# How do you solve \frac { 7} { x + 6} + \frac { 5} { x - 6} = \frac { 6} { ( x + 6) ( x - 6) }?

May 21, 2017

$x = \frac{3}{2}$

$3$ is the numerator and $2$ is the denominator.

#### Explanation:

Multiply all terms to get $\left(x + 6\right) \left(x - 6\right)$ as the denominators and cancel:

7(x−6)+5(x+6)=6

12x−12=6" "(Simplify both sides of the equation)

12x−12+12=6+12" "(Add $12$ to both sides)

$12 x = 18$

$\frac{12 x}{12} = \frac{18}{12} \text{ }$(Divide both sides by $12$)

$x = \frac{3}{2}$

Check the answer. (Plug it in to make sure it works)

$x = \frac{3}{2} \text{ }$(Works in original equation)

$x = \frac{3}{2}$

($3$ is the numerator and $2$ is the denominator)

May 21, 2017

$x = \frac{3}{2}$

#### Explanation:

First, rule out the solutions $x = 6$ and $x = - 6$ since they will result in indeterminate forms (dividing by 0).

Now, multiply both sides by $\left(x + 6\right) \left(x - 6\right)$

$7 \left(x - 6\right) + 5 \left(x + 6\right) = 6 \text{ } x \notin \left\{- 6 , 6\right\}$

From here, distribute and solve normally, keeping in mind the extraneous solutions -6 and 6.

$7 x - 42 + 5 x + 30 = 6 \textcolor{w h i t e}{\text{XXXXX}} x \notin \left\{- 6 , 6\right\}$
$\textcolor{w h i t e}{\text{XXX/XX"12x-12=6 color(white)"XXXXX}} x \notin \left\{- 6 , 6\right\}$
$\textcolor{w h i t e}{\text{XXXXXXXX.." 12x=18 color(white)"XXXX.}} x \notin \left\{- 6 , 6\right\}$
$\textcolor{w h i t e}{\text{XXXXXXXXXX}} x = \frac{3}{2}$