Precalculus Examples

$\frac{4}{x - 4} > \frac{4}{x + 2}$

Step 1

Subtract $\frac{4}{x + 2}$ from both sides of the inequality.

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

Step 2

Simplify $\frac{4}{x - 4} - \frac{4}{x + 2}$ .

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Step 2.1

To write $\frac{4}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 2.2

To write $- \frac{4}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

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

Step 2.3

Write each expression with a common denominator of $(x - 4) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{4}{x - 4}$ by $\frac{x + 2}{x + 2}$ .

$\frac{4 (x + 2)}{(x - 4) (x + 2)} - \frac{4}{x + 2} \cdot \frac{x - 4}{x - 4} > 0$

Step 2.3.2

Multiply $\frac{4}{x + 2}$ by $\frac{x - 4}{x - 4}$ .

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

Step 2.3.3

Reorder the factors of $(x - 4) (x + 2)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $4$ out of $4 (x + 2) - 4 (x - 4)$ .

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Step 2.5.1.1

Factor $4$ out of $- 4 (x - 4)$ .

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

Step 2.5.1.2

Factor $4$ out of $4 (x + 2) + 4 (- (x - 4))$ .

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $- 1$ by $- 4$ .

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

Step 2.5.4

Subtract $x$ from $x$ .

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

Step 2.5.5

Add $0$ and $2$ .

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

Step 2.5.6

Add $2$ and $4$ .

$\frac{4 \cdot 6}{(x + 2) (x - 4)} > 0$

Step 2.6

Multiply $4$ by $6$ .

$\frac{24}{(x + 2) (x - 4)} > 0$

Step 3

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

$x + 2 = 0$

$x - 4 = 0$

Step 4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Add $4$ to both sides of the equation.

$x = 4$

Step 6

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

$x = - 2$

$x = 4$

Step 7

Consolidate the solutions.

$x = - 2, 4$

Step 8

Find the domain of $\frac{24}{(x + 2) (x - 4)}$ .

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Step 8.1

Set the denominator in $\frac{24}{(x + 2) (x - 4)}$ equal to $0$ to find where the expression is undefined.

$(x + 2) (x - 4) = 0$

Step 8.2

Solve for $x$ .

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Step 8.2.1

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

$x + 2 = 0$

$x - 4 = 0$

Step 8.2.2

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

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Step 8.2.2.1

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

$x + 2 = 0$

Step 8.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8.2.3

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

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Step 8.2.3.1

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

$x - 4 = 0$

Step 8.2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 8.2.4

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

$x = - 2, 4$

Step 8.3

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

$(- \infty, - 2) \cup (- 2, 4) \cup (4, \infty)$

Step 9

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 4$

$x > 4$

Step 10

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 10.1

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

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Step 10.1.1

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

$x = - 4$

Step 10.1.2

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

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

Step 10.1.3

The left side $- 0.5$ is greater than the right side $- 2$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $- 2 < x < 4$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 2 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 10.2.2

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

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

Step 10.2.3

The left side $- 1$ is not greater than the right side $2$ , which means that the given statement is false.

False

Step 10.3

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

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Step 10.3.1

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

$x = 6$

Step 10.3.2

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

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

Step 10.3.3

The left side $2$ is greater than the right side $0.5$ , which means that the given statement is always true.

True

Step 10.4

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

$x < - 2$ True

$- 2 < x < 4$ False

$x > 4$ True

$x < - 2$ True

$- 2 < x < 4$ False

$x > 4$ True

Step 11

The solution consists of all of the true intervals.

$x < - 2$ or $x > 4$

Step 12

Convert the inequality to interval notation.

$(- \infty, - 2) \cup (4, \infty)$

Step 13