Convert to Interval Notation ((x+9)(x-11))/(x+7)<=0

$\frac{(x + 9) (x - 11)}{x + 7} \leq 0$

Step 1

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

$x + 9 = 0$

$x - 11 = 0$

$x + 7 = 0$

Step 2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 3

Add $11$ to both sides of the equation.

$x = 11$

Step 4

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 5

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

$x = - 9$

$x = 11$

$x = - 7$

Step 6

Consolidate the solutions.

$x = - 9, 11, - 7$

Step 7

Find the domain of $\frac{(x + 9) (x - 11)}{x + 7}$ .

Set the denominator in $\frac{(x + 9) (x - 11)}{x + 7}$ equal to $0$ to find where the expression is undefined.

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

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

$(- \infty, - 7) \cup (- 7, \infty)$

Step 8

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 7$

$- 7 < x < 11$

$x > 11$

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 $x < - 9$ 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.

$\frac{((- 12) + 9) ((- 12) - 11)}{(- 12) + 7} \leq 0$

The left side $- 13.8$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 9 < x < - 7$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < - 7$ and see if this value makes the original inequality true.

$x = - 8$

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

$\frac{((- 8) + 9) ((- 8) - 11)}{(- 8) + 7} \leq 0$

The left side $19$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 7 < x < 11$ to see if it makes the inequality true.

Choose a value on the interval $- 7 < x < 11$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{((0) + 9) ((0) - 11)}{(0) + 7} \leq 0$

The left side $- 14.14285714$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 11$ to see if it makes the inequality true.

Choose a value on the interval $x > 11$ and see if this value makes the original inequality true.

$x = 14$

Replace $x$ with $14$ in the original inequality.

$\frac{((14) + 9) ((14) - 11)}{(14) + 7} \leq 0$

The left side $3.\overline{285714}$ 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.

$x < - 9$ True

$- 9 < x < - 7$ False

$- 7 < x < 11$ True

$x > 11$ False

$x < - 9$ True

$- 9 < x < - 7$ False

$- 7 < x < 11$ True

$x > 11$ False

Step 10

The solution consists of all of the true intervals.

$x \leq - 9$ or $- 7 < x \leq 11$

Step 11

Convert the inequality to interval notation.

$(- \infty, - 9] \cup (- 7, 11]$

Step 12

Convert to Interval Notation ((x+9)(x-11))/(x+7)<=0

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