Solve the Inequality for x x^2>2x

$x^{2} > 2 x$

Subtract $2 x$ from 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 $0$ .

$x - 2 = 0$

Add $2$ to 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.

$x < 0$

$0 < x < 2$

$x > 2$

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 $x < 0$ 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.

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

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

True

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

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

$x = 1$

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

${(1)}^{2} > 2 (1)$

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 $x > 2$ 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.

${(4)}^{2} > 2 (4)$

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

True

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

$x < 0$ True

$0 < x < 2$ False

$x > 2$ True

$x < 0$ True

$0 < x < 2$ False

$x > 2$ True

The solution consists of all of the true intervals.

$x < 0$ or $x > 2$

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > 2$

Interval Notation:

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

Solve the Inequality for x x^2>2x

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