# How do you solve 2/(x+5)>4/(x+5)+3?

##### 2 Answers
Jul 30, 2017

$x < - \frac{17}{3}$ or $x < - 5 \frac{2}{3}$

#### Explanation:

$\frac{2}{x + 5} > \frac{4}{x + 5} + 3$

Multiply every term by $\left(x + 5\right)$.

$\left(x + 5\right) \times \frac{2}{x + 5} > \left(x + 5\right) \times \frac{4}{x + 5} + 3 \left(x + 5\right)$

$\cancel{\left(x + 5\right)} \times \frac{2}{\cancel{x + 5}} > \cancel{\left(x + 5\right)} \times \frac{4}{\cancel{x + 5}} + 3 x + 15$

$2 > 4 + 3 x + 15$

$2 > 3 x + 19$

Subtract $19$ from each side.

$2 - 19 > 3 x$

$- 17 > 3 x$

Divide both sides by $3$.

$- \frac{17}{3} > \frac{3 x}{3}$

$- \frac{17}{3} > \frac{\cancel{3} x}{\cancel{3}}$

$- \frac{17}{3} > x$ or $x < - \frac{17}{3}$

Jul 30, 2017

The solution is $x \in \left(- \frac{17}{3} , - 5\right)$

#### Explanation:

You cannot do crossing over

$\frac{2}{x + 5} > \frac{4}{x + 5} + 3$

Let's perform some simplifications

$\frac{2}{x + 5} - \frac{4}{x + 5} - 3 > 0$

$\frac{2 - 4 - 3 \left(x + 5\right)}{x + 5} > 0$

$\frac{2 - 4 - 3 x - 15}{x + 5} > 0$

$\frac{- 3 x - 17}{x + 5} > 0$

Let $f \left(x\right) = \frac{- 3 x - 17}{x + 5}$

Let's build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a}$$- \frac{17}{3}$$\textcolor{w h i t e}{a a a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$- 3 x - 17$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$x + 5$$\textcolor{w h i t e}{a a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$

Therefore,

$f \left(x\right) > 0$ when $x \in \left(- \frac{17}{3} , - 5\right)$

graph{2/(x+5)-4/(x+5)-3 [-24.25, 11.8, -10.45, 7.57]}