Calculus Examples

$\frac{x^{2} + 8 x}{x - 4} > 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} + 8 x = 0$

$x - 4 = 0$

Step 2

Factor $x$ out of $x^{2} + 8 x$ .

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Step 2.1

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

$x \cdot x + 8 x = 0$

Step 2.2

Factor $x$ out of $8 x$ .

$x \cdot x + x \cdot 8 = 0$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot 8$ .

$x (x + 8) = 0$

Step 3

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

$x = 0$

$x + 8 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 5.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 6

The final solution is all the values that make $x (x + 8) = 0$ true.

$x = 0, - 8$

Step 7

Add $4$ to both sides of the equation.

$x = 4$

Step 8

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0, - 8$

$x = 4$

Step 9

Consolidate the solutions.

$x = 0, - 8, 4$

Step 10

Find the domain of $\frac{x^{2} + 8 x}{x - 4}$ .

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Step 10.1

Set the denominator in $\frac{x^{2} + 8 x}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 10.2

Add $4$ to both sides of the equation.

$x = 4$

Step 10.3

The domain is all values of $x$ that make the expression defined.

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

Step 11

Use each root to create test intervals.

$x < - 8$

$- 8 < x < 0$

$0 < x < 4$

$x > 4$

Step 12

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 12.1

Test a value on the interval $x < - 8$ to see if it makes the inequality true.

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Step 12.1.1

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Step 12.1.2

Replace $x$ with $- 10$ in the original inequality.

$\frac{{(- 10)}^{2} + 8 (- 10)}{(- 10) - 4} > 0$

Step 12.1.3

The left side $- 1.\overline{428571}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $- 8 < x < 0$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- 8 < x < 0$ and see if this value makes the original inequality true.

$x = - 4$

Step 12.2.2

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

$\frac{{(- 4)}^{2} + 8 (- 4)}{(- 4) - 4} > 0$

Step 12.2.3

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 12.3.2

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

$\frac{{(2)}^{2} + 8 (2)}{(2) - 4} > 0$

Step 12.3.3

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

False

Step 12.4

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

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Step 12.4.1

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

$x = 6$

Step 12.4.2

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

$\frac{{(6)}^{2} + 8 (6)}{(6) - 4} > 0$

Step 12.4.3

The left side $42$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 12.5

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

$x < - 8$ False

$- 8 < x < 0$ True

$0 < x < 4$ False

$x > 4$ True

$x < - 8$ False

$- 8 < x < 0$ True

$0 < x < 4$ False

$x > 4$ True

Step 13

The solution consists of all of the true intervals.

$- 8 < x < 0$ or $x > 4$

Step 14

Convert the inequality to interval notation.

$(- 8, 0) \cup (4, \infty)$

Step 15