Solve the Inequality for x (x+5)(x+3)(x-7)<0

$(x + 5) (x + 3) (x - 7) < 0$

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

$x + 5 = 0$

$x + 3 = 0$

$x - 7 = 0$

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.

$x = - 5$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $(x + 5) (x + 3) (x - 7) < 0$ true.

$x = - 5, - 3, 7$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - 3$

$- 3 < x < 7$

$x > 7$

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.

$((- 8) + 5) ((- 8) + 3) ((- 8) - 7) < 0$

The left side $- 225$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$((- 4) + 5) ((- 4) + 3) ((- 4) - 7) < 0$

The left side $11$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) + 5) ((0) + 3) ((0) - 7) < 0$

The left side $- 105$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$((10) + 5) ((10) + 3) ((10) - 7) < 0$

The left side $585$ is not less 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.

$x < - 5$ True

$- 5 < x < - 3$ False

$- 3 < x < 7$ True

$x > 7$ False

$x < - 5$ True

$- 5 < x < - 3$ False

$- 3 < x < 7$ True

$x > 7$ False

The solution consists of all of the true intervals.

$x < - 5$ or $- 3 < x < 7$

The result can be shown in multiple forms.

Inequality Form:

$x < - 5 or - 3 < x < 7$

Interval Notation:

$(- \infty, - 5) \cup (- 3, 7)$

Solve the Inequality for x (x+5)(x+3)(x-7)<0

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