Question

How do you solve this?

`(x - 1)/(x+5 )> 2`

Answer 1

The solution is `x in (-11, -5)`

Explanation:

Solve this inequality with a sign chart

`(x-1)/(x+5)>2`

`(x-1)/(x+5)-2>0`

`((x-1)-2(x+5))/(x+5)>0`

`(x-1-2x-10)/(x+5)>0`

`(-x-11)/(x+5)>0`

`(x+11)/(x+5)<0`

Let `f(x)=(x+11)/(x+5)`

Let 's build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-11` `color(white)(aaaaaaa)` `-5` `color(white)(aaaaa)` `+oo` ####

`color(white)(aaaa)` `x+11` `color(white)(aaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaaa)` `+` ####

`color(white)(aaaa)` `x+5` `color(white)(aaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` ###color(white)(aaaa)#`+`

Therefore,

`f(x)<0` when `x in (-11, -5)`

graph{(x+11)/(x+5) [-36.52, 36.54, -18.27, 18.28]}

Answer 2

`-11 < x<-5`

Explanation:

`"one way is to multiply both sides of the inequality by"`

`(x+5)^2" which is positive"to(x!=-5)`

`rArr(x-1)(x+5)>2(x+5)^2`

`"expanding both sides gives"`

`x^2+4x-5>2x^2+20x+50`

`"subtract "x^2+4x-5" from both sides"`

`rArr0>x^2+16x+55=(x+11)(x+5)`

`"the right side is a quadratic with positive leading"`
`"coefficient and zeros "x=-11" and "x=-5`

`"It is negative when "-11< x < -5`
graph{(x+11)(x+5) [-20, 20, -10, 10]}