Precalculus Examples

$(x - 5) (x - 2) (x + 1) > 0$

Step 1

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

$x - 5 = 0$

$x - 2 = 0$

$x + 1 = 0$

Step 2

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

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Step 2.1

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

$x - 5 = 0$

Step 2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 3

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

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Step 3.1

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

$x - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

The final solution is all the values that make $(x - 5) (x - 2) (x + 1) > 0$ true.

$x = 5, 2, - 1$

Step 6

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 2$

$2 < x < 5$

$x > 5$

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 $x < - 1$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 7.1.2

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

$((- 4) - 5) ((- 4) - 2) ((- 4) + 1) > 0$

Step 7.1.3

The left side $- 162$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- 1 < x < 2$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 1 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

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

$((0) - 5) ((0) - 2) ((0) + 1) > 0$

Step 7.2.3

The left side $10$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

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

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Step 7.3.1

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

$x = 4$

Step 7.3.2

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

$((4) - 5) ((4) - 2) ((4) + 1) > 0$

Step 7.3.3

The left side $- 10$ is not greater than the right side $0$ , which means that the given statement is false.

False

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.

$((8) - 5) ((8) - 2) ((8) + 1) > 0$

Step 7.4.3

The left side $162$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.5

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

$x < - 1$ False

$- 1 < x < 2$ True

$2 < x < 5$ False

$x > 5$ True

$x < - 1$ False

$- 1 < x < 2$ True

$2 < x < 5$ False

$x > 5$ True

Step 8

The solution consists of all of the true intervals.

$- 1 < x < 2$ or $x > 5$

Step 9

Convert the inequality to interval notation.

$(- 1, 2) \cup (5, \infty)$

Step 10