Precalculus Examples

$(x + 4) (x + 9) (5 - x) \geq 0$

Step 1

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

$x + 4 = 0$

$x + 9 = 0$

$5 - x = 0$

Step 2

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3

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

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Step 3.1

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

$x + 9 = 0$

Step 3.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 4

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

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Step 4.1

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

$5 - x = 0$

Step 4.2

Solve $5 - x = 0$ for $x$ .

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Step 4.2.1

Subtract $5$ from both sides of the equation.

$- x = - 5$

Step 4.2.2

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 4.2.2.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 5

The final solution is all the values that make $(x + 4) (x + 9) (5 - x) \geq 0$ true.

$x = - 4, - 9, 5$

Step 6

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 4$

$- 4 < x < 5$

$x > 5$

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.

$((- 12) + 4) ((- 12) + 9) (5 - (- 12)) \geq 0$

Step 7.1.3

The left side $408$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $- 9 < x < - 4$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 9 < x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 7.2.2

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

$((- 6) + 4) ((- 6) + 9) (5 - (- 6)) \geq 0$

Step 7.2.3

The left side $- 66$ is less than the right side $0$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $- 4 < x < 5$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $- 4 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

Step 7.3.2

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

$((0) + 4) ((0) + 9) (5 - (0)) \geq 0$

Step 7.3.3

The left side $180$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

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

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Step 7.4.1

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

$x = 8$

Step 7.4.2

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

$((8) + 4) ((8) + 9) (5 - (8)) \geq 0$

Step 7.4.3

The left side $- 612$ is 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.

$x < - 9$ True

$- 9 < x < - 4$ False

$- 4 < x < 5$ True

$x > 5$ False

$x < - 9$ True

$- 9 < x < - 4$ False

$- 4 < x < 5$ True

$x > 5$ False

Step 8

The solution consists of all of the true intervals.

$x \leq - 9$ or $- 4 \leq x \leq 5$

Step 9

Convert the inequality to interval notation.

$(- \infty, - 9] \cup [- 4, 5]$

Step 10