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

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

Evaluate

$$2x$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{1+x}{1+x^{-1}}.

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

Cancel out x+1 in both numerator and denominator.

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

Divide 1 by \frac{1}{x} by multiplying 1 by the reciprocal of \frac{1}{x}.

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

Factor the expressions that are not already factored in \frac{1-x}{x^{-1}-1}.

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

Extract the negative sign in 1-x.

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

Cancel out -\left(x-1\right) in both numerator and denominator.

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

Divide 1 by \frac{1}{x} by multiplying 1 by the reciprocal of \frac{1}{x}.

$$x+x$$

Combine x and x to get 2x.

$$2x$$

Expand

$$2x$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{1+x}{1+x^{-1}}.

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

Cancel out x+1 in both numerator and denominator.

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

Divide 1 by \frac{1}{x} by multiplying 1 by the reciprocal of \frac{1}{x}.

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

Factor the expressions that are not already factored in \frac{1-x}{x^{-1}-1}.

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

Extract the negative sign in 1-x.

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

Cancel out -\left(x-1\right) in both numerator and denominator.

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

Divide 1 by \frac{1}{x} by multiplying 1 by the reciprocal of \frac{1}{x}.

$$x+x$$

Combine x and x to get 2x.

$$2x$$