Precalculus Examples

$g (x) = \ln (25 x - x^{2})$

Step 1

Set the argument in $\ln (25 x - x^{2})$ greater than $0$ to find where the expression is defined.

$25 x - x^{2} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$25 x - x^{2} = 0$

Step 2.2

Factor $x$ out of $25 x - x^{2}$ .

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Step 2.2.1

Factor $x$ out of $25 x$ .

$x \cdot 25 - x^{2} = 0$

Step 2.2.2

Factor $x$ out of $- x^{2}$ .

$x \cdot 25 + x (- x) = 0$

Step 2.2.3

Factor $x$ out of $x \cdot 25 + x (- x)$ .

$x (25 - x) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$25 - x = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $25 - x$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $25 - x$ equal to $0$ .

$25 - x = 0$

Step 2.5.2

Solve $25 - x = 0$ for $x$ .

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Step 2.5.2.1

Subtract $25$ from both sides of the equation.

$- x = - 25$

Step 2.5.2.2

Divide each term in $- x = - 25$ by $- 1$ and simplify.

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Step 2.5.2.2.1

Divide each term in $- x = - 25$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 25}{- 1}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 25}{- 1}$

Step 2.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 25}{- 1}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Divide $- 25$ by $- 1$ .

$x = 25$

Step 2.6

The final solution is all the values that make $x (25 - x) = 0$ true.

$x = 0, 25$

Step 2.7

Use each root to create test intervals.

$x < 0$

$0 < x < 25$

$x > 25$

Step 2.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 2.8.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 2.8.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 2.8.1.2

Replace $x$ with $- 2$ in the original inequality.

$25 (- 2) - {(- 2)}^{2} > 0$

Step 2.8.1.3

The left side $- 54$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.8.2

Test a value on the interval $0 < x < 25$ to see if it makes the inequality true.

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Step 2.8.2.1

Choose a value on the interval $0 < x < 25$ and see if this value makes the original inequality true.

$x = 2$

Step 2.8.2.2

Replace $x$ with $2$ in the original inequality.

$25 (2) - {(2)}^{2} > 0$

Step 2.8.2.3

The left side $46$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.3

Test a value on the interval $x > 25$ to see if it makes the inequality true.

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Step 2.8.3.1

Choose a value on the interval $x > 25$ and see if this value makes the original inequality true.

$x = 28$

Step 2.8.3.2

Replace $x$ with $28$ in the original inequality.

$25 (28) - {(28)}^{2} > 0$

Step 2.8.3.3

The left side $- 84$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 25$ True

$x > 25$ False

$x < 0$ False

$0 < x < 25$ True

$x > 25$ False

Step 2.9

The solution consists of all of the true intervals.

$0 < x < 25$

Step 3

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

Interval Notation:

$(0, 25)$

Set-Builder Notation:

${x | 0 < x < 25}$

Step 4