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

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

Evaluate

$$\frac{3}{x+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{x}{x-1} times \frac{x+1}{x+1}. Multiply \frac{2}{x+1} times \frac{x-1}{x-1}.

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

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

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

Do the multiplications in x\left(x+1\right)+2\left(x-1\right).

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

Combine like terms in x^{2}+x+2x-2.

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

Factor x^{2}-1.

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

Since \frac{x^{2}+3x-2}{\left(x-1\right)\left(x+1\right)} and \frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}+3x-2-\left(x^{2}+1\right).

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

Combine like terms in x^{2}+3x-2-x^{2}-1.

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

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

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

Cancel out x-1 in both numerator and denominator.

$$\frac{3}{x+1}$$

Expand

$$\frac{3}{x+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{x}{x-1} times \frac{x+1}{x+1}. Multiply \frac{2}{x+1} times \frac{x-1}{x-1}.

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

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

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

Do the multiplications in x\left(x+1\right)+2\left(x-1\right).

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

Combine like terms in x^{2}+x+2x-2.

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

Factor x^{2}-1.

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

Since \frac{x^{2}+3x-2}{\left(x-1\right)\left(x+1\right)} and \frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}+3x-2-\left(x^{2}+1\right).

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

Combine like terms in x^{2}+3x-2-x^{2}-1.

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

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

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

Cancel out x-1 in both numerator and denominator.

$$\frac{3}{x+1}$$