To find the value of \(-24 \frac{4}{5} - 6 \frac{7}{10}\) using the additive inverse, we can rewrite the subtraction as addition of the additive inverse.
First, we convert the mixed numbers into improper fractions:
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For \(-24 \frac{4}{5}\): \[ -24 \frac{4}{5} = -\left(24 \cdot 5 + 4\right) / 5 = -\frac{120 + 4}{5} = -\frac{124}{5} \]
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For \(-6 \frac{7}{10}\): \[ -6 \frac{7}{10} = -\left(6 \cdot 10 + 7\right) / 10 = -\frac{60 + 7}{10} = -\frac{67}{10} \]
Now, we convert \(-\frac{67}{10}\) to have a common denominator with \(-\frac{124}{5}\):
- The common denominator for 10 and 5 is 10. Thus, we convert \(-\frac{124}{5}\): \[ -\frac{124}{5} = -\frac{124 \cdot 2}{5 \cdot 2} = -\frac{248}{10} \]
Now, we can add the two fractions: \[ -\frac{248}{10} - \frac{67}{10} = -\frac{248 + 67}{10} = -\frac{315}{10} \]
Now, we can simplify \(-\frac{315}{10}\):
- Divide both the numerator and the denominator by 5: \[ -\frac{315 \div 5}{10 \div 5} = -\frac{63}{2} \]
Now convert this back to a mixed number: \[ -\frac{63}{2} = -31 \frac{1}{2} \]
Thus, the final answer is: \[ \boxed{-31 \frac{1}{2}} \]