Pre-Algebra Examples

$(x + 3) (x - 2) (x - 1) < 0$

Step 1

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

$x + 3 = 0$

$x - 2 = 0$

$x - 1 = 0$

Step 2

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

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

Add $1$ to both sides of the equation.

$x = 1$

Step 5

The final solution is all the values that make $(x + 3) (x - 2) (x - 1) < 0$ true.

$x = - 3, 2, 1$

Step 6

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 1$

$1 < x < 2$

$x > 2$

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 $x < - 3$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

Step 7.1.2

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

$((- 6) + 3) ((- 6) - 2) ((- 6) - 1) < 0$

Step 7.1.3

The left side $- 168$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $- 3 < x < 1$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 3 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

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

$((0) + 3) ((0) - 2) ((0) - 1) < 0$

Step 7.2.3

The left side $6$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.3

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

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Step 7.3.1

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

$x = 1.5$

Step 7.3.2

Replace $x$ with $1.5$ in the original inequality.

$((1.5) + 3) ((1.5) - 2) ((1.5) - 1) < 0$

Step 7.3.3

The left side $- 1.125$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

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

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Step 7.4.1

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

$x = 4$

Step 7.4.2

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

$((4) + 3) ((4) - 2) ((4) - 1) < 0$

Step 7.4.3

The left side $42$ is not 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.

$x < - 3$ True

$- 3 < x < 1$ False

$1 < x < 2$ True

$x > 2$ False

$x < - 3$ True

$- 3 < x < 1$ False

$1 < x < 2$ True

$x > 2$ False

Step 8

The solution consists of all of the true intervals.

$x < - 3$ or $1 < x < 2$

Step 9