Find the Domain 1/(x^2-16)-6/(x-4)=1/(x+4)

$\frac{1}{x^{2} - 16} - \frac{6}{x - 4} = \frac{1}{x + 4}$

Step 1

Set the denominator in $\frac{1}{x^{2} - 16}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 16 = 0$

Step 2

Solve for $x$ .

Add $16$ to both sides of the equation.

$x^{2} = 16$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{16}$

Simplify $\pm \sqrt{16}$ .

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

Step 3

Set the denominator in $\frac{6}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 4

Add $4$ to both sides of the equation.

$x = 4$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4) \cup (- 4, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 4, - 4}$

Step 6