Solve the Inequality for x x(2-3x)<=0

$x (2 - 3 x) \leq 0$

Step 1

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

$x = 0$

$2 - 3 x = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set $2 - 3 x$ equal to $0$ and solve for $x$ .

Set $2 - 3 x$ equal to $0$ .

$2 - 3 x = 0$

Solve $2 - 3 x = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$- 3 x = - 2$

Divide each term in $- 3 x = - 2$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 2$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 2}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 2}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{2}{3}$

Step 4

The final solution is all the values that make $x (2 - 3 x) \leq 0$ true.

$x = 0, \frac{2}{3}$

Step 5

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{2}{3}$

$x > \frac{2}{3}$

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.

$(- 2) (2 - 3 \cdot - 2) \leq 0$

The left side $- 16$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < \frac{2}{3}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{2}{3}$ and see if this value makes the original inequality true.

$x = 0.33$

Replace $x$ with $0.33$ in the original inequality.

$(0.33) (2 - 3 \cdot 0.33) \leq 0$

The left side $0.3333$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{2}{3}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{2}{3}$ and see if this value makes the original inequality true.

$x = 3$

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

$(3) (2 - 3 \cdot 3) \leq 0$

The left side $- 21$ is less 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.

$x < 0$ True

$0 < x < \frac{2}{3}$ False

$x > \frac{2}{3}$ True

$x < 0$ True

$0 < x < \frac{2}{3}$ False

$x > \frac{2}{3}$ True

Step 7

The solution consists of all of the true intervals.

$x \leq 0$ or $x \geq \frac{2}{3}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0 or x \geq \frac{2}{3}$

Interval Notation:

$(- \infty, 0] \cup [\frac{2}{3}, \infty)$

Step 9

Solve the Inequality for x x(2-3x)<=0

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