To solve the expression \(-12 \frac{2}{3} - (-8 \frac{5}{6})\), we can first rewrite it using the additive inverse property. The additive inverse of \(-8 \frac{5}{6}\) is \(8 \frac{5}{6}\). Thus, the expression becomes:
\[ -12 \frac{2}{3} + 8 \frac{5}{6} \]
Next, we need to convert both mixed numbers into improper fractions.
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Convert \(-12 \frac{2}{3}\) to an improper fraction: \[ -12 \frac{2}{3} = -\left(12 \times 3 + 2\right)/3 = -\left(36 + 2\right)/3 = -\frac{38}{3} \]
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Convert \(8 \frac{5}{6}\) to an improper fraction: \[ 8 \frac{5}{6} = \left(8 \times 6 + 5\right)/6 = \left(48 + 5\right)/6 = \frac{53}{6} \]
Now, replace the mixed numbers with the improper fractions in the expression:
\[ -\frac{38}{3} + \frac{53}{6} \]
To add these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. We will 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 the fractions:
\[ -\frac{76}{6} + \frac{53}{6} = \frac{-76 + 53}{6} = \frac{-23}{6} \]
Next, we can convert \(-\frac{23}{6}\) back to a mixed number. To do this, we divide 23 by 6:
\[ 23 \div 6 = 3 \quad \text{(with a remainder of 5)} \]
So, \(-\frac{23}{6}\) can be expressed as:
\[ -3 \frac{5}{6} \]
Thus, the final answer to the expression \(-12 \frac{2}{3} - (-8 \frac{5}{6})\) is:
\[ \boxed{-3 \frac{5}{6}} \]