Precalculus Examples

$f (x) = \ln (\sqrt{5 x - 6})$

Step 1

Set the argument in $\ln (\sqrt{5 x - 6})$ greater than $0$ to find where the expression is defined.

$\sqrt{5 x - 6} > 0$

Step 2

Solve for $x$ .

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Step 2.1

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

${\sqrt{5 x - 6}}^{2} > 0^{2}$

Step 2.2

Simplify each side of the inequality.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 6}$ as ${(5 x - 6)}^{\frac{1}{2}}$ .

${({(5 x - 6)}^{\frac{1}{2}})}^{2} > 0^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(5 x - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(5 x - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(5 x - 6)}^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(5 x - 6)}^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(5 x - 6)}^{1} > 0^{2}$

Step 2.2.2.1.2

Simplify.

$5 x - 6 > 0^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$5 x - 6 > 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Add $6$ to both sides of the inequality.

$5 x > 6$

Step 2.3.2

Divide each term in $5 x > 6$ by $5$ and simplify.

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Step 2.3.2.1

Divide each term in $5 x > 6$ by $5$ .

$\frac{5 x}{5} > \frac{6}{5}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{6}{5}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{6}{5}$

Step 2.4

Find the domain of $\sqrt{5 x - 6}$ .

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Step 2.4.1

Set the radicand in $\sqrt{5 x - 6}$ greater than or equal to $0$ to find where the expression is defined.

$5 x - 6 \geq 0$

Step 2.4.2

Solve for $x$ .

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Step 2.4.2.1

Add $6$ to both sides of the inequality.

$5 x \geq 6$

Step 2.4.2.2

Divide each term in $5 x \geq 6$ by $5$ and simplify.

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Step 2.4.2.2.1

Divide each term in $5 x \geq 6$ by $5$ .

$\frac{5 x}{5} \geq \frac{6}{5}$

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 $5$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{6}{5}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{6}{5}$

Step 2.4.3

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

$[\frac{6}{5}, \infty)$

Step 2.5

The solution consists of all of the true intervals.

$x > \frac{6}{5}$

Step 3

Set the radicand in $\sqrt{5 x - 6}$ greater than or equal to $0$ to find where the expression is defined.

$5 x - 6 \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $6$ to both sides of the inequality.

$5 x \geq 6$

Step 4.2

Divide each term in $5 x \geq 6$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x \geq 6$ by $5$ .

$\frac{5 x}{5} \geq \frac{6}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{6}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{6}{5}$

Step 5

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

Interval Notation:

$(\frac{6}{5}, \infty)$

Set-Builder Notation:

${x | x > \frac{6}{5}}$

Step 6