Manipulating Rational Expressions

We want to add the given rational expressions. $$\begin{array}{c}\frac{5x}{x^2-9}+\frac{2}{x+4}\end{array}$$ To do so, we will start by factoring the denominators. Let's factor the first one. Note that the second denominator is already factored. $$\begin{array}{c}\frac{5x}{x^2-9}+\frac{2}{x+4}\\ \Updownarrow \\ \frac{5x}{(x+3)(x-3)}+\frac{2}{x+4}\end{array}$$ The least common denominator (LCD) is $(x+3)(x-3)(x+4).$ We can add the expressions by rewriting each of them using the LCD. We will now identify the restrictions from the denominator of the simplified expression and from any other denominator used. For simplicity, we will use their factored forms.

Denominator	Restrictions on the denominator	Restrictions on the variable
$(x+3)(x-3)$	$x+3\ne 0$ and $x-3\ne 0$	$x\ne \text{-}3$ and $x\ne 3$
$x+4$	$x+4\ne 0$	$x\ne \text{-}4$
$(x+3)(x-3)(x+4)$	$x+3\ne 0,$ $x-3\ne 0$ and $x+4\ne 0$	$x\ne \text{-}3,$ $x\ne 3$ and $x\ne \text{-}4$

We found three restrictions on the variable. $$\begin{array}{c}x\ne \text{-}4,x\ne \text{-}3,\text{ and }x\ne 3\end{array}$$