# How do you solve 7/(x+3)<-5?

Jul 29, 2017

$\left(- \frac{22}{5} , - 3\right)$

#### Explanation:

$\text{add 5 to both sides}$

$\Rightarrow \frac{7}{x + 3} + 5 < 0$

$\text{express as a single rational function}$

$\frac{7 + 5 \left(x + 3\right)}{x + 3} < 0$

$\Rightarrow \frac{5 x + 22}{x + 3} < 0$

$\text{the zeros of the numerator/denominator are}$

$\text{numerator "x=-22/5," denominator } x = - 3$

These indicate where the rational function may change sign.

$\text{the intervals for consideration are}$

$x < - \frac{22}{5} , \textcolor{w h i t e}{x} - \frac{22}{5} < x < - 3 , \textcolor{w h i t e}{x} x > - 3$

$\text{consider a "color(blue)"test point "" in each interval}$

$\text{we want to find where the function is negative}$

$\text{substitute each test point into the function and consider}$
$\text{it's sign}$

$\textcolor{m a \ge n t a}{\text{x = - 5"to(-)/(-)tocolor(red)" positive}}$

$\textcolor{m a \ge n t a}{\text{x = - 4 "to(+)/(-)tocolor(blue)" negative}}$

$\textcolor{m a \ge n t a}{\text{x = 4"to(+)/(+)tocolor(red)" positive}}$

$\Rightarrow - \frac{22}{5} < x < - 3 \text{ or } \left(- \frac{22}{5} , - 3\right)$