Algebra Examples

${(x + 2)}^{2} (x + 5) (x + 9) < 0$

Step 1

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

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

$x + 5 = 0$

$x + 9 = 0$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

$x + 2 = 0$

Step 2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 3.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 5

The final solution is all the values that make ${(x + 2)}^{2} (x + 5) (x + 9) < 0$ true.

$x = - 2, - 5, - 9$

Step 6

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 5$

$- 5 < x < - 2$

$x > - 2$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $x < - 9$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Step 7.1.2

Replace $x$ with $- 12$ in the original inequality.

${((- 12) + 2)}^{2} ((- 12) + 5) ((- 12) + 9) < 0$

Step 7.1.3

The left side $2100$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- 9 < x < - 5$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 9 < x < - 5$ and see if this value makes the original inequality true.

$x = - 7$

Step 7.2.2

Replace $x$ with $- 7$ in the original inequality.

${((- 7) + 2)}^{2} ((- 7) + 5) ((- 7) + 9) < 0$

Step 7.2.3

The left side $- 100$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

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

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Step 7.3.1

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

$x = - 4$

Step 7.3.2

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

${((- 4) + 2)}^{2} ((- 4) + 5) ((- 4) + 9) < 0$

Step 7.3.3

The left side $20$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.4

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

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Step 7.4.1

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

$x = 0$

Step 7.4.2

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

${((0) + 2)}^{2} ((0) + 5) ((0) + 9) < 0$

Step 7.4.3

The left side $180$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.5

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 9$ False

$- 9 < x < - 5$ True

$- 5 < x < - 2$ False

$x > - 2$ False

$x < - 9$ False

$- 9 < x < - 5$ True

$- 5 < x < - 2$ False

$x > - 2$ False

Step 8

The solution consists of all of the true intervals.

$- 9 < x < - 5$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < - 5$

Interval Notation:

$(- 9, - 5)$

Step 10