Solve x-1/x+1-x+1/x-1*(1/2-x/4-frac{1}{4x})

$$\frac { x - 1 } { x + 1 } - \frac { x + 1 } { x - 1 } \cdot ( \frac { 1 } { 2 } - \frac { x } { 4 } - \frac { 1 } { 4 x } )$$

Evaluate

$$\frac{\left(x-1\right)\left(x^{2}+6x+1\right)}{4x\left(x+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{1}{2} times \frac{2}{2}.

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

Since \frac{2}{4} and \frac{x}{4} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 4x is 4x. Multiply \frac{2-x}{4} times \frac{x}{x}.

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

Since \frac{\left(2-x\right)x}{4x} and \frac{1}{4x} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Multiply \frac{x+1}{x-1} times \frac{2x-x^{2}-1}{4x} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{\left(x+1\right)\left(2x-x^{2}-1\right)}{\left(x-1\right)\times 4x}.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and -4x is -4x\left(x+1\right). Multiply \frac{x-1}{x+1} times \frac{-4x}{-4x}. Multiply \frac{\left(x-1\right)\left(x+1\right)}{-4x} times \frac{x+1}{x+1}.

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

Since \frac{\left(x-1\right)\left(-4\right)x}{-4x\left(x+1\right)} and \frac{\left(x-1\right)\left(x+1\right)\left(x+1\right)}{-4x\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-1\right)\left(-4\right)x-\left(x-1\right)\left(x+1\right)\left(x+1\right).

$$\frac{-4x^{2}+4x-x^{3}-2x^{2}-x+x^{2}+2x+1}{-4x\left(x+1\right)}$$

Combine like terms in -4x^{2}+4x-x^{3}-2x^{2}-x+x^{2}+2x+1.

$$\frac{-5x^{2}+5x-x^{3}+1}{-4x\left(x+1\right)}$$

Expand -4x\left(x+1\right).

$$\frac{-5x^{2}+5x-x^{3}+1}{-4x^{2}-4x}$$

Expand

$$\frac{x^{3}+5x^{2}-5x-1}{4x\left(x+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{1}{2} times \frac{2}{2}.

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

Since \frac{2}{4} and \frac{x}{4} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 4x is 4x. Multiply \frac{2-x}{4} times \frac{x}{x}.

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

Since \frac{\left(2-x\right)x}{4x} and \frac{1}{4x} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Multiply \frac{x+1}{x-1} times \frac{2x-x^{2}-1}{4x} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{\left(x+1\right)\left(2x-x^{2}-1\right)}{\left(x-1\right)\times 4x}.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and -4x is -4x\left(x+1\right). Multiply \frac{x-1}{x+1} times \frac{-4x}{-4x}. Multiply \frac{\left(x-1\right)\left(x+1\right)}{-4x} times \frac{x+1}{x+1}.

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

Since \frac{\left(x-1\right)\left(-4\right)x}{-4x\left(x+1\right)} and \frac{\left(x-1\right)\left(x+1\right)\left(x+1\right)}{-4x\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-1\right)\left(-4\right)x-\left(x-1\right)\left(x+1\right)\left(x+1\right).

$$\frac{-4x^{2}+4x-x^{3}-2x^{2}-x+x^{2}+2x+1}{-4x\left(x+1\right)}$$

Combine like terms in -4x^{2}+4x-x^{3}-2x^{2}-x+x^{2}+2x+1.

$$\frac{-5x^{2}+5x-x^{3}+1}{-4x\left(x+1\right)}$$

Expand -4x\left(x+1\right).

$$\frac{-5x^{2}+5x-x^{3}+1}{-4x^{2}-4x}$$