To solve \(-12 \frac{2}{3} - (-8 \frac{5}{6})\) using the additive inverse, we can first rewrite the expression:
- The additive inverse of \(-8 \frac{5}{6}\) is \(8 \frac{5}{6}\).
- Thus, the expression becomes:
\(-12 \frac{2}{3} + 8 \frac{5}{6}\).
Now, we need to perform the addition.
Step 1: Convert the mixed numbers to improper fractions.
\(-12 \frac{2}{3}\): \[ -12 \frac{2}{3} = -\left(12 \times 3 + 2\right)/3 = -\left(36 + 2\right)/3 = -\frac{38}{3} \]
\(8 \frac{5}{6}\): \[ 8 \frac{5}{6} = \left(8 \times 6 + 5\right)/6 = \left(48 + 5\right)/6 = \frac{53}{6} \]
Step 2: Find a common denominator and add.
The common denominator for \(\frac{38}{3}\) and \(\frac{53}{6}\) is \(6\).
First, convert \(-\frac{38}{3}\) to have a denominator of \(6\): \[ -\frac{38}{3} = -\frac{38 \times 2}{3 \times 2} = -\frac{76}{6} \]
Now, we can add \(-\frac{76}{6}\) and \(\frac{53}{6}\): \[ -\frac{76}{6} + \frac{53}{6} = \frac{-76 + 53}{6} = \frac{-23}{6} \]
Step 3: Convert \(\frac{-23}{6}\) back to a mixed number.
To convert \(\frac{-23}{6}\) to a mixed number:
- Divide \(23\) by \(6\): \(23 \div 6 = 3\) remainder \(5\).
- Thus, \(\frac{-23}{6} = -3 \frac{5}{6}\).
Therefore, the final answer is: \[ \boxed{-3 \frac{5}{6}} \]