To solve the expression \(-12 \frac{2}{3} - (-8 \frac{5}{6})\), we can first rewrite it using the additive inverse.
The additive inverse of \(-8 \frac{5}{6}\) is \(8 \frac{5}{6}\). Therefore, the expression becomes:
\[ -12 \frac{2}{3} + 8 \frac{5}{6} \]
Now, we need to add these two mixed numbers together. First, we can convert them to improper fractions.
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Convert \(-12 \frac{2}{3}\) to an improper fraction: \[ -12 \frac{2}{3} = -\left(12 \cdot 3 + 2\right)/3 = -\left(36 + 2\right)/3 = -\frac{38}{3} \]
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Convert \(8 \frac{5}{6}\) to an improper fraction: \[ 8 \frac{5}{6} = \left(8 \cdot 6 + 5\right)/6 = \frac{48 + 5}{6} = \frac{53}{6} \]
Now, we will add \(-\frac{38}{3}\) and \(\frac{53}{6}\).
Before adding, we need a common denominator. The least common multiple of 3 and 6 is 6.
Convert \(-\frac{38}{3}\) to have a denominator of 6: \[ -\frac{38}{3} = -\frac{38 \cdot 2}{3 \cdot 2} = -\frac{76}{6} \]
Now we can add the fractions: \[ -\frac{76}{6} + \frac{53}{6} = \frac{-76 + 53}{6} = \frac{-23}{6} \]
Now, we convert \(\frac{-23}{6}\) back to a mixed number: \(-23 \div 6 = -3\) remainder \(-5\), so this can be written as: \[ -3 \frac{5}{6} \]
Thus, the final answer is: \[ \boxed{-3 \frac{5}{6}} \]