Finite Math Examples

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

Step 1

Set the argument in $\ln ((\frac{1}{\sqrt{- x}}) + 3)$ greater than $0$ to find where the expression is defined.

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

Step 2

Solve for $x$ .

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Step 2.1

Solve for $\sqrt{- x}$ .

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Step 2.1.1

Subtract $3$ from both sides of the inequality.

$\frac{1}{\sqrt{- x}} > - 3$

Step 2.1.2

Multiply both sides by $\sqrt{- x}$ .

$\frac{1}{\sqrt{- x}} \sqrt{- x} = - 3 \sqrt{- x}$

Step 2.1.3

Simplify the left side.

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Step 2.1.3.1

Cancel the common factor of $\sqrt{- x}$ .

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Step 2.1.3.1.1

Cancel the common factor.

$\frac{1}{\sqrt{- x}} \sqrt{- x} = - 3 \sqrt{- x}$

Step 2.1.3.1.2

Rewrite the expression.

$1 = - 3 \sqrt{- x}$

Step 2.1.4

Solve for $\sqrt{- x}$ .

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Step 2.1.4.1

Rewrite the equation as $- 3 \sqrt{- x} = 1$ .

$- 3 \sqrt{- x} = 1$

Step 2.1.4.2

Divide each term in $- 3 \sqrt{- x} = 1$ by $- 3$ and simplify.

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Step 2.1.4.2.1

Divide each term in $- 3 \sqrt{- x} = 1$ by $- 3$ .

$\frac{- 3 \sqrt{- x}}{- 3} = \frac{1}{- 3}$

Step 2.1.4.2.2

Simplify the left side.

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Step 2.1.4.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.1.4.2.2.1.1

Cancel the common factor.

$\frac{- 3 \sqrt{- x}}{- 3} = \frac{1}{- 3}$

Step 2.1.4.2.2.1.2

Divide $\sqrt{- x}$ by $1$ .

$\sqrt{- x} = \frac{1}{- 3}$

Step 2.1.4.2.3

Simplify the right side.

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Step 2.1.4.2.3.1

Move the negative in front of the fraction.

$\sqrt{- x} = - \frac{1}{3}$

Step 2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- x}}^{2} = {(- \frac{1}{3})}^{2}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

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

${({(- x)}^{\frac{1}{2}})}^{2} = {(- \frac{1}{3})}^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

${(- x)}^{\frac{1}{2} \cdot 2} = {(- \frac{1}{3})}^{2}$

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${(- x)}^{\frac{1}{2} \cdot 2} = {(- \frac{1}{3})}^{2}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(- x)}^{1} = {(- \frac{1}{3})}^{2}$

Step 2.3.2.1.2

Simplify.

$- x = {(- \frac{1}{3})}^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.3.1.1.1

Apply the product rule to $- \frac{1}{3}$ .

$- x = {(- 1)}^{2} {(\frac{1}{3})}^{2}$

Step 2.3.3.1.1.2

Apply the product rule to $\frac{1}{3}$ .

$- x = {(- 1)}^{2} \frac{1^{2}}{3^{2}}$

Step 2.3.3.1.2

Raise $- 1$ to the power of $2$ .

$- x = 1 \frac{1^{2}}{3^{2}}$

Step 2.3.3.1.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$- x = \frac{1^{2}}{3^{2}}$

Step 2.3.3.1.4

One to any power is one.

$- x = \frac{1}{3^{2}}$

Step 2.3.3.1.5

Raise $3$ to the power of $2$ .

$- x = \frac{1}{9}$

Step 2.4

Divide each term in $- x = \frac{1}{9}$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- x = \frac{1}{9}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\frac{1}{9}}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\frac{1}{9}}{- 1}$

Step 2.4.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{1}{9}}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative one from the denominator of $\frac{\frac{1}{9}}{- 1}$ .

$x = - 1 \cdot \frac{1}{9}$

Step 2.4.3.2

Rewrite $- 1 \cdot \frac{1}{9}$ as $- \frac{1}{9}$ .

$x = - \frac{1}{9}$

Step 2.5

Find the domain of $(\frac{1}{\sqrt{- x}}) + 3$ .

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Step 2.5.1

Set the radicand in $\sqrt{- x}$ greater than or equal to $0$ to find where the expression is defined.

$- x \geq 0$

Step 2.5.2

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

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Step 2.5.2.1

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Step 2.5.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $0$ by $- 1$ .

$x \leq 0$

Step 2.5.3

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

$\sqrt{- x} = 0$

Step 2.5.4

Solve for $x$ .

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Step 2.5.4.1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2.5.4.2

Simplify each side of the equation.

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Step 2.5.4.2.1

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

${({(- x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.5.4.2.2

Simplify the left side.

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Step 2.5.4.2.2.1

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

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Step 2.5.4.2.2.1.1

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

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Step 2.5.4.2.2.1.1.1

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

${(- x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.5.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.4.2.2.1.1.2.1

Cancel the common factor.

${(- x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.5.4.2.2.1.1.2.2

Rewrite the expression.

${(- x)}^{1} = 0^{2}$

Step 2.5.4.2.2.1.2

Simplify.

$- x = 0^{2}$

Step 2.5.4.2.3

Simplify the right side.

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Step 2.5.4.2.3.1

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

$- x = 0$

Step 2.5.4.3

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 2.5.4.3.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 2.5.4.3.2

Simplify the left side.

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Step 2.5.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 2.5.4.3.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 2.5.4.3.3

Simplify the right side.

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Step 2.5.4.3.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 2.5.5

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

$(- \infty, 0)$

Step 2.6

Use each root to create test intervals.

$x < - \frac{1}{9}$

$- \frac{1}{9} < x < 0$

$x > 0$

Step 2.7

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 2.7.1

Test a value on the interval $x < - \frac{1}{9}$ to see if it makes the inequality true.

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Step 2.7.1.1

Choose a value on the interval $x < - \frac{1}{9}$ and see if this value makes the original inequality true.

$x = - 3$

Step 2.7.1.2

Replace $x$ with $- 3$ in the original inequality.

$(\frac{1}{\sqrt{- (- 3)}}) + 3 > 0$

Step 2.7.1.3

The left side $3.57735026$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.7.2

Test a value on the interval $- \frac{1}{9} < x < 0$ to see if it makes the inequality true.

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Step 2.7.2.1

Choose a value on the interval $- \frac{1}{9} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.06$

Step 2.7.2.2

Replace $x$ with $- 0.06$ in the original inequality.

$(\frac{1}{\sqrt{- (- 0.06)}}) + 3 > 0$

Step 2.7.2.3

The left side $7.0824829$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.7.3

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

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Step 2.7.3.1

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

$x = 2$

Step 2.7.3.2

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

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

Step 2.7.3.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 2.7.4

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

$x < - \frac{1}{9}$ True

$- \frac{1}{9} < x < 0$ True

$x > 0$ False

$x < - \frac{1}{9}$ True

$- \frac{1}{9} < x < 0$ True

$x > 0$ False

Step 2.8

The solution consists of all of the true intervals.

$x < - \frac{1}{9}$ or $- \frac{1}{9} < x < 0$

Step 3

Set the radicand in $\sqrt{- x}$ greater than or equal to $0$ to find where the expression is defined.

$- x \geq 0$

Step 4

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 1$ .

$x \leq 0$

Step 5

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

$\sqrt{- x} = 0$

Step 6

Solve for $x$ .

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Step 6.1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

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

${({(- x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

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

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Step 6.2.2.1.1.1

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

${(- x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(- x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(- x)}^{1} = 0^{2}$

Step 6.2.2.1.2

Simplify.

$- x = 0^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$- x = 0$

Step 6.3

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 6.3.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 7

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

Interval Notation:

$(- \infty, - \frac{1}{9}) \cup (- \frac{1}{9}, 0)$

Set-Builder Notation:

${x | x < 0, x \neq - \frac{1}{9}}$

Step 8