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\frac{3}{6}-\frac{4}{6}-\frac{1\times 6+1}{6}
Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{2}{3} to fractions with denominator 6.
\frac{3-4}{6}-\frac{1\times 6+1}{6}
Since \frac{3}{6} and \frac{4}{6} have the same denominator, subtract them by subtracting their numerators.
-\frac{1}{6}-\frac{1\times 6+1}{6}
Subtract 4 from 3 to get -1.
-\frac{1}{6}-\frac{6+1}{6}
Multiply 1 and 6 to get 6.
-\frac{1}{6}-\frac{7}{6}
Add 6 and 1 to get 7.
\frac{-1-7}{6}
Since -\frac{1}{6} and \frac{7}{6} have the same denominator, subtract them by subtracting their numerators.
\frac{-8}{6}
Subtract 7 from -1 to get -8.
-\frac{4}{3}
Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.