Question

How do you solve `1/(x+1)>2/(x-1)`?

Answer 1

`x<-3`

Explanation:

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

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

`((x-1)-2(x+1))/(x^2-1)>0` [ `x!=+-1`]

`-x-3>0 -> x+3<0`#

`:. x<-3`

Answer 2

The solution is `x in ]-oo, -3[uu]-1,1[`

Explanation:

We cannot do crossing over

Let's rearrange the equation

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

`2/(x-1)-1/(x+1)<0`

`(2x+2-x+1)/((x+1)(x-1))<0`

`(x+3)/((x+1)(x-1))<0`

Let `f(x)=(x+3)/((x+1)(x-1))`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaaaaa)` `-1` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `+` `color(white)(aaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+1` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `+` `color(white)(aaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)<0` when `x in ]-oo, -3[uu]-1,1[`