Precalculus Examples

$f (x) = \sqrt{\frac{3}{2 x - 18}}$

Step 1

Set the radicand in $\sqrt{\frac{3}{2 x - 18}}$ greater than or equal to $0$ to find where the expression is defined.

$\frac{3}{2 x - 18} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$2 x - 18 = 0$

Step 2.2

Add $18$ to both sides of the equation.

$2 x = 18$

Step 2.3

Divide each term in $2 x = 18$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x = 18$ by $2$ .

$\frac{2 x}{2} = \frac{18}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{18}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $18$ by $2$ .

$x = 9$

Step 2.4

Find the domain of $\frac{3}{2 x - 18}$ .

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Step 2.4.1

Set the denominator in $\frac{3}{2 x - 18}$ equal to $0$ to find where the expression is undefined.

$2 x - 18 = 0$

Step 2.4.2

Solve for $x$ .

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Step 2.4.2.1

Add $18$ to both sides of the equation.

$2 x = 18$

Step 2.4.2.2

Divide each term in $2 x = 18$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x = 18$ by $2$ .

$\frac{2 x}{2} = \frac{18}{2}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{18}{2}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{2}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Divide $18$ by $2$ .

$x = 9$

Step 2.4.3

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

$(- \infty, 9) \cup (9, \infty)$

Step 2.5

The solution consists of all of the true intervals.

$x > 9$

Step 3

Set the denominator in $\frac{3}{2 x - 18}$ equal to $0$ to find where the expression is undefined.

$2 x - 18 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $18$ to both sides of the equation.

$2 x = 18$

Step 4.2

Divide each term in $2 x = 18$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x = 18$ by $2$ .

$\frac{2 x}{2} = \frac{18}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{18}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $18$ by $2$ .

$x = 9$

Step 5

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

Interval Notation:

$(9, \infty)$

Set-Builder Notation:

${x | x > 9}$

Step 6