Solve for x x^2+2x<0

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

Step 1

Convert the inequality to an equation.

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

Step 2

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

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

$x \cdot x + 2 x = 0$

Factor $x$ out of $2 x$ .

$x \cdot x + x \cdot 2 = 0$

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$x (x + 2) = 0$

Step 3

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

$x = 0$

$x + 2 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

The final solution is all the values that make $x (x + 2) = 0$ true.

$x = 0, - 2$

Step 7

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$x > 0$

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 $x < - 2$ 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.

${(- 4)}^{2} + 2 (- 4) < 0$

The left side $8$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Replace $x$ with $- 1$ in the original inequality.

${(- 1)}^{2} + 2 (- 1) < 0$

The left side $- 1$ 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.

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

The left side $8$ 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.

$x < - 2$ False

$- 2 < x < 0$ True

$x > 0$ False

$x < - 2$ False

$- 2 < x < 0$ True

$x > 0$ False

Step 9

The solution consists of all of the true intervals.

$- 2 < x < 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 0$

Interval Notation:

$(- 2, 0)$

Step 11

Solve for x x^2+2x<0

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