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

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

Evaluate

$$\frac{4}{1-x^{2}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{4}{\left(x+1\right)\left(-x+1\right)}$$

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

$$\frac{4}{-x^{2}+1}$$

Expand

$$\frac{4}{1-x^{2}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{4}{\left(x+1\right)\left(-x+1\right)}$$

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

$$\frac{4}{-x^{2}+1}$$