Solve (1/x+1+x-2/x^2-1)

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

Evaluate

$$\frac{2x-3}{x^{2}-1}$$

Short Solution Steps

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right). Multiply \frac{1}{x+1} times \frac{x-1}{x-1}.

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

Since \frac{x-1}{\left(x-1\right)\left(x+1\right)} and \frac{x-2}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-1+x-2.

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

Expand \left(x-1\right)\left(x+1\right).

$$\frac{2x-3}{x^{2}-1}$$

Expand

$$\frac{2x-3}{x^{2}-1}$$

Short Solution Steps

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right). Multiply \frac{1}{x+1} times \frac{x-1}{x-1}.

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

Since \frac{x-1}{\left(x-1\right)\left(x+1\right)} and \frac{x-2}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-1+x-2.

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

Expand \left(x-1\right)\left(x+1\right).

$$\frac{2x-3}{x^{2}-1}$$