Manipulating Rational Expressions

We want to subtract the given expressions. To do so, we will first determine the least common denominator (LCD). $$\begin{array}{c}\frac{9}{x-3}+\frac{2x}{x+1}\end{array}$$ Since the denominators are already factored, we can immediately determine the LCD as their product, which is $(x-3)(x+1).$ Now, we can subtract the expressions by rewriting each of them using the LCD.

Denominator	Restrictions on the denominator	Restrictions on the variable
$x-3$	$x-3\ne 0$	$x\ne 3$
$x+1$	$x+1\ne 0$	$x\ne \text{-}1$
$(x-3)(x+1)$	$x-3\ne 0$ and $x+1\ne 0$	$x\ne 3$ and $x\ne \text{-}1$

We found two restrictions on the variable. $$\begin{array}{c}x\ne \text{-}1\text{and}x\ne 3\end{array}$$