you're selling the t-shirts at $15 a pop, so if you sell say Q quantity of shirts, your Revenue is 15Q.
your costs are a one-time setup fee of 50 bucks plus 25 bucks for shipping, so 75 bucks fixed cost, and you paid $6.50 to make each t-shirt, so if you make Q quantity of shirts, your costs will then be the 75 bucks plus 6.5Q.
the break-even point is where the revenue = costs, namely where R(Q) = C(Q).
[tex] \bf \begin{cases}
R(Q)=15Q\\
C(Q)=6.5Q+75
\end{cases}\implies \stackrel{revenue}{15Q}~=~\stackrel{costs}{6.5Q+75}
\\\\\\
8.5Q=75\implies Q=\cfrac{75}{8.5}\implies Q\approx 8.8235\implies \stackrel{rounded~up}{Q=8} [/tex]
we used 8, because we can't use 9, since it's less than 9 shirts, is 8.8, then again 8.8 shirts is 8 shirts and then you grab another one and ripped it up by 0.8, which we're not going to do, well, unless one client wants that.
