Finite Math Examples

$x^{2} + 2 x > 0$

Step 1

Solve $x < - 2 or x > 0$ .

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Step 1.1

Convert the inequality to an equation.

$x^{2} + 2 x = 0$

Step 1.2

Factor $x$ out of $x^{2} + 2 x$ .

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Step 1.2.1

Factor $x$ out of $x^{2}$ .

$x \cdot x + 2 x = 0$

Step 1.2.2

Factor $x$ out of $2 x$ .

$x \cdot x + x \cdot 2 = 0$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$x (x + 2) = 0$

Step 1.3

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

$x = 0$

$x + 2 = 0$

Step 1.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.5

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

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Step 1.5.1

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

$x + 2 = 0$

Step 1.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.6

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

$x = 0, - 2$

Step 1.7

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$x > 0$

Step 1.8

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 1.8.1

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

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Step 1.8.1.1

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

$x = - 4$

Step 1.8.1.2

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

${(- 4)}^{2} + 2 (- 4) > 0$

Step 1.8.1.3

The left side $8$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.8.2

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

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Step 1.8.2.1

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 1.8.2.2

Replace $x$ with $- 1$ in the original inequality.

${(- 1)}^{2} + 2 (- 1) > 0$

Step 1.8.2.3

The left side $- 1$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 1.8.3

Test a value on the interval $x > 0$ to see if it makes the inequality true.

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Step 1.8.3.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 1.8.3.2

Replace $x$ with $2$ in the original inequality.

${(2)}^{2} + 2 (2) > 0$

Step 1.8.3.3

The left side $8$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.8.4

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

$x < - 2$ True

$- 2 < x < 0$ False

$x > 0$ True

$x < - 2$ True

$- 2 < x < 0$ False

$x > 0$ True

Step 1.9

The solution consists of all of the true intervals.

$x < - 2$ or $x > 0$

Step 2

Use the inequality $x < - 2 or x > 0$ to build the set notation.

${x | x < - 2 or x > 0}$

Step 3