Solve x-3/x^2-1divx^2-2x-3/x^2+2x+1+1/x-1=

$$\frac { x - 3 } { x ^ { 2 } - 1 } \div \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } + 2 x + 1 } + \frac { 1 } { x - 1 } =$$

Evaluate

$$\frac{2}{x-1}$$

Short Solution Steps

Divide \frac{x-3}{x^{2}-1} by \frac{x^{2}-2x-3}{x^{2}+2x+1} by multiplying \frac{x-3}{x^{2}-1} by the reciprocal of \frac{x^{2}-2x-3}{x^{2}+2x+1}.

$$\frac{\left(x-3\right)\left(x^{2}+2x+1\right)}{\left(x^{2}-1\right)\left(x^{2}-2x-3\right)}+\frac{1}{x-1}$$

Factor the expressions that are not already factored in \frac{\left(x-3\right)\left(x^{2}+2x+1\right)}{\left(x^{2}-1\right)\left(x^{2}-2x-3\right)}.

$$\frac{\left(x-3\right)\left(x+1\right)^{2}}{\left(x-3\right)\left(x-1\right)\left(x+1\right)^{2}}+\frac{1}{x-1}$$

Cancel out \left(x-3\right)\left(x+1\right)^{2} in both numerator and denominator.

$$\frac{1}{x-1}+\frac{1}{x-1}$$

Combine \frac{1}{x-1} and \frac{1}{x-1} to get 2\times \frac{1}{x-1}.

$$2\times \frac{1}{x-1}$$

Express 2\times \frac{1}{x-1} as a single fraction.

$$\frac{2}{x-1}$$

Expand

$$\frac{2}{x-1}$$

Short Solution Steps

Divide \frac{x-3}{x^{2}-1} by \frac{x^{2}-2x-3}{x^{2}+2x+1} by multiplying \frac{x-3}{x^{2}-1} by the reciprocal of \frac{x^{2}-2x-3}{x^{2}+2x+1}.

$$\frac{\left(x-3\right)\left(x^{2}+2x+1\right)}{\left(x^{2}-1\right)\left(x^{2}-2x-3\right)}+\frac{1}{x-1}$$

Factor the expressions that are not already factored in \frac{\left(x-3\right)\left(x^{2}+2x+1\right)}{\left(x^{2}-1\right)\left(x^{2}-2x-3\right)}.

$$\frac{\left(x-3\right)\left(x+1\right)^{2}}{\left(x-3\right)\left(x-1\right)\left(x+1\right)^{2}}+\frac{1}{x-1}$$

Cancel out \left(x-3\right)\left(x+1\right)^{2} in both numerator and denominator.

$$\frac{1}{x-1}+\frac{1}{x-1}$$

Combine \frac{1}{x-1} and \frac{1}{x-1} to get 2\times \frac{1}{x-1}.

$$2\times \frac{1}{x-1}$$

Express 2\times \frac{1}{x-1} as a single fraction.

$$\frac{2}{x-1}$$