Solve the Inequality for x 3x-x^2>0

$3 x - x^{2} > 0$

Step 1

Convert the inequality to an equation.

$3 x - x^{2} = 0$

Step 2

Factor $x$ out of $3 x - x^{2}$ .

Factor $x$ out of $3 x$ .

$x \cdot 3 - x^{2} = 0$

Factor $x$ out of $- x^{2}$ .

$x \cdot 3 + x (- x) = 0$

Factor $x$ out of $x \cdot 3 + x (- x)$ .

$x (3 - x) = 0$

Step 3

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

$x = 0$

$3 - x = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

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

$3 - x = 0$

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

Subtract $3$ from both sides of the equation.

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

Step 6

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

$x = 0, 3$

Step 7

Use each root to create test intervals.

$x < 0$

$0 < x < 3$

$x > 3$

Step 8

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.

$3 (- 2) - {(- 2)}^{2} > 0$

The left side $- 10$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 3$ to see if it makes the inequality true.

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

$x = 2$

Replace $x$ with $2$ in the original inequality.

$3 (2) - {(2)}^{2} > 0$

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 $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$3 (6) - {(6)}^{2} > 0$

The left side $- 18$ is not greater 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

$0 < x < 3$ True

$x > 3$ False

$x < 0$ False

$0 < x < 3$ True

$x > 3$ False

Step 9

The solution consists of all of the true intervals.

$0 < x < 3$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 3$

Interval Notation:

$(0, 3)$

Step 11

Solve the Inequality for x 3x-x^2>0

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