Solve the Inequality for x x^2-5x<0

$x^{2} - 5 x < 0$

Step 1

Convert the inequality to an equation.

$x^{2} - 5 x = 0$

Step 2

Factor $x$ out of $x^{2} - 5 x$ .

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

$x \cdot x - 5 x = 0$

Factor $x$ out of $- 5 x$ .

$x \cdot x + x \cdot - 5 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$x (x - 5) = 0$

Step 3

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

$x = 0$

$x - 5 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 6

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 7

Use each root to create test intervals.

$x < 0$

$0 < x < 5$

$x > 5$

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.

${(- 2)}^{2} - 5 \cdot - 2 < 0$

The left side $14$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 5$ and see if this value makes the original inequality true.

$x = 2$

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

${(2)}^{2} - 5 \cdot 2 < 0$

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

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

${(8)}^{2} - 5 \cdot 8 < 0$

The left side $24$ 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

$0 < x < 5$ True

$x > 5$ False

$x < 0$ False

$0 < x < 5$ True

$x > 5$ False

Step 9

The solution consists of all of the true intervals.

$0 < x < 5$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 5$

Interval Notation:

$(0, 5)$

Step 11

Solve the Inequality for x x^2-5x<0

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