Solve (-1/x-1)(x/x+1+1)div(1-3x^2/1-x^2)

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

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in x+x+1.

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

Multiply -\frac{1}{x-1} times \frac{2x+1}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Factor 1-x^{2}.

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

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

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

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

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

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

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

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

$$\frac{\frac{-\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)}}{\frac{-4x^{2}+1}{\left(x-1\right)\left(-x-1\right)}}$$

Divide \frac{-\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)} by \frac{-4x^{2}+1}{\left(x-1\right)\left(-x-1\right)} by multiplying \frac{-\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)} by the reciprocal of \frac{-4x^{2}+1}{\left(x-1\right)\left(-x-1\right)}.

$$\frac{-\left(2x+1\right)\left(x-1\right)\left(-x-1\right)}{\left(x-1\right)\left(x+1\right)\left(-4x^{2}+1\right)}$$

Extract the negative sign in -x-1.

$$\frac{-\left(-1\right)\left(x-1\right)\left(x+1\right)\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)\left(-4x^{2}+1\right)}$$

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

$$\frac{-\left(-1\right)\left(2x+1\right)}{-4x^{2}+1}$$

Factor the expressions that are not already factored.

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

Extract the negative sign in 1+2x.

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

Cancel out -2x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

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

Combine like terms in x+x+1.

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

Multiply -\frac{1}{x-1} times \frac{2x+1}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Factor 1-x^{2}.

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

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

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

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

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

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

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

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

$$\frac{\frac{-\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)}}{\frac{-4x^{2}+1}{\left(x-1\right)\left(-x-1\right)}}$$

Divide \frac{-\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)} by \frac{-4x^{2}+1}{\left(x-1\right)\left(-x-1\right)} by multiplying \frac{-\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)} by the reciprocal of \frac{-4x^{2}+1}{\left(x-1\right)\left(-x-1\right)}.

$$\frac{-\left(2x+1\right)\left(x-1\right)\left(-x-1\right)}{\left(x-1\right)\left(x+1\right)\left(-4x^{2}+1\right)}$$

Extract the negative sign in -x-1.

$$\frac{-\left(-1\right)\left(x-1\right)\left(x+1\right)\left(2x+1\right)}{\left(x-1\right)\left(x+1\right)\left(-4x^{2}+1\right)}$$

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

$$\frac{-\left(-1\right)\left(2x+1\right)}{-4x^{2}+1}$$

Factor the expressions that are not already factored.

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

Extract the negative sign in 1+2x.

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

Cancel out -2x-1 in both numerator and denominator.

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