To determine which of the given options illustrates the identity property of 0, let's first understand the identity property of 0. The identity property of 0 states that adding 0 to any number results in the number itself.
### Analyzing Each Option:
1. Option 1:
[tex]\[
-3 \frac{1}{4} + 6 \frac{1}{2} = 3 \frac{1}{4}
\][/tex]
This option involves the addition of two non-zero numbers. Therefore, it does not demonstrate the identity property of 0. Here, we are performing a regular arithmetic addition.
2. Option 2:
[tex]\[
0 + \left(-7 \frac{5}{6}\right) = -7 \frac{5}{6}
\][/tex]
This options involves adding 0 to [tex]\(-7 \frac{5}{6}\)[/tex]. According to the identity property of 0, adding 0 to any number should result in the same number, which is exactly what is shown here. Thus, this equation satisfies the identity property of 0.
3. Option 3:
[tex]\[
\frac{22}{7} + \left(-\frac{22}{7}\right) = 0
\][/tex]
This option shows the addition of a number and its opposite, which equals 0. It does not illustrate the identity property of 0 because we are not adding 0 to a number; instead, we're showing the additive inverse property.
4. Option 4:
[tex]\[
\frac{1}{2} + \frac{1}{2} = 1
\][/tex]
This option also involves the addition of two non-zero numbers. Therefore, it does not demonstrate the identity property of 0.
By examining each option, only Option 2 correctly demonstrates the identity property of 0, where adding 0 to [tex]\(-7 \frac{5}{6}\)[/tex] results in [tex]\(-7 \frac{5}{6}\)[/tex].
Hence, the correct answer is:
[tex]\[
0 + \left(-7 \frac{5}{6}\right) = -7 \frac{5}{6}
\][/tex]
Thus, the option that shows an example of the identity property of 0 is Option 2.
