Finite Math Examples

$\ln (\frac{5 + x}{3 - x})$

Step 1

Set the argument in $\ln (\frac{5 + x}{3 - x})$ greater than $0$ to find where the expression is defined.

$\frac{5 + x}{3 - x} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$5 + x = 0$

$3 - x = 0$

Step 2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.3

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 2.4

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 2.4.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 2.5

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = - 5$

$x = 3$

Step 2.6

Consolidate the solutions.

$x = - 5, 3$

Step 2.7

Find the domain of $\frac{5 + x}{3 - x}$ .

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Step 2.7.1

Set the denominator in $\frac{5 + x}{3 - x}$ equal to $0$ to find where the expression is undefined.

$3 - x = 0$

Step 2.7.2

Solve for $x$ .

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Step 2.7.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 2.7.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 2.7.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 2.7.2.2.2

Simplify the left side.

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Step 2.7.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 2.7.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 2.7.2.2.3

Simplify the right side.

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Step 2.7.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 2.7.3

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

$(- \infty, 3) \cup (3, \infty)$

Step 2.8

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 3$

$x > 3$

Step 2.9

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 2.9.1

Test a value on the interval $x < - 5$ to see if it makes the inequality true.

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Step 2.9.1.1

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

Step 2.9.1.2

Replace $x$ with $- 8$ in the original inequality.

$\frac{5 - 8}{3 - (- 8)} > 0$

Step 2.9.1.3

The left side $- 0.\overline{27}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.9.2

Test a value on the interval $- 5 < x < 3$ to see if it makes the inequality true.

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Step 2.9.2.1

Choose a value on the interval $- 5 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

Step 2.9.2.2

Replace $x$ with $0$ in the original inequality.

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

Step 2.9.2.3

The left side $1.\overline{6}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.9.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 2.9.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 2.9.3.2

Replace $x$ with $6$ in the original inequality.

$\frac{5 + 6}{3 - (6)} > 0$

Step 2.9.3.3

The left side $- 3.\overline{6}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.9.4

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

$x < - 5$ False

$- 5 < x < 3$ True

$x > 3$ False

$x < - 5$ False

$- 5 < x < 3$ True

$x > 3$ False

Step 2.10

The solution consists of all of the true intervals.

$- 5 < x < 3$

Step 3

Set the denominator in $\frac{5 + x}{3 - x}$ equal to $0$ to find where the expression is undefined.

$3 - x = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 4.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 5

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

Interval Notation:

$(- 5, 3)$

Set-Builder Notation:

${x | - 5 < x < 3}$

Step 6