Question

How do you solve `8/(x-1)<1`?

Answer 1

The answer is `x in ] -oo,1 [ uu ] 9, +oo[`

Explanation:

We cannot do crossing over

So,

`8/(x-1)<1`, `=>`, `8/(x-1)-1<0`

`(8-(x-1))/(x-1)<0`

`(9-x)/(x-1)<0`

Let `f(x)=(9-x)/(x-1)`

We can do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `1` `color(white)(aaaa)` `9` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `9-x` `color(white)(aaaaa)` `+` `color(white)(aaa)` `+` `color(white)(aaa)` `-`

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

Therefore,

`f(x)<0`, when `x in ] -oo,1 [ uu ] 9, +oo[`