Solve the Inequality for x 2/(x+4)>=1/(x+1)

$\frac{2}{x + 4} \geq \frac{1}{x + 1}$

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

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

Simplify $\frac{2}{x + 4} - \frac{1}{x + 1}$ .

To write $\frac{2}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

To write $- \frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

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

Multiply $\frac{2}{x + 4}$ by $\frac{x + 1}{x + 1}$ .

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

Multiply $\frac{1}{x + 1}$ by $\frac{x + 4}{x + 4}$ .

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

Reorder the factors of $(x + 4) (x + 1)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $4$ .

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

Subtract $x$ from $2 x$ .

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

Subtract $4$ from $2$ .

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

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

$x - 2 = 0$

$x + 1 = 0$

$x + 4 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Subtract $1$ from both sides of the equation.

$x = - 1$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$x = 2$

$x = - 1$

$x = - 4$

Consolidate the solutions.

$x = 2, - 1, - 4$

Find the domain of $\frac{x - 2}{(x + 1) (x + 4)}$ .

Set the denominator in $\frac{x - 2}{(x + 1) (x + 4)}$ equal to $0$ to find where the expression is undefined.

$(x + 1) (x + 4) = 0$

Solve for $x$ .

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

$x + 1 = 0$

$x + 4 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x + 1) (x + 4) = 0$ true.

$x = - 1, - 4$

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

$(- \infty, - 4) \cup (- 4, - 1) \cup (- 1, \infty)$

Use each root to create test intervals.

$x < - 4$

$- 4 < x < - 1$

$- 1 < x < 2$

$x > 2$

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.

$\frac{2}{(- 6) + 4} \geq \frac{1}{(- 6) + 1}$

The left side $- 1$ is less than the right side $- 0.2$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < - 1$ and see if this value makes the original inequality true.

$x = - 2$

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

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

The left side $1$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

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

The left side $0.5$ is less than the right side $1$ , which means that the given statement is false.

False

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.

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

The left side $0.25$ is greater than the right side $0.2$ , which means that the given statement is always true.

True

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

$x < - 4$ False

$- 4 < x < - 1$ True

$- 1 < x < 2$ False

$x > 2$ True

$x < - 4$ False

$- 4 < x < - 1$ True

$- 1 < x < 2$ False

$x > 2$ True

The solution consists of all of the true intervals.

$- 4 < x < - 1$ or $x \geq 2$

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < - 1 or x \geq 2$

Interval Notation:

$(- 4, - 1) \cup [2, \infty)$

Solve the Inequality for x 2/(x+4)>=1/(x+1)

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