Solve the Inequality for x (x^2+6x+8)/(x-2)<=0

$\frac{x^{2} + 6 x + 8}{x - 2} \leq 0$

Step 1

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

$x^{2} + 6 x + 8 = 0$

$x - 2 = 0$

Step 2

Factor $x^{2} + 6 x + 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $6$ .

$2, 4$

Write the factored form using these integers.

$(x + 2) (x + 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 4 = 0$

Step 4

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6

The final solution is all the values that make $(x + 2) (x + 4) = 0$ true.

$x = - 2, - 4$

Step 7

Add $2$ to both sides of the equation.

$x = 2$

Step 8

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

$x = - 2, - 4$

$x = 2$

Step 9

Consolidate the solutions.

$x = - 2, - 4, 2$

Step 10

Find the domain of $\frac{x^{2} + 6 x + 8}{x - 2}$ .

Set the denominator in $\frac{x^{2} + 6 x + 8}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$(- \infty, 2) \cup (2, \infty)$

Step 11

Use each root to create test intervals.

$x < - 4$

$- 4 < x < - 2$

$- 2 < x < 2$

$x > 2$

Step 12

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 $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

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

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 4 < x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < - 2$ and see if this value makes the original inequality true.

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

$\frac{{(- 3)}^{2} + 6 (- 3) + 8}{(- 3) - 2} \leq 0$

The left side $0.2$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 0$

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

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

The left side $- 4$ 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.

$\frac{{(4)}^{2} + 6 (4) + 8}{(4) - 2} \leq 0$

The left side $24$ 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.

$x < - 4$ True

$- 4 < x < - 2$ False

$- 2 < x < 2$ True

$x > 2$ False

$x < - 4$ True

$- 4 < x < - 2$ False

$- 2 < x < 2$ True

$x > 2$ False

Step 13

The solution consists of all of the true intervals.

$x \leq - 4$ or $- 2 \leq x < 2$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 4 or - 2 \leq x < 2$

Interval Notation:

$(- \infty, - 4] \cup [- 2, 2)$

Step 15

Solve the Inequality for x (x^2+6x+8)/(x-2)<=0

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