Solve the Inequality for x ((x+6)(x-1))/(x-9)<=0

$\frac{(x + 6) (x - 1)}{x - 9} \leq 0$

Step 1

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

$x + 6 = 0$

$x - 1 = 0$

$x - 9 = 0$

Step 2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 3

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Add $9$ to both sides of the equation.

$x = 9$

Step 5

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

$x = - 6$

$x = 1$

$x = 9$

Step 6

Consolidate the solutions.

$x = - 6, 1, 9$

Step 7

Find the domain of $\frac{(x + 6) (x - 1)}{x - 9}$ .

Set the denominator in $\frac{(x + 6) (x - 1)}{x - 9}$ 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 8

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 1$

$1 < x < 9$

$x > 9$

Step 9

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.

$\frac{((- 8) + 6) ((- 8) - 1)}{(- 8) - 9} \leq 0$

The left side $- 1.05882352$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 6 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$\frac{((0) + 6) ((0) - 1)}{(0) - 9} \leq 0$

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

False

Test a value on the interval $1 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 9$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$\frac{((5) + 6) ((5) - 1)}{(5) - 9} \leq 0$

The left side $- 11$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$\frac{((12) + 6) ((12) - 1)}{(12) - 9} \leq 0$

The left side $66$ is greater than the right side $0$ , which means that the given statement is false.

False

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

$x < - 6$ True

$- 6 < x < 1$ False

$1 < x < 9$ True

$x > 9$ False

$x < - 6$ True

$- 6 < x < 1$ False

$1 < x < 9$ True

$x > 9$ False

Step 10

The solution consists of all of the true intervals.

$x \leq - 6$ or $1 \leq x < 9$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 6 or 1 \leq x < 9$

Interval Notation:

$(- \infty, - 6] \cup [1, 9)$

Step 12

Solve the Inequality for x ((x+6)(x-1))/(x-9)<=0

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