Solve the Inequality for x x(x-6)<0

$x (x - 6) < 0$

Step 1

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

$x = 0$

$x - 6 = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 4

The final solution is all the values that make $x (x - 6) < 0$ true.

$x = 0, 6$

Step 5

Use each root to create test intervals.

$x < 0$

$0 < x < 6$

$x > 6$

Step 6

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.

$(- 2) ((- 2) - 6) < 0$

The left side $16$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 6$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$(3) ((3) - 6) < 0$

The left side $- 9$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$(8) ((8) - 6) < 0$

The left side $16$ 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.

$x < 0$ False

$0 < x < 6$ True

$x > 6$ False

$x < 0$ False

$0 < x < 6$ True

$x > 6$ False

Step 7

The solution consists of all of the true intervals.

$0 < x < 6$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 6$

Interval Notation:

$(0, 6)$

Step 9

Solve the Inequality for x x(x-6)<0

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