Solve the Inequality for x x/(x+6)>0

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

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

$x = 0$

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

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

$x = 0$

$x = - 6$

Consolidate the solutions.

$x = 0, - 6$

Find the domain of $\frac{x}{x + 6}$ .

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

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

$(- \infty, - 6) \cup (- 6, \infty)$

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 0$

$x > 0$

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 $x < - 6$ to see if it makes the inequality true.

Choose a value on the interval $x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

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

$\frac{- 8}{(- 8) + 6} > 0$

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

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

The left side $- 1$ is not greater 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.

$\frac{2}{(2) + 6} > 0$

The left side $0.25$ 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.

$x < - 6$ True

$- 6 < x < 0$ False

$x > 0$ True

$x < - 6$ True

$- 6 < x < 0$ False

$x > 0$ True

The solution consists of all of the true intervals.

$x < - 6$ or $x > 0$

The result can be shown in multiple forms.

Inequality Form:

$x < - 6 or x > 0$

Interval Notation:

$(- \infty, - 6) \cup (0, \infty)$

Solve the Inequality for x x/(x+6)>0

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