Sure, let's outline the constraints given Alan's situation:
1. Alan uses [tex]\(\frac{1}{3}\)[/tex] cup of oil for each tray of blueberry muffins and [tex]\(\frac{1}{2}\)[/tex] cup of oil for each tray of bran muffins. Given that he has 4 cups of oil available, he needs to satisfy the constraint for oil usage:
[tex]\[
\frac{1}{3}x + \frac{1}{2}y \leq 4
\][/tex]
2. For eggs, Alan uses 2 eggs for each tray of blueberry muffins and 1 egg for each tray of bran muffins. Given that he has 12 eggs on hand, this results in the constraint for egg usage:
[tex]\[
2x + y \leq 12
\][/tex]
3. Alan cannot bake a negative number of trays for either type of muffin, so the constraints for the number of trays baked are:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
Combining these constraints, we have the following system of inequalities summarizing the problem:
[tex]\[
\begin{array}{l}
\frac{1}{3}x + \frac{1}{2}y \leq 4 \\
2x + y \leq 12 \\
x \geq 0 \\
y \geq 0 \\
\end{array}
\][/tex]
Thus, these constraints must be satisfied in order to maximize the money raised at the bake sale.
