To evaluate the expression \( 2 \frac{6}{7} - \frac{5}{7} + 1 \frac{4}{7} \), we will perform the operations step by step.
First, we convert the mixed numbers to improper fractions.
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Convert \( 2 \frac{6}{7} \) to an improper fraction: \[ 2 \frac{6}{7} = \frac{2 \times 7 + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7} \]
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Convert \( 1 \frac{4}{7} \) to an improper fraction: \[ 1 \frac{4}{7} = \frac{1 \times 7 + 4}{7} = \frac{7 + 4}{7} = \frac{11}{7} \]
Now we substitute these values back into the expression: \[ \frac{20}{7} - \frac{5}{7} + \frac{11}{7} \]
Next, we can perform the subtraction and addition using a common denominator (which is already 7): \[ \frac{20 - 5 + 11}{7} = \frac{20 + 11 - 5}{7} = \frac{26}{7} \]
Finally, we can convert the improper fraction \( \frac{26}{7} \) back to a mixed number: \[ \frac{26}{7} = 3 \frac{5}{7} \quad \text{(since \( 26 \div 7 = 3 \) remainder \( 5 \))} \]
Thus, the final result of the expression \( 2 \frac{6}{7} - \frac{5}{7} + 1 \frac{4}{7} \) is: \[ \boxed{3 \frac{5}{7}} \]