Solve 1/x+1-1/x/1-1{2+frac{x/x-1}}

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-x-1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in 2x-2+x.

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

Divide 1 by \frac{3x-2}{x-1} by multiplying 1 by the reciprocal of \frac{3x-2}{x-1}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3x-2}{3x-2}.

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

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

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

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

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

Combine like terms in 3x-2-x+1.

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

Divide \frac{-1}{x\left(x+1\right)} by \frac{2x-1}{3x-2} by multiplying \frac{-1}{x\left(x+1\right)} by the reciprocal of \frac{2x-1}{3x-2}.

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

To find the opposite of 3x-2, find the opposite of each term.

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

The opposite of -2 is 2.

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

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

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

Apply the distributive property by multiplying each term of x^{2}+x by each term of 2x-1.

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

Combine -x^{2} and 2x^{2} to get x^{2}.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-x-1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in 2x-2+x.

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

Divide 1 by \frac{3x-2}{x-1} by multiplying 1 by the reciprocal of \frac{3x-2}{x-1}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3x-2}{3x-2}.

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

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

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

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

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

Combine like terms in 3x-2-x+1.

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

Divide \frac{-1}{x\left(x+1\right)} by \frac{2x-1}{3x-2} by multiplying \frac{-1}{x\left(x+1\right)} by the reciprocal of \frac{2x-1}{3x-2}.

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

To find the opposite of 3x-2, find the opposite of each term.

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

The opposite of -2 is 2.

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

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

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

Apply the distributive property by multiplying each term of x^{2}+x by each term of 2x-1.

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

Combine -x^{2} and 2x^{2} to get x^{2}.

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