Precalculus Examples

$\frac{x}{x + 1} > 6 x$

Step 1

Subtract $6 x$ from both sides of the inequality.

$\frac{x}{x + 1} - 6 x > 0$

Step 2

Simplify $\frac{x}{x + 1} - 6 x$ .

Tap for more steps...

Step 2.1

To write $- 6 x$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x}{x + 1} - 6 x \cdot \frac{x + 1}{x + 1} > 0$

Step 2.2

Combine $- 6 x$ and $\frac{x + 1}{x + 1}$ .

$\frac{x}{x + 1} + \frac{- 6 x (x + 1)}{x + 1} > 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x - 6 x (x + 1)}{x + 1} > 0$

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

Factor $x$ out of $x - 6 x (x + 1)$ .

Tap for more steps...

Step 2.4.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - 6 x (x + 1)}{x + 1} > 0$

Step 2.4.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - 6 x (x + 1)}{x + 1} > 0$

Step 2.4.1.3

Factor $x$ out of $- 6 x (x + 1)$ .

$\frac{x \cdot 1 + x (- 6 (x + 1))}{x + 1} > 0$

Step 2.4.1.4

Factor $x$ out of $x \cdot 1 + x (- 6 (x + 1))$ .

$\frac{x (1 - 6 (x + 1))}{x + 1} > 0$

Step 2.4.2

Apply the distributive property.

$\frac{x (1 - 6 x - 6 \cdot 1)}{x + 1} > 0$

Step 2.4.3

Multiply $- 6$ by $1$ .

$\frac{x (1 - 6 x - 6)}{x + 1} > 0$

Step 2.4.4

Subtract $6$ from $1$ .

$\frac{x (- 6 x - 5)}{x + 1} > 0$

Step 2.5

Factor $- 1$ out of $- 6 x$ .

$\frac{x (- (6 x) - 5)}{x + 1} > 0$

Step 2.6

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

$\frac{x (- (6 x) - 1 (5))}{x + 1} > 0$

Step 2.7

Factor $- 1$ out of $- (6 x) - 1 (5)$ .

$\frac{x (- (6 x + 5))}{x + 1} > 0$

Step 2.8

Rewrite $- (6 x + 5)$ as $- 1 (6 x + 5)$ .

$\frac{x (- 1 (6 x + 5))}{x + 1} > 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{x (6 x + 5)}{x + 1} > 0$

Step 3

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

$- x = 0$

$6 x + 5 = 0$

$x + 1 = 0$

Step 4

Divide each term in $- x = 0$ by $- 1$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 5

Subtract $5$ from both sides of the equation.

$6 x = - 5$

Step 6

Divide each term in $6 x = - 5$ by $6$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $6 x = - 5$ by $6$ .

$\frac{6 x}{6} = \frac{- 5}{6}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 5}{6}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{6}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{6}$

Step 7

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 8

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

$x = 0$

$x = - \frac{5}{6}$

$x = - 1$

Step 9

Consolidate the solutions.

$x = 0, - \frac{5}{6}, - 1$

Step 10

Find the domain of $- \frac{x (6 x + 5)}{x + 1}$ .

Tap for more steps...

Step 10.1

Set the denominator in $\frac{x (6 x + 5)}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Step 10.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 10.3

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

$(- \infty, - 1) \cup (- 1, \infty)$

Step 11

Use each root to create test intervals.

$x < - 1$

$- 1 < x < - \frac{5}{6}$

$- \frac{5}{6} < x < 0$

$x > 0$

Step 12

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

Tap for more steps...

Step 12.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Tap for more steps...

Step 12.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 12.1.2

Replace $x$ with $- 4$ in the original inequality.

$\frac{- 4}{(- 4) + 1} > 6 (- 4)$

Step 12.1.3

The left side $1.\overline{3}$ is greater than the right side $- 24$ , which means that the given statement is always true.

True

Step 12.2

Test a value on the interval $- 1 < x < - \frac{5}{6}$ to see if it makes the inequality true.

Tap for more steps...

Step 12.2.1

Choose a value on the interval $- 1 < x < - \frac{5}{6}$ and see if this value makes the original inequality true.

$x = - 0.92$

Step 12.2.2

Replace $x$ with $- 0.92$ in the original inequality.

$\frac{- 0.92}{(- 0.92) + 1} > 6 (- 0.92)$

Step 12.2.3

The left side $- 11.5$ is not greater than the right side $- 5.52$ , which means that the given statement is false.

False

Step 12.3

Test a value on the interval $- \frac{5}{6} < x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 12.3.1

Choose a value on the interval $- \frac{5}{6} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.42$

Step 12.3.2

Replace $x$ with $- 0.42$ in the original inequality.

$\frac{- 0.42}{(- 0.42) + 1} > 6 (- 0.42)$

Step 12.3.3

The left side $- 0.72413793$ is greater than the right side $- 2.52$ , which means that the given statement is always true.

True

Step 12.4

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

Tap for more steps...

Step 12.4.1

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

$x = 2$

Step 12.4.2

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

$\frac{2}{(2) + 1} > 6 (2)$

Step 12.4.3

The left side $0.\overline{6}$ is not greater than the right side $12$ , which means that the given statement is false.

False

Step 12.5

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

$x < - 1$ True

$- 1 < x < - \frac{5}{6}$ False

$- \frac{5}{6} < x < 0$ True

$x > 0$ False

$x < - 1$ True

$- 1 < x < - \frac{5}{6}$ False

$- \frac{5}{6} < x < 0$ True

$x > 0$ False

Step 13

The solution consists of all of the true intervals.

$x < - 1$ or $- \frac{5}{6} < x < 0$

Step 14

Convert the inequality to interval notation.

$(- \infty, - 1) \cup (- \frac{5}{6}, 0)$

Step 15