Algebra Examples

$x^{2} + 11 x < - 18$

Step 1

Convert the inequality to an equation.

$x^{2} + 11 x = - 18$

Step 2

Add $18$ to both sides of the equation.

$x^{2} + 11 x + 18 = 0$

Step 3

Factor $x^{2} + 11 x + 18$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $11$ .

$2, 9$

Step 3.2

Write the factored form using these integers.

$(x + 2) (x + 9) = 0$

Step 4

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

$x + 2 = 0$

$x + 9 = 0$

Step 5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 5.1

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

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

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Step 6.1

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

$x + 9 = 0$

Step 6.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 7

The final solution is all the values that make $(x + 2) (x + 9) = 0$ true.

$x = - 2, - 9$

Step 8

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 2$

$x > - 2$

Step 9

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 9.1

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

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Step 9.1.1

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

$x = - 12$

Step 9.1.2

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

${(- 12)}^{2} + 11 (- 12) < - 18$

Step 9.1.3

The left side $12$ is not less than the right side $- 18$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- 9 < x < - 2$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 9 < x < - 2$ and see if this value makes the original inequality true.

$x = - 6$

Step 9.2.2

Replace $x$ with $- 6$ in the original inequality.

${(- 6)}^{2} + 11 (- 6) < - 18$

Step 9.2.3

The left side $- 30$ is less than the right side $- 18$ , which means that the given statement is always true.

True

Step 9.3

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

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Step 9.3.1

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

$x = 0$

Step 9.3.2

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

${(0)}^{2} + 11 (0) < - 18$

Step 9.3.3

The left side $0$ is not less than the right side $- 18$ , which means that the given statement is false.

False

Step 9.4

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

$x < - 9$ False

$- 9 < x < - 2$ True

$x > - 2$ False

$x < - 9$ False

$- 9 < x < - 2$ True

$x > - 2$ False

Step 10

The solution consists of all of the true intervals.

$- 9 < x < - 2$

Step 11

Convert the inequality to interval notation.

$(- 9, - 2)$

Step 12