Calculus Examples

$\frac{5}{1 - \sqrt{5 x - 9}}$

Step 1

Set the radicand in $\sqrt{5 x - 9}$ greater than or equal to $0$ to find where the expression is defined.

$5 x - 9 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $9$ to both sides of the inequality.

$5 x \geq 9$

Step 2.2

Divide each term in $5 x \geq 9$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 x \geq 9$ by $5$ .

$\frac{5 x}{5} \geq \frac{9}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{9}{5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{9}{5}$

Step 3

Set the denominator in $\frac{5}{1 - \sqrt{5 x - 9}}$ equal to $0$ to find where the expression is undefined.

$1 - \sqrt{5 x - 9} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$- \sqrt{5 x - 9} = - 1$

Step 4.2

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

${(- \sqrt{5 x - 9})}^{2} = {(- 1)}^{2}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 9}$ as ${(5 x - 9)}^{\frac{1}{2}}$ .

${(- {(5 x - 9)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify ${(- {(5 x - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1

Apply the product rule to $- {(5 x - 9)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(5 x - 9)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 4.3.2.1.2

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

$1 {({(5 x - 9)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 4.3.2.1.3

Multiply ${({(5 x - 9)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(5 x - 9)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 4.3.2.1.4

Multiply the exponents in ${({(5 x - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.4.1

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

${(5 x - 9)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 4.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.4.2.1

Cancel the common factor.

${(5 x - 9)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 4.3.2.1.4.2.2

Rewrite the expression.

${(5 x - 9)}^{1} = {(- 1)}^{2}$

Step 4.3.2.1.5

Simplify.

$5 x - 9 = {(- 1)}^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

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

$5 x - 9 = 1$

Step 4.4

Solve for $x$ .

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Step 4.4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.4.1.1

Add $9$ to both sides of the equation.

$5 x = 1 + 9$

Step 4.4.1.2

Add $1$ and $9$ .

$5 x = 10$

Step 4.4.2

Divide each term in $5 x = 10$ by $5$ and simplify.

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Step 4.4.2.1

Divide each term in $5 x = 10$ by $5$ .

$\frac{5 x}{5} = \frac{10}{5}$

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $5$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{10}{5}$

Step 4.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{5}$

Step 4.4.2.3

Simplify the right side.

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Step 4.4.2.3.1

Divide $10$ by $5$ .

$x = 2$

Step 5

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

Interval Notation:

$[\frac{9}{5}, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \geq \frac{9}{5}, x \neq 2}$

Step 6