To add the mixed numbers \(-3 \frac{1}{6}\) and \(5 \frac{3}{4}\), we first convert them to improper fractions.
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Convert \(-3 \frac{1}{6}\): \[ -3 \frac{1}{6} = -\left(3 \times 6 + 1\right)/6 = -\left(18 + 1\right)/6 = -19/6 \]
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Convert \(5 \frac{3}{4}\): \[ 5 \frac{3}{4} = 5 \times 4 + 3/4 = (20 + 3)/4 = 23/4 \]
Now we add the two improper fractions: \[ -\frac{19}{6} + \frac{23}{4} \]
To add these fractions, we need a common denominator. The least common multiple of 6 and 4 is 12.
Convert both fractions to have a denominator of 12: \[ -\frac{19}{6} = -\frac{19 \times 2}{6 \times 2} = -\frac{38}{12} \] \[ \frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12} \]
Now add the two fractions: \[ -\frac{38}{12} + \frac{69}{12} = \frac{69 - 38}{12} = \frac{31}{12} \]
Now, we convert \(\frac{31}{12}\) back to a mixed number: \[ 31 \div 12 = 2 \quad \text{(whole number part)} \] with a remainder of \(31 - 2 \times 12 = 31 - 24 = 7\).
Thus, we can express \(\frac{31}{12}\) as: \[ 2 \frac{7}{12} \]
Therefore, the final answer is: \[ \boxed{2 \frac{7}{12}} \]