Trigonometry Examples

$x (1 - \ln (x)) > 0$

Step 1

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

$x = 0$

$1 - \ln (x) = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set $1 - \ln (x)$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $1 - \ln (x)$ equal to $0$ .

$1 - \ln (x) = 0$

Step 3.2

Solve $1 - \ln (x) = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$- \ln (x) = - 1$

Step 3.2.2

Divide each term in $- \ln (x) = - 1$ by $- 1$ and simplify.

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Step 3.2.2.1

Divide each term in $- \ln (x) = - 1$ by $- 1$ .

$\frac{- \ln (x)}{- 1} = \frac{- 1}{- 1}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (x)}{1} = \frac{- 1}{- 1}$

Step 3.2.2.2.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- 1}{- 1}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 1$ by $- 1$ .

$\ln (x) = 1$

Step 3.2.3

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{1}$

Step 3.2.4

Rewrite $\ln (x) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{1} = x$

Step 3.2.5

Rewrite the equation as $x = e^{1}$ .

$x = e$

Step 4

The final solution is all the values that make $x (1 - \ln (x)) > 0$ true.

$x = 0, e$

Step 5

Find the domain of $x (1 - \ln (x))$ .

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Step 5.1

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

$x > 0$

Step 5.2

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

$(0, \infty)$

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < e$

$x > e$

Step 7

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 7.1

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

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Step 7.1.1

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

$x = - 2$

Step 7.1.2

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

$(- 2) (1 - \ln (- 2)) > 0$

Step 7.1.3

Determine if the inequality is true.

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Step 7.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 7.1.3.2

The left side has no solution, which means that the given statement is false.

False

Step 7.2

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

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Step 7.2.1

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

$x = 1$

Step 7.2.2

Replace $x$ with $1$ in the original inequality.

$(1) (1 - \ln (1)) > 0$

Step 7.2.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

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

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Step 7.3.1

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

$x = 5$

Step 7.3.2

Replace $x$ with $5$ in the original inequality.

$(5) (1 - \ln (5)) > 0$

Step 7.3.3

The left side $- 3.04718956$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.4

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

$x < 0$ False

$0 < x < e$ True

$x > e$ False

$x < 0$ False

$0 < x < e$ True

$x > e$ False

Step 8

The solution consists of all of the true intervals.

$0 < x < e$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$0 < x < e$

Interval Notation:

$(0, e)$

Step 10