Solve the Inequality for x x^2<=4x

$x^{2} \leq 4 x$

Step 1

Subtract $4 x$ from both sides of the inequality.

$x^{2} - 4 x \leq 0$

Step 2

Convert the inequality to an equation.

$x^{2} - 4 x = 0$

Step 3

Factor $x$ out of $x^{2} - 4 x$ .

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

$x \cdot x - 4 x = 0$

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

Step 4

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

$x = 0$

$x - 4 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 7

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

Step 8

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

Step 9

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.

${(- 2)}^{2} \leq 4 (- 2)$

The left side $4$ is greater than the right side $- 8$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

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

${(2)}^{2} \leq 4 (2)$

The left side $4$ is less than the right side $8$ , which means that the given statement is always true.

True

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

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

${(6)}^{2} \leq 4 (6)$

The left side $36$ is greater than the right side $24$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 10

The solution consists of all of the true intervals.

$0 \leq x \leq 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$0 \leq x \leq 4$

Interval Notation:

$[0, 4]$

Step 12

Solve the Inequality for x x^2<=4x

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