Solve the Inequality for x 3+4/x>=(x+2)/x

$3 + \frac{4}{x} \geq \frac{x + 2}{x}$

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

$3 + \frac{4}{x} - \frac{x + 2}{x} \geq 0$

Simplify $3 + \frac{4}{x} - \frac{x + 2}{x}$ .

Combine the numerators over the common denominator.

$3 + \frac{4 - (x + 2)}{x} \geq 0$

Simplify each term.

Apply the distributive property.

$3 + \frac{4 - x - 1 \cdot 2}{x} \geq 0$

Multiply $- 1$ by $2$ .

$3 + \frac{4 - x - 2}{x} \geq 0$

Subtract $2$ from $4$ .

$3 + \frac{- x + 2}{x} \geq 0$

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 x}{x} + \frac{- x + 2}{x} \geq 0$

Combine the numerators over the common denominator.

$\frac{3 x - x + 2}{x} \geq 0$

Simplify the numerator.

Subtract $x$ from $3 x$ .

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

Factor $2$ out of $2 x + 2$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $2$ .

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

Factor $2$ out of $2 (x) + 2 (1)$ .

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

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

$x = 0$

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = 0$

$x = - 1$

Consolidate the solutions.

$x = 0, - 1$

Find the domain of $\frac{2 (x + 1)}{x}$ .

Set the denominator in $\frac{2 (x + 1)}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

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

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

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 0$

$x > 0$

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 $x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

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

$3 + \frac{4}{- 4} \geq \frac{(- 4) + 2}{- 4}$

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

True

Test a value on the interval $- 1 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 0$ and see if this value makes the original inequality true.

$x = - 0.5$

Replace $x$ with $- 0.5$ in the original inequality.

$3 + \frac{4}{- 0.5} \geq \frac{(- 0.5) + 2}{- 0.5}$

The left side $- 5$ is less than the right side $- 3$ , which means that the given statement is false.

False

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

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

$x = 2$

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

$3 + \frac{4}{2} \geq \frac{(2) + 2}{2}$

The left side $5$ is greater than the right side $2$ , which means that the given statement is always true.

True

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

$x < - 1$ True

$- 1 < x < 0$ False

$x > 0$ True

$x < - 1$ True

$- 1 < x < 0$ False

$x > 0$ True

The solution consists of all of the true intervals.

$x \leq - 1$ or $x > 0$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 1 or x > 0$

Interval Notation:

$(- \infty, - 1] \cup (0, \infty)$

Solve the Inequality for x 3+4/x>=(x+2)/x

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