Solve the Inequality for x (x-4)(x+1)(x-6)^2>=0

$(x - 4) (x + 1) {(x - 6)}^{2} \geq 0$

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

$x - 4 = 0$

$x + 1 = 0$

${(x - 6)}^{2} = 0$

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.

$x = 4$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set ${(x - 6)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 6)}^{2}$ equal to $0$ .

${(x - 6)}^{2} = 0$

Solve ${(x - 6)}^{2} = 0$ for $x$ .

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $(x - 4) (x + 1) {(x - 6)}^{2} \geq 0$ true.

$x = 4, - 1, 6$

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 4$

$4 < x < 6$

$x > 6$

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 $x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$((- 4) - 4) ((- 4) + 1) {((- 4) - 6)}^{2} \geq 0$

The left side $2400$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$((0) - 4) ((0) + 1) {((0) - 6)}^{2} \geq 0$

The left side $- 144$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $4 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 6$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$((5) - 4) ((5) + 1) {((5) - 6)}^{2} \geq 0$

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

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

$((8) - 4) ((8) + 1) {((8) - 6)}^{2} \geq 0$

The left side $144$ 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.

$x < - 1$ True

$- 1 < x < 4$ False

$4 < x < 6$ True

$x > 6$ True

$x < - 1$ True

$- 1 < x < 4$ False

$4 < x < 6$ True

$x > 6$ True

The solution consists of all of the true intervals.

$x \leq - 1$ or $4 \leq x \leq 6$ or $x \geq 6$

Combine the intervals.

$x \leq - 1 or x \geq 4$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 1 or x \geq 4$

Interval Notation:

$(- \infty, - 1] \cup [4, \infty)$

Solve the Inequality for x (x-4)(x+1)(x-6)^2>=0

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