Solve the Inequality for x x(x-1)(x+2)>0

$x (x - 1) (x + 2) > 0$

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

$x = 0$

$x - 1 = 0$

$x + 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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 $x (x - 1) (x + 2) > 0$ true.

$x = 0, 1, - 2$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 1$

$x > 1$

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.

$(- 4) ((- 4) - 1) ((- 4) + 2) > 0$

The left side $- 40$ is not greater 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.

$(- 1) ((- 1) - 1) ((- 1) + 2) > 0$

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Replace $x$ with $0.5$ in the original inequality.

$(0.5) ((0.5) - 1) ((0.5) + 2) > 0$

The left side $- 0.625$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 1$ to see if it makes the inequality true.

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$(4) ((4) - 1) ((4) + 2) > 0$

The left side $72$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - 2$ False

$- 2 < x < 0$ True

$0 < x < 1$ False

$x > 1$ True

$x < - 2$ False

$- 2 < x < 0$ True

$0 < x < 1$ False

$x > 1$ True

The solution consists of all of the true intervals.

$- 2 < x < 0$ or $x > 1$

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 0 or x > 1$

Interval Notation:

$(- 2, 0) \cup (1, \infty)$

Solve the Inequality for x x(x-1)(x+2)>0

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