To evaluate the expression \(-5 \frac{7}{12} + 3 \frac{9}{12} - 2 \frac{5}{12}\), we first convert each mixed number to an improper fraction.
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Convert \(-5 \frac{7}{12}\): \[ -5 \frac{7}{12} = -\left(5 \times 12 + 7\right) \div 12 = -\left(60 + 7\right) \div 12 = -\frac{67}{12} \]
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Convert \(3 \frac{9}{12}\): \[ 3 \frac{9}{12} = 3 \times 12 + 9 \div 12 = \frac{36 + 9}{12} = \frac{45}{12} \]
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Convert \(-2 \frac{5}{12}\): \[ -2 \frac{5}{12} = -\left(2 \times 12 + 5\right) \div 12 = -\left(24 + 5\right) \div 12 = -\frac{29}{12} \]
Now our expression becomes: \[ -\frac{67}{12} + \frac{45}{12} - \frac{29}{12} \]
We can combine these fractions since they have a common denominator: \[ -\frac{67 + 29 - 45}{12} = -\frac{67 + 29 - 45}{12} = -\frac{67 - 45 + 29}{12} \]
Calculating the numerator: \[ -67 + 45 + 29 = -67 + 74 = 7 \]
Putting it all together: \[ -\frac{7}{12} \]
Thus, the value of the expression \(-5 \frac{7}{12} + 3 \frac{9}{12} - 2 \frac{5}{12}\) is: \[ \boxed{-\frac{7}{12}} \]