The outside of a closed glass display case measures 22 inches by 15 inches by 12 inches. The glass is [tex]$\frac{1}{2}$[/tex] inch thick. How much air is contained in the case?

To find out how much air is contained in the glass display case, we need to calculate the volume of the space inside the case. Here are the steps:

1. Identify the external dimensions:
- Length: 22 inches
- Width: 15 inches
- Height: 12 inches

2. Consider the thickness of the glass:
- The glass is [tex]$\frac{1}{2}$[/tex] inch thick. Since the thickness is on all sides, we need to reduce the dimensions of the internal space by 1 inch (0.5 inch from each side).

3. Calculate the internal dimensions:
- Internal Length = External Length - 2 [tex]$\times$[/tex] Thickness
[tex]\[
22 - 2 \times 0.5 = 21 \text{ inches}
\][/tex]
- Internal Width = External Width - 2 [tex]$\times$[/tex] Thickness
[tex]\[
15 - 2 \times 0.5 = 14 \text{ inches}
\][/tex]
- Internal Height = External Height - 2 [tex]$\times$[/tex] Thickness
[tex]\[
12 - 2 \times 0.5 = 11 \text{ inches}
\][/tex]

4. Calculate the internal volume:
- Volume = Internal Length [tex]$\times$[/tex] Internal Width [tex]$\times$[/tex] Internal Height
[tex]\[
21 \times 14 \times 11 = 3234 \text{ cubic inches}
\][/tex]

So, the amount of air contained in the case is 3234 cubic inches.