Precalculus Examples

$\frac{x^{2} + 9 x - 22}{x + 4} \geq 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} + 9 x - 22 = 0$

$x + 4 = 0$

Step 2

Factor $x^{2} + 9 x - 22$ using the AC method.

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Step 2.1

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 $- 22$ and whose sum is $9$ .

$- 2, 11$

Step 2.2

Write the factored form using these integers.

$(x - 2) (x + 11) = 0$

Step 3

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

$x - 2 = 0$

$x + 11 = 0$

Step 4

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

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Step 4.1

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

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Step 5.2

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 6

The final solution is all the values that make $(x - 2) (x + 11) = 0$ true.

$x = 2, - 11$

Step 7

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 8

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 2, - 11$

$x = - 4$

Step 9

Consolidate the solutions.

$x = 2, - 11, - 4$

Step 10

Find the domain of $\frac{x^{2} + 9 x - 22}{x + 4}$ .

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Step 10.1

Set the denominator in $\frac{x^{2} + 9 x - 22}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Step 10.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 10.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 4) \cup (- 4, \infty)$

Step 11

Use each root to create test intervals.

$x < - 11$

$- 11 < x < - 4$

$- 4 < x < 2$

$x > 2$

Step 12

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 12.1

Test a value on the interval $x < - 11$ to see if it makes the inequality true.

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Step 12.1.1

Choose a value on the interval $x < - 11$ and see if this value makes the original inequality true.

$x = - 14$

Step 12.1.2

Replace $x$ with $- 14$ in the original inequality.

$\frac{{(- 14)}^{2} + 9 (- 14) - 22}{(- 14) + 4} \geq 0$

Step 12.1.3

The left side $- 4.8$ is less than the right side $0$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $- 11 < x < - 4$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- 11 < x < - 4$ and see if this value makes the original inequality true.

$x = - 8$

Step 12.2.2

Replace $x$ with $- 8$ in the original inequality.

$\frac{{(- 8)}^{2} + 9 (- 8) - 22}{(- 8) + 4} \geq 0$

Step 12.2.3

The left side $7.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $- 4 < x < 2$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $- 4 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 12.3.2

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

$\frac{{(0)}^{2} + 9 (0) - 22}{(0) + 4} \geq 0$

Step 12.3.3

The left side $- 5.5$ is less than the right side $0$ , which means that the given statement is false.

False

Step 12.4

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

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Step 12.4.1

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

$x = 4$

Step 12.4.2

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

$\frac{{(4)}^{2} + 9 (4) - 22}{(4) + 4} \geq 0$

Step 12.4.3

The left side $3.75$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 12.5

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

$x < - 11$ False

$- 11 < x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

$x < - 11$ False

$- 11 < x < - 4$ True

$- 4 < x < 2$ False

$x > 2$ True

Step 13

The solution consists of all of the true intervals.

$- 11 \leq x < - 4$ or $x \geq 2$

Step 14

Convert the inequality to interval notation.

$[- 11, - 4) \cup [2, \infty)$

Step 15