Finite Math Examples

$- \sqrt{\frac{6}{x^{2} - 1}}$

Step 1

Set the denominator in $\frac{6}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 1 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 2.2

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

$x = \pm \sqrt{1}$

Step 2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$x = 1$

Step 2.4.2

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

$x = - 1$

Step 2.4.3

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

$x = 1, - 1$

Step 3

Set the radicand in $\sqrt{\frac{6}{x^{2} - 1}}$ less than $0$ to find where the expression is undefined.

$\frac{6}{x^{2} - 1} < 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

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

$x^{2} - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 4.3

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

$x = \pm \sqrt{1}$

Step 4.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 4.5

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

Tap for more steps...

Step 4.5.1

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

$x = 1$

Step 4.5.2

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

$x = - 1$

Step 4.5.3

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

$x = 1, - 1$

Step 4.6

Consolidate the solutions.

$x = 1, - 1$

Step 4.7

Find the domain of $\frac{6}{x^{2} - 1}$ .

Tap for more steps...

Step 4.7.1

Set the denominator in $\frac{6}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 1 = 0$

Step 4.7.2

Solve for $x$ .

Tap for more steps...

Step 4.7.2.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 4.7.2.2

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

$x = \pm \sqrt{1}$

Step 4.7.2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 4.7.2.4

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

Tap for more steps...

Step 4.7.2.4.1

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

$x = 1$

Step 4.7.2.4.2

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

$x = - 1$

Step 4.7.2.4.3

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

$x = 1, - 1$

Step 4.7.3

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

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Step 4.8

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 1$

$x > 1$

Step 4.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 4.9.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Tap for more steps...

Step 4.9.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 4.9.1.2

Replace $x$ with $- 4$ in the original inequality.

$\frac{6}{{(- 4)}^{2} - 1} < 0$

Step 4.9.1.3

The left side $0.4$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 4.9.2

Test a value on the interval $- 1 < x < 1$ to see if it makes the inequality true.

Tap for more steps...

Step 4.9.2.1

Choose a value on the interval $- 1 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 4.9.2.2

Replace $x$ with $0$ in the original inequality.

$\frac{6}{{(0)}^{2} - 1} < 0$

Step 4.9.2.3

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 4.9.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

Tap for more steps...

Step 4.9.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 4.9.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{6}{{(4)}^{2} - 1} < 0$

Step 4.9.3.3

The left side $0.4$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 4.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1$ False

$- 1 < x < 1$ True

$x > 1$ False

$x < - 1$ False

$- 1 < x < 1$ True

$x > 1$ False

Step 4.10

The solution consists of all of the true intervals.

$- 1 < x < 1$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$- 1 \leq x \leq 1$

$[- 1, 1]$

Step 6