Finite Math Examples

$\frac{\sqrt{1 + \frac{4}{x^{2}}}}{1 + \frac{4}{x}}$

Step 1

Set the denominator in $\frac{4}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x = \pm \sqrt{0}$

Step 2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.2.2

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

$x = \pm 0$

Step 2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Set the denominator in $\frac{\sqrt{1 + \frac{4}{x^{2}}}}{1 + \frac{4}{x}}$ equal to $0$ to find where the expression is undefined.

$1 + \frac{4}{x} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$\frac{4}{x} = - 1$

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 4.2.2

The LCM of one and any expression is the expression.

$x$

Step 4.3

Multiply each term in $\frac{4}{x} = - 1$ by $x$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $\frac{4}{x} = - 1$ by $x$ .

$\frac{4}{x} x = - x$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{4}{x} x = - x$

Step 4.3.2.1.2

Rewrite the expression.

$4 = - x$

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as $- x = 4$ .

$- x = 4$

Step 4.4.2

Divide each term in $- x = 4$ by $- 1$ and simplify.

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Step 4.4.2.1

Divide each term in $- x = 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{4}{- 1}$

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{4}{- 1}$

Step 4.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{4}{- 1}$

Step 4.4.2.3

Simplify the right side.

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Step 4.4.2.3.1

Divide $4$ by $- 1$ .

$x = - 4$

Step 5

Set the radicand in $\sqrt{1 + \frac{4}{x^{2}}}$ less than $0$ to find where the expression is undefined.

$1 + \frac{4}{x^{2}} < 0$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $1$ from both sides of the inequality.

$\frac{4}{x^{2}} < - 1$

Step 6.2

Multiply both sides by $x^{2}$ .

$\frac{4}{x^{2}} x^{2} = - x^{2}$

Step 6.3

Simplify the left side.

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Step 6.3.1

Cancel the common factor of $x^{2}$ .

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Step 6.3.1.1

Cancel the common factor.

$\frac{4}{x^{2}} x^{2} = - x^{2}$

Step 6.3.1.2

Rewrite the expression.

$4 = - x^{2}$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Rewrite the equation as $- x^{2} = 4$ .

$- x^{2} = 4$

Step 6.4.2

Divide each term in $- x^{2} = 4$ by $- 1$ and simplify.

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Step 6.4.2.1

Divide each term in $- x^{2} = 4$ by $- 1$ .

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

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.4.2.2.2

Divide $x^{2}$ by $1$ .

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

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Divide $4$ by $- 1$ .

$x^{2} = - 4$

Step 6.4.3

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

$x = \pm \sqrt{- 4}$

Step 6.4.4

Simplify $\pm \sqrt{- 4}$ .

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Step 6.4.4.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 6.4.4.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 6.4.4.3

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

$x = \pm i \cdot \sqrt{4}$

Step 6.4.4.4

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

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

Step 6.4.4.5

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

$x = \pm i \cdot 2$

Step 6.4.4.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 6.4.5

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

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Step 6.4.5.1

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

$x = 2 i$

Step 6.4.5.2

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

$x = - 2 i$

Step 6.4.5.3

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

$x = 2 i, - 2 i$

Step 6.5

Find the domain of $1 + \frac{4}{x^{2}}$ .

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Step 6.5.1

Set the denominator in $\frac{4}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 6.5.2

Solve for $x$ .

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Step 6.5.2.1

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

$x = \pm \sqrt{0}$

Step 6.5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.5.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.5.2.2.2

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

$x = \pm 0$

Step 6.5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.5.3

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

$(- \infty, 0) \cup (0, \infty)$

Step 6.6

Use each root to create test intervals.

$x < 0$

$x > 0$

Step 6.7

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 6.7.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 6.7.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 6.7.1.2

Replace $x$ with $- 2$ in the original inequality.

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

Step 6.7.1.3

The left side $2$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.7.2

Test a value on the interval $x > 0$ to see if it makes the inequality true.

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Step 6.7.2.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 6.7.2.2

Replace $x$ with $2$ in the original inequality.

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

Step 6.7.2.3

The left side $2$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.7.3

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

$x < 0$ False

$x > 0$ False

$x < 0$ False

$x > 0$ False

Step 6.8

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 7

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 $0$ .

$x = - 4, x = 0$

Step 8