Manipulating Rational Expressions

We want to subtract the given expressions. To do so, we will first determine the least common denominator (LCD). $$\begin{array}{c}\frac{3}{x+4}-\frac{1}{x+6}\end{array}$$ Since the denominators are already factored, we can immediately determine the LCD as their product, which is $(x+4)(x+6).$ Now, we can subtract 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
$x+4$	$x+4\ne 0$	$x\ne \text{-}4$
$x+6$	$x+6\ne 0$	$x\ne \text{-}6$
$(x+4)(x+6)$	$x+4\ne 0$ and $x+6\ne 0$	$x\ne \text{-}4$ and $x\ne \text{-}6$

We found two restrictions on the variable. $$\begin{array}{c}x\ne \text{-}4\text{and}x\ne \text{-}6\end{array}$$