Solve the Inequality for x 3/(x+5)>2

$\frac{3}{x + 5} > 2$

Subtract $2$ from both sides of the inequality.

$\frac{3}{x + 5} - 2 > 0$

Simplify $\frac{3}{x + 5} - 2$ .

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3}{x + 5} - 2 \cdot \frac{x + 5}{x + 5} > 0$

Combine $- 2$ and $\frac{x + 5}{x + 5}$ .

$\frac{3}{x + 5} + \frac{- 2 (x + 5)}{x + 5} > 0$

Combine the numerators over the common denominator.

$\frac{3 - 2 (x + 5)}{x + 5} > 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 - 2 x - 2 \cdot 5}{x + 5} > 0$

Multiply $- 2$ by $5$ .

$\frac{3 - 2 x - 10}{x + 5} > 0$

Subtract $10$ from $3$ .

$\frac{- 2 x - 7}{x + 5} > 0$

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

$\frac{- (2 x) - 7}{x + 5} > 0$

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{- (2 x) - 1 (7)}{x + 5} > 0$

Factor $- 1$ out of $- (2 x) - 1 (7)$ .

$\frac{- (2 x + 7)}{x + 5} > 0$

Rewrite $- (2 x + 7)$ as $- 1 (2 x + 7)$ .

$\frac{- 1 (2 x + 7)}{x + 5} > 0$

Move the negative in front of the fraction.

$- \frac{2 x + 7}{x + 5} > 0$

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

$2 x + 7 = 0$

$x + 5 = 0$

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Divide each term in $2 x = - 7$ by $2$ and simplify.

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

$x = - \frac{7}{2}$

$x = - 5$

Consolidate the solutions.

$x = - \frac{7}{2}, - 5$

Find the domain of $- \frac{2 x + 7}{x + 5}$ .

Set the denominator in $\frac{2 x + 7}{x + 5}$ equal to $0$ to find where the expression is undefined.

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

$(- \infty, - 5) \cup (- 5, \infty)$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < - \frac{7}{2}$

$x > - \frac{7}{2}$

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 $x < - 5$ to see if it makes the inequality true.

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

$x = - 8$

Replace $x$ with $- 8$ in the original inequality.

$\frac{3}{(- 8) + 5} > 2$

The left side $- 1$ is not greater than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < - \frac{7}{2}$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < - \frac{7}{2}$ and see if this value makes the original inequality true.

$x = - 4$

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

$\frac{3}{(- 4) + 5} > 2$

The left side $3$ is greater than the right side $2$ , which means that the given statement is always true.

True

Test a value on the interval $x > - \frac{7}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > - \frac{7}{2}$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{3}{(0) + 5} > 2$

The left side $0.6$ is not greater than the right side $2$ , which means that the given statement is false.

False

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

$x < - 5$ False

$- 5 < x < - \frac{7}{2}$ True

$x > - \frac{7}{2}$ False

$x < - 5$ False

$- 5 < x < - \frac{7}{2}$ True

$x > - \frac{7}{2}$ False

The solution consists of all of the true intervals.

$- 5 < x < - \frac{7}{2}$

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < - \frac{7}{2}$

Interval Notation:

$(- 5, - \frac{7}{2})$

Solve the Inequality for x 3/(x+5)>2

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