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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

$$\frac{2}{\frac{2x-4+2}{x-2}}+\frac{2}{2-\frac{2}{x-2}}$$

Combine like terms in 2x-4+2.

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

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

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

Factor the expressions that are not already factored in \frac{2\left(x-2\right)}{2x-2}.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

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

$$\frac{x-2}{x-1}+\frac{2}{\frac{2x-4-2}{x-2}}$$

Combine like terms in 2x-4-2.

$$\frac{x-2}{x-1}+\frac{2}{\frac{2x-6}{x-2}}$$

Divide 2 by \frac{2x-6}{x-2} by multiplying 2 by the reciprocal of \frac{2x-6}{x-2}.

$$\frac{x-2}{x-1}+\frac{2\left(x-2\right)}{2x-6}$$

Factor the expressions that are not already factored in \frac{2\left(x-2\right)}{2x-6}.

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

Cancel out 2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x-3 is \left(x-3\right)\left(x-1\right). Multiply \frac{x-2}{x-1} times \frac{x-3}{x-3}. Multiply \frac{x-2}{x-3} times \frac{x-1}{x-1}.

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

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

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

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

$$\frac{x^{2}-3x-2x+6+x^{2}-x-2x+2}{\left(x-3\right)\left(x-1\right)}$$

Combine like terms in x^{2}-3x-2x+6+x^{2}-x-2x+2.

$$\frac{2x^{2}-8x+8}{\left(x-3\right)\left(x-1\right)}$$

Expand \left(x-3\right)\left(x-1\right).

$$\frac{2x^{2}-8x+8}{x^{2}-4x+3}$$

Factor

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