Solve the Inequality for x 3/(2-x)<=1/(x+4)

$\frac{3}{2 - x} \leq \frac{1}{x + 4}$

Subtract $\frac{1}{x + 4}$ from both sides of the inequality.

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

Simplify $\frac{3}{2 - x} - \frac{1}{x + 4}$ .

To write $\frac{3}{2 - x}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

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

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

Write each expression with a common denominator of $(2 - x) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{3 (x + 4)}{(2 - x) (x + 4)} - \frac{1}{x + 4} \cdot \frac{2 - x}{2 - x} \leq 0$

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

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

Reorder the factors of $(x + 4) (2 - x)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $4$ .

$\frac{3 x + 12 - (2 - x)}{(2 - x) (x + 4)} \leq 0$

Apply the distributive property.

$\frac{3 x + 12 - 1 \cdot 2 - - x}{(2 - x) (x + 4)} \leq 0$

Multiply $- 1$ by $2$ .

$\frac{3 x + 12 - 2 - - x}{(2 - x) (x + 4)} \leq 0$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$\frac{3 x + 12 - 2 + 1 x}{(2 - x) (x + 4)} \leq 0$

Multiply $x$ by $1$ .

$\frac{3 x + 12 - 2 + x}{(2 - x) (x + 4)} \leq 0$

Add $3 x$ and $x$ .

$\frac{4 x + 12 - 2}{(2 - x) (x + 4)} \leq 0$

Subtract $2$ from $12$ .

$\frac{4 x + 10}{(2 - x) (x + 4)} \leq 0$

Factor $2$ out of $4 x + 10$ .

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 10}{(2 - x) (x + 4)} \leq 0$

Factor $2$ out of $10$ .

$\frac{2 (2 x) + 2 (5)}{(2 - x) (x + 4)} \leq 0$

Factor $2$ out of $2 (2 x) + 2 (5)$ .

$\frac{2 (2 x + 5)}{(2 - x) (x + 4)} \leq 0$

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

$2 x + 5 = 0$

$2 - x = 0$

$x + 4 = 0$

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

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.

$x = - \frac{5}{2}$

$x = 2$

$x = - 4$

Consolidate the solutions.

$x = - \frac{5}{2}, 2, - 4$

Find the domain of $\frac{2 (2 x + 5)}{(2 - x) (x + 4)}$ .

Set the denominator in $\frac{2 (2 x + 5)}{(2 - x) (x + 4)}$ equal to $0$ to find where the expression is undefined.

$(2 - x) (x + 4) = 0$

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 $0$ .

$2 - x = 0$

$x + 4 = 0$

Set $2 - x$ equal to $0$ and solve for $x$ .

Set $2 - x$ equal to $0$ .

$2 - x = 0$

Solve $2 - x = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

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 $(2 - x) (x + 4) = 0$ true.

$x = 2, - 4$

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

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

Use each root to create test intervals.

$x < - 4$

$- 4 < x < - \frac{5}{2}$

$- \frac{5}{2} < x < 2$

$x > 2$

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.

$\frac{3}{2 - (- 6)} \leq \frac{1}{(- 6) + 4}$

The left side $0.375$ is greater than the right side $- 0.5$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < - \frac{5}{2}$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < - \frac{5}{2}$ and see if this value makes the original inequality true.

$x = - 3$

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

$\frac{3}{2 - (- 3)} \leq \frac{1}{(- 3) + 4}$

The left side $0.6$ is less than the right side $1$ , which means that the given statement is always true.

True

Test a value on the interval $- \frac{5}{2} < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{5}{2} < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{3}{2 - (0)} \leq \frac{1}{(0) + 4}$

The left side $1.5$ is greater than the right side $0.25$ , 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.

$\frac{3}{2 - (4)} \leq \frac{1}{(4) + 4}$

The left side $- 1.5$ is less than the right side $0.125$ , which means that the given statement is always true.

True

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

$x < - 4$ False

$- 4 < x < - \frac{5}{2}$ True

$- \frac{5}{2} < x < 2$ False

$x > 2$ True

$x < - 4$ False

$- 4 < x < - \frac{5}{2}$ True

$- \frac{5}{2} < x < 2$ False

$x > 2$ True

The solution consists of all of the true intervals.

$- 4 < x \leq - \frac{5}{2}$ or $x > 2$

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x \leq - \frac{5}{2} or x > 2$

Interval Notation:

$(- 4, - \frac{5}{2}] \cup (2, \infty)$

Solve the Inequality for x 3/(2-x)<=1/(x+4)

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