Solve the Inequality for x x^2>-2x

$x^{2} > - 2 x$

Add $2 x$ to both sides of the inequality.

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

Convert the inequality to an equation.

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

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 $x \cdot x + x \cdot 2$ .

$x (x + 2) = 0$

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

$x = 0$

$x + 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

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.

$x = - 2$

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

$x = 0, - 2$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$x > 0$

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.

${(- 4)}^{2} > - 2 \cdot - 4$

The left side $16$ is greater than the right side $8$ , which means that the given statement is always true.

True

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.

${(- 1)}^{2} > - 2 \cdot - 1$

The left side $1$ is not greater than the right side $2$ , which means that the given statement is false.

False

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.

${(2)}^{2} > - 2 \cdot 2$

The left side $4$ is greater than the right side $- 4$ , which means that the given statement is always true.

True

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

$x < - 2$ True

$- 2 < x < 0$ False

$x > 0$ True

$x < - 2$ True

$- 2 < x < 0$ False

$x > 0$ True

The solution consists of all of the true intervals.

$x < - 2$ or $x > 0$

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 0$

Interval Notation:

$(- \infty, - 2) \cup (0, \infty)$

Solve the Inequality for x x^2>-2x

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