To solve the expression \(-12 \frac{2}{3} - (-8 \frac{5}{6})\) using the additive inverse, we can change the subtraction of the negative into addition:
\[ -12 \frac{2}{3} + 8 \frac{5}{6} \]
Now, let's first convert the mixed numbers to improper fractions.
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Convert \(-12 \frac{2}{3}\): \[ -12 \frac{2}{3} = -\left(12 \times 3 + 2\right)/3 = -\frac{36 + 2}{3} = -\frac{38}{3} \]
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Convert \(8 \frac{5}{6}\): \[ 8 \frac{5}{6} = \left(8 \times 6 + 5\right)/6 = \frac{48 + 5}{6} = \frac{53}{6} \]
Next, we need to add these two improper fractions:
\[ -\frac{38}{3} + \frac{53}{6} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 6 is 6. We will convert \(-\frac{38}{3}\) to a fraction with a denominator of 6:
\[ -\frac{38}{3} = -\frac{38 \times 2}{3 \times 2} = -\frac{76}{6} \]
Now we can add the two 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. Since \(-23\) is less than \(-18\) and \(-6\) (which would be \(-3\)), we can express \(-\frac{23}{6}\) as:
\[ -\frac{23}{6} = -3 \frac{5}{6} \]
So, the final answer as a mixed number is:
\[ \boxed{-3 \frac{5}{6}} \]