Solve the Inequality for x (x-7)^3(x+2)^2(x-13)<0

${(x - 7)}^{3} {(x + 2)}^{2} (x - 13) < 0$

Step 1

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

${(x - 7)}^{3} = 0$

${(x + 2)}^{2} = 0$

$x - 13 = 0$

Step 2

Set ${(x - 7)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 7)}^{3}$ equal to $0$ .

${(x - 7)}^{3} = 0$

Solve ${(x - 7)}^{3} = 0$ for $x$ .

Set the $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 3

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

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

${(x + 2)}^{2} = 0$

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

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

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

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

$x - 13 = 0$

Add $13$ to both sides of the equation.

$x = 13$

Step 5

The final solution is all the values that make ${(x - 7)}^{3} {(x + 2)}^{2} (x - 13) < 0$ true.

$x = 7, - 2, 13$

Step 6

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 7$

$7 < x < 13$

$x > 13$

Step 7

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 $x < - 2$ to see if it makes the inequality true.

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

$x = - 4$

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

${((- 4) - 7)}^{3} {((- 4) + 2)}^{2} ((- 4) - 13) < 0$

The left side $90508$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

${((0) - 7)}^{3} {((0) + 2)}^{2} ((0) - 13) < 0$

The left side $17836$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $7 < x < 13$ to see if it makes the inequality true.

Choose a value on the interval $7 < x < 13$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

${((10) - 7)}^{3} {((10) + 2)}^{2} ((10) - 13) < 0$

The left side $- 11664$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

${((16) - 7)}^{3} {((16) + 2)}^{2} ((16) - 13) < 0$

The left side $708588$ 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.

$x < - 2$ False

$- 2 < x < 7$ False

$7 < x < 13$ True

$x > 13$ False

$x < - 2$ False

$- 2 < x < 7$ False

$7 < x < 13$ True

$x > 13$ False

Step 8

The solution consists of all of the true intervals.

$7 < x < 13$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$7 < x < 13$

Interval Notation:

$(7, 13)$

Step 10

Solve the Inequality for x (x-7)^3(x+2)^2(x-13)<0

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