Solve the Inequality for x 3x^2+9x<0

$3 x^{2} + 9 x < 0$

Step 1

Convert the inequality to an equation.

$3 x^{2} + 9 x = 0$

Step 2

Factor $3 x$ out of $3 x^{2} + 9 x$ .

Factor $3 x$ out of $3 x^{2}$ .

$3 x (x) + 9 x = 0$

Factor $3 x$ out of $9 x$ .

$3 x (x) + 3 x (3) = 0$

Factor $3 x$ out of $3 x (x) + 3 x (3)$ .

$3 x (x + 3) = 0$

Step 3

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

$x = 0$

$x + 3 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 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.

$x = - 3$

Step 6

The final solution is all the values that make $3 x (x + 3) = 0$ true.

$x = 0, - 3$

Step 7

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 0$

$x > 0$

Step 8

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 $x < - 3$ to see if it makes the inequality true.

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

$x = - 6$

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

$3 {(- 6)}^{2} + 9 (- 6) < 0$

The left side $54$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$3 {(- 2)}^{2} + 9 (- 2) < 0$

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 0$ to see if it makes the inequality true.

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$3 {(2)}^{2} + 9 (2) < 0$

The left side $30$ 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.

$x < - 3$ False

$- 3 < x < 0$ True

$x > 0$ False

$x < - 3$ False

$- 3 < x < 0$ True

$x > 0$ False

Step 9

The solution consists of all of the true intervals.

$- 3 < x < 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 0$

Interval Notation:

$(- 3, 0)$

Step 11

Solve the Inequality for x 3x^2+9x<0

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