Solve the Inequality for x 4x^2-16x>=0

$4 x^{2} - 16 x \geq 0$

Step 1

Convert the inequality to an equation.

$4 x^{2} - 16 x = 0$

Step 2

Factor $4 x$ out of $4 x^{2} - 16 x$ .

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

$4 x (x) - 16 x = 0$

Factor $4 x$ out of $- 16 x$ .

$4 x (x) + 4 x (- 4) = 0$

Factor $4 x$ out of $4 x (x) + 4 x (- 4)$ .

$4 x (x - 4) = 0$

Step 3

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

$x = 0$

$x - 4 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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 6

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

$x = 0, 4$

Step 7

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

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.

$4 {(- 2)}^{2} - 16 \cdot - 2 \geq 0$

The left side $48$ is greater than the right side $0$ , which means that the given statement is always true.

True

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.

$4 {(2)}^{2} - 16 \cdot 2 \geq 0$

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

False

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.

$4 {(6)}^{2} - 16 \cdot 6 \geq 0$

The left side $48$ is greater 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

$0 < x < 4$ False

$x > 4$ True

$x < 0$ True

$0 < x < 4$ False

$x > 4$ True

Step 9

The solution consists of all of the true intervals.

$x \leq 0$ or $x \geq 4$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0 or x \geq 4$

Interval Notation:

$(- \infty, 0] \cup [4, \infty)$

Step 11

Solve the Inequality for x 4x^2-16x>=0

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