Question

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

Answer 1

The answer is `x in ] -1,0 [`

Explanation:

We rewrite the equation as

`7-7/(y+1)<0`

`(7(y+1)-7)/(y+1)<0`

`(7y+7-7)/(y+1)<0`

`(7y)/(y+1)<0`

Let `f(y)=(7y)/(y+1)`

and `y!=-1`

We do a sign chart

`color(white)(aaaa)` `y` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `0` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaa)` `-` `color(white)(aa)` `∥` `color(white)(a)` `-` `color(white)(aa)` `+`

`color(white)(aaaa)` `y+1` `color(white)(aaaa)` `-` `color(white)(aaa)` `∥` `color(white)(a)` `+` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(y)` `color(white)(aaaaa)` `+` `color(white)(aaa)` `∥` `color(white)(a)` `-` `color(white)(aa)` `+`

Therefore,

`f(y)<0` when `x in ] -1,0 [`

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