Solve the Inequality for x x^2-3x<=0

$x^{2} - 3 x \leq 0$

Convert the inequality to an equation.

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

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

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

$x \cdot x - 3 x = 0$

Factor $x$ out of $- 3 x$ .

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

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

$x (x - 3) = 0$

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

$x = 0$

$x - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

$x = 0, 3$

Use each root to create test intervals.

$x < 0$

$0 < x < 3$

$x > 3$

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 $10$ is 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.

${(2)}^{2} - 3 \cdot 2 \leq 0$

The left side $- 2$ is less 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.

${(6)}^{2} - 3 \cdot 6 \leq 0$

The left side $18$ is 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

The solution consists of all of the true intervals.

$0 \leq x \leq 3$

The result can be shown in multiple forms.

Inequality Form:

$0 \leq x \leq 3$

Interval Notation:

$[0, 3]$

Solve the Inequality for x x^2-3x<=0

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