Pre-Algebra Examples

$(x + 5) (x - 6) < 0$

Step 1

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

$x + 5 = 0$

$x - 6 = 0$

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

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4

The final solution is all the values that make $(x + 5) (x - 6) < 0$ true.

$x = - 5, 6$

Step 5

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 6$

$x > 6$

Step 6

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 6.1

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

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Step 6.1.1

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

$x = - 8$

Step 6.1.2

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

$((- 8) + 5) ((- 8) - 6) < 0$

Step 6.1.3

The left side $42$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.2

Test a value on the interval $- 5 < x < 6$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $- 5 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

Step 6.2.2

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

$((0) + 5) ((0) - 6) < 0$

Step 6.2.3

The left side $- 30$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.3

Test a value on the interval $x > 6$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Step 6.3.2

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

$((8) + 5) ((8) - 6) < 0$

Step 6.3.3

The left side $26$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.4

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

$x < - 5$ False

$- 5 < x < 6$ True

$x > 6$ False

$x < - 5$ False

$- 5 < x < 6$ True

$x > 6$ False

Step 7

The solution consists of all of the true intervals.

$- 5 < x < 6$

Step 8