Precalculus Examples

$| \frac{x}{x + 4} | < 5$

Step 1

Write $| \frac{x}{x + 4} | < 5$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{x}{x + 4} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

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

$x = 0$

$x + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.3

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

$x = 0$

$x = - 4$

Step 1.2.4

Consolidate the solutions.

$x = 0, - 4$

Step 1.2.5

Find the domain of $| \frac{x}{x + 4} | - 5$ .

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Step 1.2.5.1

Set the denominator in $\frac{x}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 1.2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.5.3

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 1.2.6

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 0$

$x > 0$

Step 1.2.7

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 1.2.7.1

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

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Step 1.2.7.1.1

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 1.2.7.1.2

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

$\frac{- 6}{(- 6) + 4} \geq 0$

Step 1.2.7.1.3

The left side $3$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.7.2

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

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Step 1.2.7.2.1

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 1.2.7.2.2

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

$\frac{- 2}{(- 2) + 4} \geq 0$

Step 1.2.7.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.2.7.3

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

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Step 1.2.7.3.1

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

$x = 2$

Step 1.2.7.3.2

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

$\frac{2}{(2) + 4} \geq 0$

Step 1.2.7.3.3

The left side $0.\overline{3}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.7.4

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

$x < - 4$ True

$- 4 < x < 0$ False

$x > 0$ True

$x < - 4$ True

$- 4 < x < 0$ False

$x > 0$ True

Step 1.2.8

The solution consists of all of the true intervals.

$x < - 4$ or $x \geq 0$

Step 1.3

In the piece where $\frac{x}{x + 4}$ is non-negative, remove the absolute value.

$\frac{x}{x + 4} < 5$

Step 1.4

Find the domain of $\frac{x}{x + 4} < 5$ and find the intersection with $x < - 4 or x \geq 0$ .

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Step 1.4.1

Find the domain of $\frac{x}{x + 4} < 5$ .

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Step 1.4.1.1

Set the denominator in $\frac{x}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 1.4.1.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.4.1.3

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 1.4.2

Find the intersection of $x < - 4 or x \geq 0$ and $(- \infty, - 4) \cup (- 4, \infty)$ .

$x < - 4 or x \geq 0$

Step 1.5

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{x}{x + 4} < 0$

Step 1.6

Solve the inequality.

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Step 1.6.1

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

$x = 0$

$x + 4 = 0$

Step 1.6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.6.3

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

$x = 0$

$x = - 4$

Step 1.6.4

Consolidate the solutions.

$x = 0, - 4$

Step 1.6.5

Find the domain of $| \frac{x}{x + 4} | - 5$ .

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Step 1.6.5.1

Set the denominator in $\frac{x}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 1.6.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.6.5.3

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 1.6.6

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 0$

$x > 0$

Step 1.6.7

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 1.6.7.1

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

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Step 1.6.7.1.1

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 1.6.7.1.2

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

$\frac{- 6}{(- 6) + 4} < 0$

Step 1.6.7.1.3

The left side $3$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 1.6.7.2

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

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Step 1.6.7.2.1

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 1.6.7.2.2

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

$\frac{- 2}{(- 2) + 4} < 0$

Step 1.6.7.2.3

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

True

Step 1.6.7.3

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

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Step 1.6.7.3.1

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

$x = 2$

Step 1.6.7.3.2

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

$\frac{2}{(2) + 4} < 0$

Step 1.6.7.3.3

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

False

Step 1.6.7.4

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

$x < - 4$ False

$- 4 < x < 0$ True

$x > 0$ False

$x < - 4$ False

$- 4 < x < 0$ True

$x > 0$ False

Step 1.6.8

The solution consists of all of the true intervals.

$- 4 < x < 0$

Step 1.7

In the piece where $\frac{x}{x + 4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{x}{x + 4} < 5$

Step 1.8

Find the domain of $- \frac{x}{x + 4} < 5$ and find the intersection with $- 4 < x < 0$ .

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Step 1.8.1

Find the domain of $- \frac{x}{x + 4} < 5$ .

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Step 1.8.1.1

Set the denominator in $\frac{x}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 1.8.1.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.8.1.3

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 1.8.2

Find the intersection of $- 4 < x < 0$ and $(- \infty, - 4) \cup (- 4, \infty)$ .

$- 4 < x < 0$

Step 1.9

Write as a piecewise.

${\begin{cases} \frac{x}{x + 4} < 5 & x < - 4 or x \geq 0 \\ - \frac{x}{x + 4} < 5 & - 4 < x < 0 \end{cases}$

Step 2

Solve $\frac{x}{x + 4} < 5$ when $x < - 4 or x \geq 0$ .

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Step 2.1

Solve $\frac{x}{x + 4} < 5$ for $x$ .

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Step 2.1.1

Subtract $5$ from both sides of the inequality.

$\frac{x}{x + 4} - 5 < 0$

Step 2.1.2

Simplify $\frac{x}{x + 4} - 5$ .

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Step 2.1.2.1

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{x}{x + 4} - 5 \cdot \frac{x + 4}{x + 4} < 0$

Step 2.1.2.2

Combine $- 5$ and $\frac{x + 4}{x + 4}$ .

$\frac{x}{x + 4} + \frac{- 5 (x + 4)}{x + 4} < 0$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{x - 5 (x + 4)}{x + 4} < 0$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Apply the distributive property.

$\frac{x - 5 x - 5 \cdot 4}{x + 4} < 0$

Step 2.1.2.4.2

Multiply $- 5$ by $4$ .

$\frac{x - 5 x - 20}{x + 4} < 0$

Step 2.1.2.4.3

Subtract $5 x$ from $x$ .

$\frac{- 4 x - 20}{x + 4} < 0$

Step 2.1.2.4.4

Factor $4$ out of $- 4 x - 20$ .

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Step 2.1.2.4.4.1

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

$\frac{4 (- x) - 20}{x + 4} < 0$

Step 2.1.2.4.4.2

Factor $4$ out of $- 20$ .

$\frac{4 (- x) + 4 (- 5)}{x + 4} < 0$

Step 2.1.2.4.4.3

Factor $4$ out of $4 (- x) + 4 (- 5)$ .

$\frac{4 (- x - 5)}{x + 4} < 0$

Step 2.1.2.5

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) - 5)}{x + 4} < 0$

Step 2.1.2.6

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{4 (- (x) - 1 (5))}{x + 4} < 0$

Step 2.1.2.7

Factor $- 1$ out of $- (x) - 1 (5)$ .

$\frac{4 (- (x + 5))}{x + 4} < 0$

Step 2.1.2.8

Rewrite $- (x + 5)$ as $- 1 (x + 5)$ .

$\frac{4 (- 1 (x + 5))}{x + 4} < 0$

Step 2.1.2.9

Move the negative in front of the fraction.

$- \frac{4 (x + 5)}{x + 4} < 0$

Step 2.1.3

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

$x + 5 = 0$

$x + 4 = 0$

Step 2.1.4

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.1.5

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.1.6

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

$x = - 5$

$x = - 4$

Step 2.1.7

Consolidate the solutions.

$x = - 5, - 4$

Step 2.1.8

Find the domain of $- \frac{4 (x + 5)}{x + 4}$ .

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Step 2.1.8.1

Set the denominator in $\frac{4 (x + 5)}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 2.1.8.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.1.8.3

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 2.1.9

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 4$

$x > - 4$

Step 2.1.10

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 2.1.10.1

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

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Step 2.1.10.1.1

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

$x = - 8$

Step 2.1.10.1.2

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

$\frac{- 8}{(- 8) + 4} < 5$

Step 2.1.10.1.3

The left side $2$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 2.1.10.2

Test a value on the interval $- 5 < x < - 4$ to see if it makes the inequality true.

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Step 2.1.10.2.1

Choose a value on the interval $- 5 < x < - 4$ and see if this value makes the original inequality true.

$x = - 4.5$

Step 2.1.10.2.2

Replace $x$ with $- 4.5$ in the original inequality.

$\frac{- 4.5}{(- 4.5) + 4} < 5$

Step 2.1.10.2.3

The left side $9$ is not less than the right side $5$ , which means that the given statement is false.

False

Step 2.1.10.3

Test a value on the interval $x > - 4$ to see if it makes the inequality true.

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Step 2.1.10.3.1

Choose a value on the interval $x > - 4$ and see if this value makes the original inequality true.

$x = 0$

Step 2.1.10.3.2

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

$\frac{0}{(0) + 4} < 5$

Step 2.1.10.3.3

The left side $0$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 2.1.10.4

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

$x < - 5$ True

$- 5 < x < - 4$ False

$x > - 4$ True

$x < - 5$ True

$- 5 < x < - 4$ False

$x > - 4$ True

Step 2.1.11

The solution consists of all of the true intervals.

$x < - 5$ or $x > - 4$

Step 2.2

Find the intersection of $x < - 5 or x > - 4$ and $x < - 4 or x \geq 0$ .

$x < - 5$ or $x \geq 0$

Step 3

Solve $- \frac{x}{x + 4} < 5$ when $- 4 < x < 0$ .

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Step 3.1

Solve $- \frac{x}{x + 4} < 5$ for $x$ .

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Step 3.1.1

Subtract $5$ from both sides of the inequality.

$- \frac{x}{x + 4} - 5 < 0$

Step 3.1.2

Simplify $- \frac{x}{x + 4} - 5$ .

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Step 3.1.2.1

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$- \frac{x}{x + 4} - 5 \cdot \frac{x + 4}{x + 4} < 0$

Step 3.1.2.2

Combine $- 5$ and $\frac{x + 4}{x + 4}$ .

$- \frac{x}{x + 4} + \frac{- 5 (x + 4)}{x + 4} < 0$

Step 3.1.2.3

Combine the numerators over the common denominator.

$\frac{- x - 5 (x + 4)}{x + 4} < 0$

Step 3.1.2.4

Simplify the numerator.

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Step 3.1.2.4.1

Apply the distributive property.

$\frac{- x - 5 x - 5 \cdot 4}{x + 4} < 0$

Step 3.1.2.4.2

Multiply $- 5$ by $4$ .

$\frac{- x - 5 x - 20}{x + 4} < 0$

Step 3.1.2.4.3

Subtract $5 x$ from $- x$ .

$\frac{- 6 x - 20}{x + 4} < 0$

Step 3.1.2.4.4

Factor $2$ out of $- 6 x - 20$ .

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Step 3.1.2.4.4.1

Factor $2$ out of $- 6 x$ .

$\frac{2 (- 3 x) - 20}{x + 4} < 0$

Step 3.1.2.4.4.2

Factor $2$ out of $- 20$ .

$\frac{2 (- 3 x) + 2 (- 10)}{x + 4} < 0$

Step 3.1.2.4.4.3

Factor $2$ out of $2 (- 3 x) + 2 (- 10)$ .

$\frac{2 (- 3 x - 10)}{x + 4} < 0$

Step 3.1.2.5

Factor $- 1$ out of $- 3 x$ .

$\frac{2 (- (3 x) - 10)}{x + 4} < 0$

Step 3.1.2.6

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{2 (- (3 x) - 1 (10))}{x + 4} < 0$

Step 3.1.2.7

Factor $- 1$ out of $- (3 x) - 1 (10)$ .

$\frac{2 (- (3 x + 10))}{x + 4} < 0$

Step 3.1.2.8

Rewrite $- (3 x + 10)$ as $- 1 (3 x + 10)$ .

$\frac{2 (- 1 (3 x + 10))}{x + 4} < 0$

Step 3.1.2.9

Move the negative in front of the fraction.

$- \frac{2 (3 x + 10)}{x + 4} < 0$

Step 3.1.3

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

$3 x + 10 = 0$

$x + 4 = 0$

Step 3.1.4

Subtract $10$ from both sides of the equation.

$3 x = - 10$

Step 3.1.5

Divide each term in $3 x = - 10$ by $3$ and simplify.

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Step 3.1.5.1

Divide each term in $3 x = - 10$ by $3$ .

$\frac{3 x}{3} = \frac{- 10}{3}$

Step 3.1.5.2

Simplify the left side.

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Step 3.1.5.2.1

Cancel the common factor of $3$ .

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Step 3.1.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 10}{3}$

Step 3.1.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{3}$

Step 3.1.5.3

Simplify the right side.

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Step 3.1.5.3.1

Move the negative in front of the fraction.

$x = - \frac{10}{3}$

Step 3.1.6

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.1.7

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

$x = - \frac{10}{3}$

$x = - 4$

Step 3.1.8

Consolidate the solutions.

$x = - \frac{10}{3}, - 4$

Step 3.1.9

Find the domain of $- \frac{2 (3 x + 10)}{x + 4}$ .

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Step 3.1.9.1

Set the denominator in $\frac{2 (3 x + 10)}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 3.1.9.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.1.9.3

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

$(- \infty, - 4) \cup (- 4, \infty)$

Step 3.1.10

Use each root to create test intervals.

$x < - 4$

$- 4 < x < - \frac{10}{3}$

$x > - \frac{10}{3}$

Step 3.1.11

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 3.1.11.1

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

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Step 3.1.11.1.1

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 3.1.11.1.2

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

$- \frac{- 6}{(- 6) + 4} < 5$

Step 3.1.11.1.3

The left side $- 3$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 3.1.11.2

Test a value on the interval $- 4 < x < - \frac{10}{3}$ to see if it makes the inequality true.

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Step 3.1.11.2.1

Choose a value on the interval $- 4 < x < - \frac{10}{3}$ and see if this value makes the original inequality true.

$x = - 3.67$

Step 3.1.11.2.2

Replace $x$ with $- 3.67$ in the original inequality.

$- \frac{- 3.67}{(- 3.67) + 4} < 5$

Step 3.1.11.2.3

The left side $11.\overline{12}$ is not less than the right side $5$ , which means that the given statement is false.

False

Step 3.1.11.3

Test a value on the interval $x > - \frac{10}{3}$ to see if it makes the inequality true.

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Step 3.1.11.3.1

Choose a value on the interval $x > - \frac{10}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 3.1.11.3.2

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

$- \frac{0}{(0) + 4} < 5$

Step 3.1.11.3.3

The left side $0$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 3.1.11.4

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

$x < - 4$ True

$- 4 < x < - \frac{10}{3}$ False

$x > - \frac{10}{3}$ True

$x < - 4$ True

$- 4 < x < - \frac{10}{3}$ False

$x > - \frac{10}{3}$ True

Step 3.1.12

The solution consists of all of the true intervals.

$x < - 4$ or $x > - \frac{10}{3}$

Step 3.2

Find the intersection of $x < - 4 or x > - \frac{10}{3}$ and $- 4 < x < 0$ .

$- \frac{10}{3} < x < 0$

Step 4

Find the union of the solutions.

$x < - 5$ or $x > - \frac{10}{3}$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 5) \cup (- \frac{10}{3}, \infty)$

Step 6