Solve the Inequality for x x/(x+2)>=2

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

Step 1

Subtract $2$ from both sides of the inequality.

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

Step 2

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 2$ by $2$ .

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

Subtract $2 x$ from $x$ .

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

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

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

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- (x) - 1 (4)}{x + 2} \geq 0$

Factor $- 1$ out of $- (x) - 1 (4)$ .

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

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

$\frac{- 1 (x + 4)}{x + 2} \geq 0$

Move the negative in front of the fraction.

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

Step 3

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

$x + 4 = 0$

$x + 2 = 0$

Step 4

Subtract $4$ from both sides of the equation.

$x = - 4$

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.

$x = - 4$

$x = - 2$

Step 7

Consolidate the solutions.

$x = - 4, - 2$

Step 8

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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.

$x < - 4$

$- 4 < x < - 2$

$x > - 2$

Step 10

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

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

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

$x = - 6$

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

$\frac{- 6}{(- 6) + 2} \geq 2$

The left side $1.5$ is less than the right side $2$ , which means that the given statement is false.

False

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

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

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

$\frac{- 3}{(- 3) + 2} \geq 2$

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

True

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

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

$x = 0$

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

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

The left side $0$ is less than the right side $2$ , which means that the given statement is false.

False

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

$x < - 4$ False

$- 4 < x < - 2$ True

$x > - 2$ False

$x < - 4$ False

$- 4 < x < - 2$ True

$x > - 2$ False

Step 11

The solution consists of all of the true intervals.

$- 4 \leq x < - 2$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 4 \leq x < - 2$

Interval Notation:

$[- 4, - 2)$

Step 13

Solve the Inequality for x x/(x+2)>=2

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