Solve the Inequality for x ((x-9)(x+9))/x>=0

$\frac{(x - 9) (x + 9)}{x} \geq 0$

Step 1

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

$x = 0$

$x - 9 = 0$

$x + 9 = 0$

Step 2

Add $9$ to both sides of the equation.

$x = 9$

Step 3

Subtract $9$ from 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.

$x = 0$

$x = 9$

$x = - 9$

Step 5

Consolidate the solutions.

$x = 0, 9, - 9$

Step 6

Find the domain of $\frac{(x - 9) (x + 9)}{x}$ .

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

$x = 0$

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

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

Step 7

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 0$

$0 < x < 9$

$x > 9$

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 $x < - 9$ 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.

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

The left side $- 5.25$ is less than the right side $0$ , which means that the given statement is false.

False

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

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

$x = - 4$

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

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

The left side $16.25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < 9$ 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.

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

The left side $- 16.25$ 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.

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

The left side $5.25$ 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.

$x < - 9$ False

$- 9 < x < 0$ True

$0 < x < 9$ False

$x > 9$ True

$x < - 9$ False

$- 9 < x < 0$ True

$0 < x < 9$ False

$x > 9$ True

Step 9

The solution consists of all of the true intervals.

$- 9 \leq x < 0$ or $x \geq 9$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 9 \leq x < 0 or x \geq 9$

Interval Notation:

$[- 9, 0) \cup [9, \infty)$

Step 11

Solve the Inequality for x ((x-9)(x+9))/x>=0

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