Algebra Examples

$\frac{x^{2} - 36}{x - 6} \leq 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} - 36 = 0$

$x - 6 = 0$

Step 2

Add $36$ to both sides of the equation.

$x^{2} = 36$

Step 3

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

$x = \pm \sqrt{36}$

Step 4

Simplify $\pm \sqrt{36}$ .

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Step 4.1

Rewrite $36$ as $6^{2}$ .

$x = \pm \sqrt{6^{2}}$

Step 4.2

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

$x = \pm 6$

Step 5

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

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Step 5.1

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

$x = 6$

Step 5.2

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

$x = - 6$

Step 5.3

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

$x = 6, - 6$

Step 6

Add $6$ to both sides of the equation.

$x = 6$

Step 7

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 6, - 6$

$x = 6$

Step 8

Consolidate the solutions.

$x = 6, - 6, 6$

Step 9

Find the domain of $\frac{x^{2} - 36}{x - 6}$ .

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Step 9.1

Set the denominator in $\frac{x^{2} - 36}{x - 6}$ equal to $0$ to find where the expression is undefined.

$x - 6 = 0$

Step 9.2

Add $6$ to both sides of the equation.

$x = 6$

Step 9.3

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

$(- \infty, 6) \cup (6, \infty)$

Step 10

The solution consists of all of the true intervals.

$x \leq - 6$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 6$

Interval Notation:

$(- \infty, - 6]$

Step 12