Solve the Inequality for x (x+1)(x+5)>0

$(x + 1) (x + 5) > 0$

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

$x + 1 = 0$

$x + 5 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x + 1) (x + 5) > 0$ true.

$x = - 1, - 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 1$

$x > - 1$

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.

$((- 8) + 1) ((- 8) + 5) > 0$

The left side $21$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 5 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - 1$ and see if this value makes the original inequality true.

$x = - 3$

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

$((- 3) + 1) ((- 3) + 5) > 0$

The left side $- 4$ is not greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 0$

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

$((0) + 1) ((0) + 5) > 0$

The left side $5$ 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.

$x < - 5$ True

$- 5 < x < - 1$ False

$x > - 1$ True

$x < - 5$ True

$- 5 < x < - 1$ False

$x > - 1$ True

The solution consists of all of the true intervals.

$x < - 5$ or $x > - 1$

The result can be shown in multiple forms.

Inequality Form:

$x < - 5 or x > - 1$

Interval Notation:

$(- \infty, - 5) \cup (- 1, \infty)$

Solve the Inequality for x (x+1)(x+5)>0

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