Calculus Examples

$f (x) = \ln (6 - 9 x)$

Step 1

Set the argument in $\ln (6 - 9 x)$ greater than $0$ to find where the expression is defined.

$6 - 9 x > 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the inequality.

$- 9 x > - 6$

Step 2.2

Divide each term in $- 9 x > - 6$ by $- 9$ and simplify.

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Step 2.2.1

Divide each term in $- 9 x > - 6$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 x}{- 9} < \frac{- 6}{- 9}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 9$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} < \frac{- 6}{- 9}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 6}{- 9}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 6$ and $- 9$ .

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Step 2.2.3.1.1

Factor $- 3$ out of $- 6$ .

$x < \frac{- 3 (2)}{- 9}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $- 3$ out of $- 9$ .

$x < \frac{- 3 \cdot 2}{- 3 \cdot 3}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x < \frac{- 3 \cdot 2}{- 3 \cdot 3}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x < \frac{2}{3}$

Step 3

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

Interval Notation:

$(- \infty, \frac{2}{3})$

Set-Builder Notation:

${x | x < \frac{2}{3}}$

Step 4