Solve the Inequality for x x>=2/(x+1)

$x \geq \frac{2}{x + 1}$

Subtract $\frac{2}{x + 1}$ from both sides of the inequality.

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

Simplify $x - \frac{2}{x + 1}$ .

To write $x$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

$\frac{x^{2} + x - 2}{x + 1} \geq 0$

Factor $x^{2} + x - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

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

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

$x - 1 = 0$

$x + 2 = 0$

$x + 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Subtract $2$ from both sides of the equation.

$x = - 2$

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.

$x = 1$

$x = - 2$

$x = - 1$

Consolidate the solutions.

$x = 1, - 2, - 1$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

Use each root to create test intervals.

$x < - 2$

$- 2 < x < - 1$

$- 1 < x < 1$

$x > 1$

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 $x < - 2$ 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.

$x = - 4$

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

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

The left side $- 4$ is less than the right side $- 0.\overline{6}$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < - 1$ and see if this value makes the original inequality true.

$x = - 1.5$

Replace $x$ with $- 1.5$ in the original inequality.

$- 1.5 \geq \frac{2}{(- 1.5) + 1}$

The left side $- 1.5$ is greater than the right side $- 4$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

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

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

False

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.

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

The left side $4$ is greater than the right side $0.4$ , which means that the given statement is always true.

True

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

$x < - 2$ False

$- 2 < x < - 1$ True

$- 1 < x < 1$ False

$x > 1$ True

$x < - 2$ False

$- 2 < x < - 1$ True

$- 1 < x < 1$ False

$x > 1$ True

The solution consists of all of the true intervals.

$- 2 \leq x < - 1$ or $x \geq 1$

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x < - 1 or x \geq 1$

Interval Notation:

$[- 2, - 1) \cup [1, \infty)$

Solve the Inequality for x x>=2/(x+1)

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