Solve 1/x+1+1/1-x-x-3/left(x-1right)left(x+1right)

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

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in -x+1+x+1.

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

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

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

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

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

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

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

Combine like terms in -2-x+3.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

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

Combine like terms in -x+1+x+1.

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

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

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

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

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

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

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

Combine like terms in -2-x+3.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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