Solve the Inequality for x ((x+6)*(x-2))/(x+1)<=0

$\frac{(x + 6) \cdot (x - 2)}{x + 1} \leq 0$

Step 1

Simplify the left side.

Multiply $x + 6$ by $x - 2$ .

$\frac{(x + 6) (x - 2)}{x + 1} \leq 0$

Step 2

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

$x + 6 = 0$

$x - 2 = 0$

$x + 1 = 0$

Step 3

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 4

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

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

$x = - 6$

$x = 2$

$x = - 1$

Step 7

Consolidate the solutions.

$x = - 6, 2, - 1$

Step 8

Find the domain of $\frac{(x + 6) \cdot (x - 2)}{x + 1}$ .

Set the denominator in $\frac{(x + 6) \cdot (x - 2)}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$(- \infty, - 1) \cup (- 1, \infty)$

Step 9

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - 1$

$- 1 < x < 2$

$x > 2$

Step 10

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.

$\frac{((- 8) + 6) \cdot ((- 8) - 2)}{(- 8) + 1} \leq 0$

The left side $- 2.\overline{857142}$ 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.

$x = - 4$

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

$\frac{((- 4) + 6) \cdot ((- 4) - 2)}{(- 4) + 1} \leq 0$

The left side $4$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{((0) + 6) \cdot ((0) - 2)}{(0) + 1} \leq 0$

The left side $- 12$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{((4) + 6) \cdot ((4) - 2)}{(4) + 1} \leq 0$

The left side $4$ 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

$- 1 < x < 2$ True

$x > 2$ False

$x < - 6$ True

$- 6 < x < - 1$ False

$- 1 < x < 2$ True

$x > 2$ False

Step 11

The solution consists of all of the true intervals.

$x \leq - 6$ or $- 1 < x \leq 2$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 6 or - 1 < x \leq 2$

Interval Notation:

$(- \infty, - 6] \cup (- 1, 2]$

Step 13

Solve the Inequality for x ((x+6)*(x-2))/(x+1)<=0

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