Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

The simplified form of the expression $\frac{x}{x-2}+\frac{2}{2-x}$ is 1.
$\frac{x}{x-2}+\frac{2}{2-x}$ Obtain the alternative form of the expression by multiplying the rational expression $\frac{x}{x-2}+\frac{2}{2-x}$ by 1 in the form $\frac{-1}{-1}$. \begin{align} & \frac{x}{x-2}+\frac{2}{2-x}=\frac{x}{x-2}+\frac{2}{2-x}.1 \\ & =\frac{x}{x-2}+\frac{2}{2-x}.\frac{-1}{-1} \end{align} Apply the Distributive property: \begin{align} & \frac{x}{x-2}+\frac{2}{2-x}=\frac{x}{x-2}+\frac{2}{2-x}.\frac{-1}{-1} \\ & =\frac{x}{x-2}+\frac{-2}{-\left( 2-x \right)} \\ & =\frac{x}{x-2}+\frac{-2}{x-2} \end{align} Now, the denominators are same. So, add the numerators and keep the common denominator: \begin{align} & \frac{x}{x-2}+\frac{2}{2-x}=\frac{x}{x-2}+\frac{-2}{x-2} \\ & =\frac{x-2}{x-2} \end{align} Remove the factor equal to 1, $\frac{x}{x-2}+\frac{2}{2-x}=1$