$$
Since the athletic boosters charge customers $\$2.25$ per box of popcorn sold, then the income, $I$, is
$$
I=2.25x
.$$
To make a profit, the income, $I$, less the total cost, $TC$, should be greater than zero. That is,
$$\begin{aligned}
I-TC&>0
\\
2.25x-(325+0.15x)&>0
\\
2.25x-325-0.15x&>0
\\
2.1x-325&>0
.\end{aligned}
$$
Hence, the inequality that represents the boosters making a profit is
$$
2.1x-325>0
.$$
b) Using the properties of inequality, the solution to $2.1x-325>0$ is
$$\begin{aligned}
2.1x&>325
\\
x&>\frac{325}{2.1}
\\
x&>154.76
.\end{aligned}
$$
Hence, the athletic boosters will make a profit if they sell at least $155$ boxes of popcorn.
