Linear Algebra Examples

$\frac{x + 3}{x - 4} \cdot \frac{x^{2} - 2 x - 8}{x^{2} - 9}$

Step 1

Set the denominator in $\frac{x + 3}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

Set the denominator in $\frac{x^{2} - 2 x - 8}{x^{2} - 9}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 9 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $9$ to both sides of the equation.

$x^{2} = 9$

Step 4.2

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

$x = \pm \sqrt{9}$

Step 4.3

Simplify $\pm \sqrt{9}$ .

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Step 4.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 4.3.2

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

$x = \pm 3$

Step 4.4

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

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Step 4.4.1

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

$x = 3$

Step 4.4.2

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

$x = - 3$

Step 4.4.3

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

$x = 3, - 3$

Step 5

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, 3) \cup (3, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 3, - 3, 4}$

Step 6