Convert to Interval Notation x/(x-6)>0

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

Step 1

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

$x = 0$

$x - 6 = 0$

Step 2

Add $6$ to both sides of the equation.

$x = 6$

Step 3

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

$x = 0$

$x = 6$

Step 4

Consolidate the solutions.

$x = 0, 6$

Step 5

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.

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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

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

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < 6$

$x > 6$

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 $x < 0$ 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

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

Choose a value on the interval $0 < x < 6$ 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 $x > 6$ 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

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

$x < 0$ True

$0 < x < 6$ False

$x > 6$ True

$x < 0$ True

$0 < x < 6$ False

$x > 6$ True

Step 8

The solution consists of all of the true intervals.

$x < 0$ or $x > 6$

Step 9

Convert the inequality to interval notation.

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

Step 10

Convert to Interval Notation x/(x-6)>0

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