Simplify (8x)/(x-9)>=0

$\frac{8 x}{x - 9} \geq 0$

Step 1

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

$8 x = 0$

$x - 9 = 0$

Step 2

Divide each term in $8 x = 0$ by $8$ and simplify.

Divide each term in $8 x = 0$ by $8$ .

$\frac{8 x}{8} = \frac{0}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{0}{8}$

Divide $x$ by $1$ .

$x = \frac{0}{8}$

Simplify the right side.

Divide $0$ by $8$ .

$x = 0$

Step 3

Add $9$ to both sides of the equation.

$x = 9$

Step 4

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

$x = 0$

$x = 9$

Step 5

Consolidate the solutions.

$x = 0, 9$

Step 6

Find the domain of $\frac{8 x}{x - 9}$ .

Set the denominator in $\frac{8 x}{x - 9}$ equal to $0$ to find where the expression is undefined.

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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

$(- \infty, 9) \cup (9, \infty)$

Step 7

Use each root to create test intervals.

$x < 0$

$0 < x < 9$

$x > 9$

Step 8

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.

$\frac{8 (- 2)}{(- 2) - 9} \geq 0$

The left side $1.\overline{45}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 9$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{8 (4)}{(4) - 9} \geq 0$

The left side $- 6.4$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$\frac{8 (12)}{(12) - 9} \geq 0$

The left side $32$ 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

$0 < x < 9$ False

$x > 9$ True

$x < 0$ True

$0 < x < 9$ False

$x > 9$ True

Step 9

The solution consists of all of the true intervals.

$x \leq 0$ or $x > 9$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0 or x > 9$

Interval Notation:

$(- \infty, 0] \cup (9, \infty)$

Step 11

Simplify (8x)/(x-9)>=0

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