Solve the Inequality for x (x^2-4)(x+13)(x-5)^2<=0

$(x^{2} - 4) (x + 13) {(x - 5)}^{2} \leq 0$

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

$x^{2} - 4 = 0$

$x + 13 = 0$

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

Set $x^{2} - 4$ equal to $0$ and solve for $x$ .

Set $x^{2} - 4$ equal to $0$ .

$x^{2} - 4 = 0$

Solve $x^{2} - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

$x^{2} = 4$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Set $x + 13$ equal to $0$ and solve for $x$ .

Set $x + 13$ equal to $0$ .

$x + 13 = 0$

Subtract $13$ from both sides of the equation.

$x = - 13$

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{2}$ equal to $0$ .

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

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x^{2} - 4) (x + 13) {(x - 5)}^{2} \leq 0$ true.

$x = 2, - 2, - 13, 5$

Use each root to create test intervals.

$x < - 13$

$- 13 < x < - 2$

$- 2 < x < 2$

$2 < x < 5$

$x > 5$

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 $x < - 13$ to see if it makes the inequality true.

Choose a value on the interval $x < - 13$ and see if this value makes the original inequality true.

$x = - 16$

Replace $x$ with $- 16$ in the original inequality.

$({(- 16)}^{2} - 4) ((- 16) + 13) {((- 16) - 5)}^{2} \leq 0$

The left side $- 333396$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 13 < x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $- 13 < x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

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

$({(- 4)}^{2} - 4) ((- 4) + 13) {((- 4) - 5)}^{2} \leq 0$

The left side $8748$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$({(0)}^{2} - 4) ((0) + 13) {((0) - 5)}^{2} \leq 0$

The left side $- 1300$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

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

$({(4)}^{2} - 4) ((4) + 13) {((4) - 5)}^{2} \leq 0$

The left side $204$ is greater than the right side $0$ , which means that the given statement is false.

False

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.

$({(8)}^{2} - 4) ((8) + 13) {((8) - 5)}^{2} \leq 0$

The left side $11340$ 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.

$x < - 13$ True

$- 13 < x < - 2$ False

$- 2 < x < 2$ True

$2 < x < 5$ False

$x > 5$ False

$x < - 13$ True

$- 13 < x < - 2$ False

$- 2 < x < 2$ True

$2 < x < 5$ False

$x > 5$ False

The solution consists of all of the true intervals.

$x \leq - 13$ or $- 2 \leq x \leq 2$ or $x = 5$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 13 or - 2 \leq x \leq 2 or x = 5$

Interval Notation:

$(- \infty, - 13] \cup [- 2, 2] \cup [5, 5]$

Solve the Inequality for x (x^2-4)(x+13)(x-5)^2<=0

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