# How do you solve the inequality 7/(y+1)>7?

Dec 17, 2016

The answer is x in ] -1,0 [

#### Explanation:

We rewrite the equation as

$7 - \frac{7}{y + 1} < 0$

$\frac{7 \left(y + 1\right) - 7}{y + 1} < 0$

$\frac{7 y + 7 - 7}{y + 1} < 0$

$\frac{7 y}{y + 1} < 0$

Let $f \left(y\right) = \frac{7 y}{y + 1}$

and $y \ne - 1$

We do a sign chart

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a}$∥$\textcolor{w h i t e}{a}$$-$$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y + 1$$\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}$$+$$\textcolor{w h i t e}{a a}$$+$

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

Therefore,

$f \left(y\right) < 0$ when x in ] -1,0 [

graph{7x/(x+1) [-41.1, 41.1, -20.55, 20.57]}