Answer
The simplified form of the expression $\frac{x}{x-2}+\frac{2}{2-x}$ is 1.
Work Step by Step
$\frac{x}{x-2}+\frac{2}{2-x}$
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,
$\frac{x}{x-2}+\frac{2}{2-x}=1$
