Solve for x x/(x+1)>3x

$\frac{x}{x + 1} > 3 x$

Step 1

Subtract $3 x$ from both sides of the inequality.

$\frac{x}{x + 1} - 3 x > 0$

Step 2

Simplify $\frac{x}{x + 1} - 3 x$ .

To write $- 3 x$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x}{x + 1} - 3 x \cdot \frac{x + 1}{x + 1} > 0$

Combine $- 3 x$ and $\frac{x + 1}{x + 1}$ .

$\frac{x}{x + 1} + \frac{- 3 x (x + 1)}{x + 1} > 0$

Combine the numerators over the common denominator.

$\frac{x - 3 x (x + 1)}{x + 1} > 0$

Simplify the numerator.

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

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

$\frac{x^{1} - 3 x (x + 1)}{x + 1} > 0$

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

$\frac{x \cdot 1 - 3 x (x + 1)}{x + 1} > 0$

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

$\frac{x \cdot 1 + x (- 3 (x + 1))}{x + 1} > 0$

Factor $x$ out of $x \cdot 1 + x (- 3 (x + 1))$ .

$\frac{x (1 - 3 (x + 1))}{x + 1} > 0$

Apply the distributive property.

$\frac{x (1 - 3 x - 3 \cdot 1)}{x + 1} > 0$

Multiply $- 3$ by $1$ .

$\frac{x (1 - 3 x - 3)}{x + 1} > 0$

Subtract $3$ from $1$ .

$\frac{x (- 3 x - 2)}{x + 1} > 0$

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

$\frac{x (- (3 x) - 2)}{x + 1} > 0$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{x (- (3 x) - 1 (2))}{x + 1} > 0$

Factor $- 1$ out of $- (3 x) - 1 (2)$ .

$\frac{x (- (3 x + 2))}{x + 1} > 0$

Rewrite $- (3 x + 2)$ as $- 1 (3 x + 2)$ .

$\frac{x (- 1 (3 x + 2))}{x + 1} > 0$

Move the negative in front of the fraction.

$- \frac{x (3 x + 2)}{x + 1} > 0$

Step 3

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

$- x = 0$

$3 x + 2 = 0$

$x + 1 = 0$

Step 4

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $0$ by $- 1$ .

$x = 0$

Step 5

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 6

Divide each term in $3 x = - 2$ by $3$ and simplify.

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

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.

$x = 0$

$x = - \frac{2}{3}$

$x = - 1$

Step 9

Consolidate the solutions.

$x = 0, - \frac{2}{3}, - 1$

Step 10

Find the domain of $- \frac{x (3 x + 2)}{x + 1}$ .

Set the denominator in $\frac{x (3 x + 2)}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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.

$x < - 1$

$- 1 < x < - \frac{2}{3}$

$- \frac{2}{3} < x < 0$

$x > 0$

Step 12

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

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

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

$x = - 4$

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

$\frac{- 4}{(- 4) + 1} > 3 (- 4)$

The left side $1.\overline{3}$ is greater than the right side $- 12$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < - \frac{2}{3}$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < - \frac{2}{3}$ and see if this value makes the original inequality true.

$x = - 0.83$

Replace $x$ with $- 0.83$ in the original inequality.

$\frac{- 0.83}{(- 0.83) + 1} > 3 (- 0.83)$

The left side $- 4.88235294$ is not greater than the right side $- 2.49$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{2}{3} < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{2}{3} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.33$

Replace $x$ with $- 0.33$ in the original inequality.

$\frac{- 0.33}{(- 0.33) + 1} > 3 (- 0.33)$

The left side $- 0.49253731$ is greater than the right side $- 0.99$ , which means that the given statement is always true.

True

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

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

$x = 2$

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

$\frac{2}{(2) + 1} > 3 (2)$

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

False

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

$x < - 1$ True

$- 1 < x < - \frac{2}{3}$ False

$- \frac{2}{3} < x < 0$ True

$x > 0$ False

$x < - 1$ True

$- 1 < x < - \frac{2}{3}$ False

$- \frac{2}{3} < x < 0$ True

$x > 0$ False

Step 13

The solution consists of all of the true intervals.

$x < - 1$ or $- \frac{2}{3} < x < 0$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or - \frac{2}{3} < x < 0$

Interval Notation:

$(- \infty, - 1) \cup (- \frac{2}{3}, 0)$

Step 15

Solve for x x/(x+1)>3x

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