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

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

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in x+1-x+2.

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

Multiply \frac{1}{3} times \frac{3}{\left(x+1\right)\left(-x+2\right)} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply 3 by x+1.

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

Apply the distributive property by multiplying each term of 3x+3 by each term of -x+2.

$$\frac{3}{-3x^{2}+6x-3x+6}$$

Combine 6x and -3x to get 3x.

$$\frac{3}{-3x^{2}+3x+6}$$

Expand

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

Short Solution Steps

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

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

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

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

Combine like terms in x+1-x+2.

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

Multiply \frac{1}{3} times \frac{3}{\left(x+1\right)\left(-x+2\right)} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply 3 by x+1.

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

Apply the distributive property by multiplying each term of 3x+3 by each term of -x+2.

$$\frac{3}{-3x^{2}+6x-3x+6}$$

Combine 6x and -3x to get 3x.

$$\frac{3}{-3x^{2}+3x+6}$$