Solve the Inequality for x (x+8)(x-3)^2(x-7)^3<0

$(x + 8) {(x - 3)}^{2} {(x - 7)}^{3} < 0$

Step 1

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

$x + 8 = 0$

${(x - 3)}^{2} = 0$

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

Step 2

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 3)}^{2}$ equal to $0$ .

${(x - 3)}^{2} = 0$

Solve ${(x - 3)}^{2} = 0$ for $x$ .

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 4

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 5

The final solution is all the values that make $(x + 8) {(x - 3)}^{2} {(x - 7)}^{3} < 0$ true.

$x = - 8, 3, 7$

Step 6

Use each root to create test intervals.

$x < - 8$

$- 8 < x < 3$

$3 < x < 7$

$x > 7$

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 $x < - 8$ to see if it makes the inequality true.

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

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

$((- 10) + 8) {((- 10) - 3)}^{2} {((- 10) - 7)}^{3} < 0$

The left side $1660594$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 8 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) + 8) {((0) - 3)}^{2} {((0) - 7)}^{3} < 0$

The left side $- 24696$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $3 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 7$ and see if this value makes the original inequality true.

$x = 5$

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

$((5) + 8) {((5) - 3)}^{2} {((5) - 7)}^{3} < 0$

The left side $- 416$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 10$

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

$((10) + 8) {((10) - 3)}^{2} {((10) - 7)}^{3} < 0$

The left side $23814$ 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.

$x < - 8$ False

$- 8 < x < 3$ True

$3 < x < 7$ True

$x > 7$ False

$x < - 8$ False

$- 8 < x < 3$ True

$3 < x < 7$ True

$x > 7$ False

Step 8

The solution consists of all of the true intervals.

$- 8 < x < 3$ or $3 < x < 7$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < 3 or 3 < x < 7$

Interval Notation:

$(- 8, 3) \cup (3, 7)$

Step 10

Solve the Inequality for x (x+8)(x-3)^2(x-7)^3<0

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