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

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

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

$x - 1 = 0$

$x - 2 = 0$

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 $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 1) (x - 2) > 0$ true.

$x = 1, 2$

Use each root to create test intervals.

$x < 1$

$1 < x < 2$

$x > 2$

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 $x < 1$ 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.

$x = 0$

Replace $x$ with $0$ in the original inequality.

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

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 $1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 2$ and see if this value makes the original inequality true.

$x = 1.5$

Replace $x$ with $1.5$ in the original inequality.

$((1.5) - 1) ((1.5) - 2) > 0$

The left side $- 0.25$ is not greater than the right side $0$ , 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.

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

The left side $6$ 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.

$x < 1$ True

$1 < x < 2$ False

$x > 2$ True

$x < 1$ True

$1 < x < 2$ False

$x > 2$ True

The solution consists of all of the true intervals.

$x < 1$ or $x > 2$

The result can be shown in multiple forms.

Inequality Form:

$x < 1 or x > 2$

Interval Notation:

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

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

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