Algebra Examples

$(2 x + 6) (x - 4) > 0$

Step 1

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

$2 x + 6 = 0$

$x - 4 = 0$

Step 2

Set $2 x + 6$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $2 x + 6$ equal to $0$ .

$2 x + 6 = 0$

Step 2.2

Solve $2 x + 6 = 0$ for $x$ .

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Step 2.2.1

Subtract $6$ from both sides of the equation.

$2 x = - 6$

Step 2.2.2

Divide each term in $2 x = - 6$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 x = - 6$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{2}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 6$ by $2$ .

$x = - 3$

Step 3

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

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Step 3.1

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

$x - 4 = 0$

Step 3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4

The final solution is all the values that make $(2 x + 6) (x - 4) > 0$ true.

$x = - 3, 4$

Step 5

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 4$

$x > 4$

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

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Step 6.1.1

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

$x = - 6$

Step 6.1.2

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

$(2 (- 6) + 6) ((- 6) - 4) > 0$

Step 6.1.3

The left side $60$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.2

Test a value on the interval $- 3 < x < 4$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $- 3 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 6.2.2

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

$(2 (0) + 6) ((0) - 4) > 0$

Step 6.2.3

The left side $- 24$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 6.3

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

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Step 6.3.1

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

$x = 6$

Step 6.3.2

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

$(2 (6) + 6) ((6) - 4) > 0$

Step 6.3.3

The left side $36$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.4

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

$x < - 3$ True

$- 3 < x < 4$ False

$x > 4$ True

$x < - 3$ True

$- 3 < x < 4$ False

$x > 4$ True

Step 7

The solution consists of all of the true intervals.

$x < - 3$ or $x > 4$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - 3 or x > 4$

Interval Notation:

$(- \infty, - 3) \cup (4, \infty)$

Step 9