Precalculus Examples

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

Step 1

Subtract $2$ from both sides of the inequality.

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Combine $- 2$ and $\frac{x + 2}{x + 2}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $- 2$ by $2$ .

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

Step 2.4.3

Subtract $2 x$ from $x$ .

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

Step 2.4.4

Subtract $4$ from $- 4$ .

$\frac{- x - 8}{x + 2} \geq 0$

Step 2.5

Factor $- 1$ out of $- x$ .

$\frac{- (x) - 8}{x + 2} \geq 0$

Step 2.6

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- (x) - 1 (8)}{x + 2} \geq 0$

Step 2.7

Factor $- 1$ out of $- (x) - 1 (8)$ .

$\frac{- (x + 8)}{x + 2} \geq 0$

Step 2.8

Rewrite $- (x + 8)$ as $- 1 (x + 8)$ .

$\frac{- 1 (x + 8)}{x + 2} \geq 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{x + 8}{x + 2} \geq 0$

Step 3

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

$x + 8 = 0$

$x + 2 = 0$

Step 4

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 5

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

$x = - 8$

$x = - 2$

Step 7

Consolidate the solutions.

$x = - 8, - 2$

Step 8

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

Tap for more steps...

Step 8.1

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

$x + 2 = 0$

Step 8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8.3

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

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

Step 9

Use each root to create test intervals.

$x < - 8$

$- 8 < x < - 2$

$x > - 2$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 10.1

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

Tap for more steps...

Step 10.1.1

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

$x = - 10$

Step 10.1.2

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

$\frac{(- 10) - 4}{(- 10) + 2} \geq 2$

Step 10.1.3

The left side $1.75$ is less than the right side $2$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $- 8 < x < - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 10.2.1

Choose a value on the interval $- 8 < x < - 2$ and see if this value makes the original inequality true.

$x = - 5$

Step 10.2.2

Replace $x$ with $- 5$ in the original inequality.

$\frac{(- 5) - 4}{(- 5) + 2} \geq 2$

Step 10.2.3

The left side $3$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 10.3

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

Tap for more steps...

Step 10.3.1

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

$x = 0$

Step 10.3.2

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

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

Step 10.3.3

The left side $- 2$ is less than the right side $2$ , which means that the given statement is false.

False

Step 10.4

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

$x < - 8$ False

$- 8 < x < - 2$ True

$x > - 2$ False

$x < - 8$ False

$- 8 < x < - 2$ True

$x > - 2$ False

Step 11

The solution consists of all of the true intervals.

$- 8 \leq x < - 2$

Step 12

Convert the inequality to interval notation.

$[- 8, - 2)$

Step 13