Solve 1/2(frac{1{2x^-1})}+2(1/2x-1)/2(frac{1{2x}-1)+1}=

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

Evaluate

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

Short Solution Steps

Express 2\times \frac{1}{2x^{-1}} as a single fraction.

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

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

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

Cancel out 2 and 2.

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

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

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

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

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

Express 2\times \frac{1-2x}{2x} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

Express 2\times \frac{1-2x}{2x} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in -1+\frac{1}{x}-2x+1.

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

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

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

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

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

Do the multiplications in 1-2xx.

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

Express \frac{\frac{1-2x^{2}}{x}}{-x+1} as a single fraction.

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

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

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

Expand

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

Short Solution Steps

Express 2\times \frac{1}{2x^{-1}} as a single fraction.

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

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

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

Cancel out 2 and 2.

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

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

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

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

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

Express 2\times \frac{1-2x}{2x} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

Express 2\times \frac{1-2x}{2x} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in -1+\frac{1}{x}-2x+1.

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

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

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

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

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

Do the multiplications in 1-2xx.

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

Express \frac{\frac{1-2x^{2}}{x}}{-x+1} as a single fraction.

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

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

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