To solve \( -12 \frac{3}{4} - (-8 \frac{5}{6}) \), we can first rewrite the expression using the additive inverse. When we subtract a negative number, it is equivalent to adding its positive counterpart.
Thus, we can rewrite the problem as follows:
\[ -12 \frac{3}{4} + 8 \frac{5}{6} \]
Next, let's convert the mixed numbers to improper fractions.
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Convert \( -12 \frac{3}{4} \): \[ -12 \frac{3}{4} = -\left(12 \times 4 + 3\right) = -\left(48 + 3\right) = -\frac{51}{4} \]
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Convert \( 8 \frac{5}{6} \): \[ 8 \frac{5}{6} = 8 \times 6 + 5 = 48 + 5 = \frac{53}{6} \]
Now substitute these back into the expression:
\[ -\frac{51}{4} + \frac{53}{6} \]
To add these fractions, we need a common denominator. The least common multiple of 4 and 6 is 12.
Convert each fraction to have a denominator of 12:
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Convert \( -\frac{51}{4} \): \[ -\frac{51}{4} = -\frac{51 \times 3}{4 \times 3} = -\frac{153}{12} \]
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Convert \( \frac{53}{6} \): \[ \frac{53}{6} = \frac{53 \times 2}{6 \times 2} = \frac{106}{12} \]
Now we can add the two fractions:
\[ -\frac{153}{12} + \frac{106}{12} = \frac{-153 + 106}{12} = \frac{-47}{12} \]
Now, we convert \( \frac{-47}{12} \) into a mixed number.
Since \( 47 \) divided by \( 12 \) is \( 3 \) with a remainder of \( 11 \), we can express this as:
\[ -\frac{47}{12} = -3 \frac{11}{12} \]
Thus, the final answer is:
\[ \boxed{-3 \frac{11}{12}} \]