Solve the Inequality for x x/(x+1)>=0

$\frac{x}{x + 1} \geq 0$

Step 1

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

$x = 0$

$x + 1 = 0$

Step 2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3

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

$x = 0$

$x = - 1$

Step 4

Consolidate the solutions.

$x = 0, - 1$

Step 5

Find the domain of $\frac{x}{x + 1}$ .

Set the denominator in $\frac{x}{x + 1}$ 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 6

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 0$

$x > 0$

Step 7

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.

$\frac{- 4}{(- 4) + 1} \geq 0$

The left side $1.\overline{3}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 0$ and see if this value makes the original inequality true.

$x = - 0.5$

Replace $x$ with $- 0.5$ in the original inequality.

$\frac{- 0.5}{(- 0.5) + 1} \geq 0$

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

False

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.

$\frac{2}{(2) + 1} \geq 0$

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

True

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

$x < - 1$ True

$- 1 < x < 0$ False

$x > 0$ True

$x < - 1$ True

$- 1 < x < 0$ False

$x > 0$ True

Step 8

The solution consists of all of the true intervals.

$x < - 1$ or $x \geq 0$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x \geq 0$

Interval Notation:

$(- \infty, - 1) \cup [0, \infty)$

Step 10

Solve the Inequality for x x/(x+1)>=0

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