Algebra Examples

$y = \ln (\frac{12 - 4 x}{5})$

Step 1

Set the argument in $\ln (\frac{12 - 4 x}{5})$ greater than $0$ to find where the expression is defined.

$\frac{12 - 4 x}{5} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides by $5$ .

$\frac{12 - 4 x}{5} \cdot 5 > 0 \cdot 5$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{12 - 4 x}{5} \cdot 5$ .

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Step 2.2.1.1.1

Factor $4$ out of $12 - 4 x$ .

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Step 2.2.1.1.1.1

Factor $4$ out of $12$ .

$\frac{4 (3) - 4 x}{5} \cdot 5 > 0 \cdot 5$

Step 2.2.1.1.1.2

Factor $4$ out of $- 4 x$ .

$\frac{4 (3) + 4 (- x)}{5} \cdot 5 > 0 \cdot 5$

Step 2.2.1.1.1.3

Factor $4$ out of $4 (3) + 4 (- x)$ .

$\frac{4 (3 - x)}{5} \cdot 5 > 0 \cdot 5$

Step 2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$\frac{4 (3 - x)}{5} \cdot 5 > 0 \cdot 5$

Step 2.2.1.1.2.2

Rewrite the expression.

$4 (3 - x) > 0 \cdot 5$

Step 2.2.1.1.3

Apply the distributive property.

$4 \cdot 3 + 4 (- x) > 0 \cdot 5$

Step 2.2.1.1.4

Simplify the expression.

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Step 2.2.1.1.4.1

Multiply $4$ by $3$ .

$12 + 4 (- x) > 0 \cdot 5$

Step 2.2.1.1.4.2

Multiply $- 1$ by $4$ .

$12 - 4 x > 0 \cdot 5$

Step 2.2.1.1.4.3

Reorder $12$ and $- 4 x$ .

$- 4 x + 12 > 0 \cdot 5$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $0$ by $5$ .

$- 4 x + 12 > 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Subtract $12$ from both sides of the inequality.

$- 4 x > - 12$

Step 2.3.2

Divide each term in $- 4 x > - 12$ by $- 4$ and simplify.

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Step 2.3.2.1

Divide each term in $- 4 x > - 12$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{- 12}{- 4}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{- 12}{- 4}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 12}{- 4}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $- 12$ by $- 4$ .

$x < 3$

Step 3

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

Interval Notation:

$(- \infty, 3)$

Set-Builder Notation:

${x | x < 3}$

Step 4