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

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

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps for Solving Linear Equation

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

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

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

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

Combine like terms in x-1+1.

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

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

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

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

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

Variable x cannot be equal to 0 since division by zero is not defined. Divide \frac{x}{x-1} by \frac{x-1}{x} by multiplying \frac{x}{x-1} by the reciprocal of \frac{x-1}{x}.

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

Multiply x and x to get x^{2}.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}=x^{2}-2x+1$$

Subtract x^{2} from both sides.

$$x^{2}-x^{2}=-2x+1$$

Combine x^{2} and -x^{2} to get 0.

$$0=-2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$-2x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-2x=-1$$

Divide both sides by -2.

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

Fraction \frac{-1}{-2} can be simplified to \frac{1}{2} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{1}{2}$$