Solve the Inequality for x ((x+3)(x+5))/(x+2)>=0

$\frac{(x + 3) (x + 5)}{x + 2} \geq 0$

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

$x + 3 = 0$

$x + 5 = 0$

$x + 2 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Subtract $5$ from both sides of the equation.

$x = - 5$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

$x = - 3$

$x = - 5$

$x = - 2$

Consolidate the solutions.

$x = - 3, - 5, - 2$

Find the domain of $\frac{(x + 3) (x + 5)}{x + 2}$ .

Set the denominator in $\frac{(x + 3) (x + 5)}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

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

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 3$

$- 3 < x < - 2$

$x > - 2$

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 $x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

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

$\frac{((- 8) + 3) ((- 8) + 5)}{(- 8) + 2} \geq 0$

The left side $- 2.5$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - 3$ and see if this value makes the original inequality true.

$x = - 4$

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

$\frac{((- 4) + 3) ((- 4) + 5)}{(- 4) + 2} \geq 0$

The left side $0.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 3 < x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < - 2$ and see if this value makes the original inequality true.

$x = - 2.5$

Replace $x$ with $- 2.5$ in the original inequality.

$\frac{((- 2.5) + 3) ((- 2.5) + 5)}{(- 2.5) + 2} \geq 0$

The left side $- 2.5$ is less than the right side $0$ , which means that the given statement is false.

False

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.

$x = 0$

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

$\frac{((0) + 3) ((0) + 5)}{(0) + 2} \geq 0$

The left side $7.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - 5$ False

$- 5 < x < - 3$ True

$- 3 < x < - 2$ False

$x > - 2$ True

$x < - 5$ False

$- 5 < x < - 3$ True

$- 3 < x < - 2$ False

$x > - 2$ True

The solution consists of all of the true intervals.

$- 5 \leq x \leq - 3$ or $x > - 2$

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq - 3 or x > - 2$

Interval Notation:

$[- 5, - 3] \cup (- 2, \infty)$

Solve the Inequality for x ((x+3)(x+5))/(x+2)>=0

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