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\frac{\left(x^{2}-1\right)\times 3}{6\left(x-1\right)}
Divide \frac{x^{2}-1}{6} by \frac{x-1}{3} by multiplying \frac{x^{2}-1}{6} by the reciprocal of \frac{x-1}{3}.
\frac{x^{2}-1}{2\left(x-1\right)}
Cancel out 3 in both numerator and denominator.
\frac{\left(x-1\right)\left(x+1\right)}{2\left(x-1\right)}
Factor the expressions that are not already factored.
\frac{x+1}{2}
Cancel out x-1 in both numerator and denominator.
\frac{\left(x^{2}-1\right)\times 3}{6\left(x-1\right)}
Divide \frac{x^{2}-1}{6} by \frac{x-1}{3} by multiplying \frac{x^{2}-1}{6} by the reciprocal of \frac{x-1}{3}.
\frac{x^{2}-1}{2\left(x-1\right)}
Cancel out 3 in both numerator and denominator.
\frac{\left(x-1\right)\left(x+1\right)}{2\left(x-1\right)}
Factor the expressions that are not already factored.
\frac{x+1}{2}
Cancel out x-1 in both numerator and denominator.