Answer
$160$
Work Step by Step
Lets start by setting up the following data for company 1 based on the problem description.
$$
\begin{aligned}
\textbf{faxed setup charge: b} &=150 \\
\textbf{cost per T-shirt: m} &=4.5 \\
\textbf{ size of order} &= x \\
\textbf{total cost of T-shirt with logo} &= y_1
\end{aligned}
$$ The linear model for this option can be written as:
$$\begin{aligned}
y &=mx+b\\
y_1&= 4.5x+150.
\end{aligned}$$ We can do the same set up company 2 as follows:
$$
\begin{aligned}
\textbf{faxed setup charge: b} &=230 \\
\textbf{cost per T-shirt: m} &=4 \\
\textbf{ size of order} &= x \\
\textbf{total cost of T-shirt with logo} &= y_2.
\end{aligned}
$$ The linear model for this option can be written as: $$\begin{aligned}
y &=mx+b\\
y_2&= 4x+230.
\end{aligned}$$ Out target is to find the size of the number of T-shirt ordered, $x$, that will lead to the same cost charged at either company. We can do this by setting the two linear functions equal and solving for $x$. $$\begin{aligned}
y_1 &=y_2\\
4.5x+150&= 4x+230\\
4.5x-4x&= 230-150\\
0.5x & = 80\\
x&= \frac{80}{0.5}\\
&=160.
\end{aligned}$$ Check:
$$\begin{aligned}
y_1 &=4.5\cdot 160+150 = \$870\\
y_2&= 4\cdot 160+230 =\$870.
\end{aligned}$$ The size of the order should be $160$ so that the charge at either T-shirt company be the same.
