Solve the Inequality for x (x-2)/(x+4)<=0

$\frac{x - 2}{x + 4} \leq 0$

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

$x - 2 = 0$

$x + 4 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$x = 2$

$x = - 4$

Consolidate the solutions.

$x = 2, - 4$

Find the domain of $\frac{x - 2}{x + 4}$ .

Set the denominator in $\frac{x - 2}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$(- \infty, - 4) \cup (- 4, \infty)$

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 2$

$x > 2$

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.

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

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 $- 4 < x < 2$ 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.

$x = 0$

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

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

The left side $- 0.5$ 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.

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

The left side $0.25$ 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.

$x < - 4$ False

$- 4 < x < 2$ True

$x > 2$ False

$x < - 4$ False

$- 4 < x < 2$ True

$x > 2$ False

The solution consists of all of the true intervals.

$- 4 < x \leq 2$

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x \leq 2$

Interval Notation:

$(- 4, 2]$

Solve the Inequality for x (x-2)/(x+4)<=0

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