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

$\frac{x - 3}{x + 4} > 3$

Subtract $3$ from both sides of the inequality.

$\frac{x - 3}{x + 4} - 3 > 0$

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

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

$\frac{x - 3}{x + 4} - 3 \cdot \frac{x + 4}{x + 4} > 0$

Combine $- 3$ and $\frac{x + 4}{x + 4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{x - 3 - 3 x - 3 \cdot 4}{x + 4} > 0$

Multiply $- 3$ by $4$ .

$\frac{x - 3 - 3 x - 12}{x + 4} > 0$

Subtract $3 x$ from $x$ .

$\frac{- 2 x - 3 - 12}{x + 4} > 0$

Subtract $12$ from $- 3$ .

$\frac{- 2 x - 15}{x + 4} > 0$

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) - 15}{x + 4} > 0$

Rewrite $- 15$ as $- 1 (15)$ .

$\frac{- (2 x) - 1 (15)}{x + 4} > 0$

Factor $- 1$ out of $- (2 x) - 1 (15)$ .

$\frac{- (2 x + 15)}{x + 4} > 0$

Rewrite $- (2 x + 15)$ as $- 1 (2 x + 15)$ .

$\frac{- 1 (2 x + 15)}{x + 4} > 0$

Move the negative in front of the fraction.

$- \frac{2 x + 15}{x + 4} > 0$

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

$2 x + 15 = 0$

$x + 4 = 0$

Subtract $15$ from both sides of the equation.

$2 x = - 15$

Divide each term in $2 x = - 15$ by $2$ and simplify.

Divide each term in $2 x = - 15$ by $2$ .

$\frac{2 x}{2} = \frac{- 15}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 15}{2}$

Divide $x$ by $1$ .

$x = \frac{- 15}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{15}{2}$

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.

$x = - \frac{15}{2}$

$x = - 4$

Consolidate the solutions.

$x = - \frac{15}{2}, - 4$

Find the domain of $- \frac{2 x + 15}{x + 4}$ .

Set the denominator in $\frac{2 x + 15}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

Use each root to create test intervals.

$x < - \frac{15}{2}$

$- \frac{15}{2} < x < - 4$

$x > - 4$

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 $x < - \frac{15}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{15}{2}$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

$\frac{(- 10) - 3}{(- 10) + 4} > 3$

The left side $2.1\overline{6}$ is not greater than the right side $3$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{15}{2} < x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{15}{2} < x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

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

$\frac{(- 6) - 3}{(- 6) + 4} > 3$

The left side $4.5$ is greater than the right side $3$ , which means that the given statement is always true.

True

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.

$x = 0$

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

$\frac{(0) - 3}{(0) + 4} > 3$

The left side $- 0.75$ is not greater than the right side $3$ , which means that the given statement is false.

False

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

$x < - \frac{15}{2}$ False

$- \frac{15}{2} < x < - 4$ True

$x > - 4$ False

$x < - \frac{15}{2}$ False

$- \frac{15}{2} < x < - 4$ True

$x > - 4$ False

The solution consists of all of the true intervals.

$- \frac{15}{2} < x < - 4$

The result can be shown in multiple forms.

Inequality Form:

$- \frac{15}{2} < x < - 4$

Interval Notation:

$(- \frac{15}{2}, - 4)$

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

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