Finite Math Examples

$f (x) = 6 \ln ({(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3}})$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$6 \ln (\sqrt[3]{{(\frac{x^{2} + 2}{x^{2} - 2})}^{1}})$

Step 1.2

Anything raised to $1$ is the base itself.

$6 \ln (\sqrt[3]{\frac{x^{2} + 2}{x^{2} - 2}})$

Step 2

Set the argument in $\ln (\sqrt[3]{\frac{x^{2} + 2}{x^{2} - 2}})$ greater than $0$ to find where the expression is defined.

$\sqrt[3]{\frac{x^{2} + 2}{x^{2} - 2}} > 0$

Step 3

Solve for $x$ .

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Step 3.1

To remove the radical on the left side of the inequality, cube both sides of the inequality.

${\sqrt[3]{\frac{x^{2} + 2}{x^{2} - 2}}}^{3} > 0^{3}$

Step 3.2

Simplify each side of the inequality.

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x^{2} + 2}{x^{2} - 2}}$ as ${(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3}}$ .

${({(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3}})}^{3} > 0^{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify ${({(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.2.1.1

Multiply the exponents in ${({(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3} \cdot 3} > 0^{3}$

Step 3.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

${(\frac{x^{2} + 2}{x^{2} - 2})}^{\frac{1}{3} \cdot 3} > 0^{3}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

${(\frac{x^{2} + 2}{x^{2} - 2})}^{1} > 0^{3}$

Step 3.2.2.1.2

Simplify.

$\frac{x^{2} + 2}{x^{2} - 2} > 0^{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{x^{2} + 2}{x^{2} - 2} > 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

$x^{2} + 2 = 0$

$x^{2} - 2 = 0$

Step 3.3.2

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 3.3.3

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

$x = \pm \sqrt{- 2}$

Step 3.3.4

Simplify $\pm \sqrt{- 2}$ .

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Step 3.3.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 3.3.4.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 3.3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{2}$

Step 3.3.5

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

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Step 3.3.5.1

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

$x = i \sqrt{2}$

Step 3.3.5.2

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

$x = - i \sqrt{2}$

Step 3.3.5.3

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

$x = i \sqrt{2}, - i \sqrt{2}$

Step 3.3.6

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 3.3.7

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

$x = \pm \sqrt{2}$

Step 3.3.8

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

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Step 3.3.8.1

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

$x = \sqrt{2}$

Step 3.3.8.2

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

$x = - \sqrt{2}$

Step 3.3.8.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 3.3.9

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

$x = i \sqrt{2}, - i \sqrt{2}$

$x = \sqrt{2}, - \sqrt{2}$

Step 3.3.10

Consolidate the solutions.

$x = \sqrt{2}, - \sqrt{2}$

Step 3.4

Find the domain of $\sqrt[3]{\frac{x^{2} + 2}{x^{2} - 2}}$ .

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Step 3.4.1

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

$x^{2} - 2 = 0$

Step 3.4.2

Solve for $x$ .

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Step 3.4.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 3.4.2.2

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

$x = \pm \sqrt{2}$

Step 3.4.2.3

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

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Step 3.4.2.3.1

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

$x = \sqrt{2}$

Step 3.4.2.3.2

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

$x = - \sqrt{2}$

Step 3.4.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 3.4.3

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

$(- \infty, - \sqrt{2}) \cup (- \sqrt{2}, \sqrt{2}) \cup (\sqrt{2}, \infty)$

Step 3.5

Use each root to create test intervals.

$x < - \sqrt{2}$

$- \sqrt{2} < x < \sqrt{2}$

$x > \sqrt{2}$

Step 3.6

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

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Step 3.6.1

Test a value on the interval $x < - \sqrt{2}$ to see if it makes the inequality true.

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Step 3.6.1.1

Choose a value on the interval $x < - \sqrt{2}$ and see if this value makes the original inequality true.

$x = - 4$

Step 3.6.1.2

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

$\sqrt[3]{\frac{{(- 4)}^{2} + 2}{{(- 4)}^{2} - 2}} > 0$

Step 3.6.1.3

The left side $1.08738037$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.6.2

Test a value on the interval $- \sqrt{2} < x < \sqrt{2}$ to see if it makes the inequality true.

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Step 3.6.2.1

Choose a value on the interval $- \sqrt{2} < x < \sqrt{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 3.6.2.2

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

$\sqrt[3]{\frac{{(0)}^{2} + 2}{{(0)}^{2} - 2}} > 0$

Step 3.6.2.3

The left side $- 1$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.6.3

Test a value on the interval $x > \sqrt{2}$ to see if it makes the inequality true.

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Step 3.6.3.1

Choose a value on the interval $x > \sqrt{2}$ and see if this value makes the original inequality true.

$x = 4$

Step 3.6.3.2

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

$\sqrt[3]{\frac{{(4)}^{2} + 2}{{(4)}^{2} - 2}} > 0$

Step 3.6.3.3

The left side $1.08738037$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.6.4

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

$x < - \sqrt{2}$ True

$- \sqrt{2} < x < \sqrt{2}$ False

$x > \sqrt{2}$ True

$x < - \sqrt{2}$ True

$- \sqrt{2} < x < \sqrt{2}$ False

$x > \sqrt{2}$ True

Step 3.7

The solution consists of all of the true intervals.

$x < - \sqrt{2}$ or $x > \sqrt{2}$

Step 4

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

$x^{2} - 2 = 0$

Step 5

Solve for $x$ .

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Step 5.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 5.2

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

$x = \pm \sqrt{2}$

Step 5.3

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

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Step 5.3.1

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

$x = \sqrt{2}$

Step 5.3.2

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

$x = - \sqrt{2}$

Step 5.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 6

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

Interval Notation:

$(- \infty, - \sqrt{2}) \cup (\sqrt{2}, \infty)$

Set-Builder Notation:

${x | x < - \sqrt{2}, x > \sqrt{2}}$

Step 7