Question

How do you solve `\frac { x - 1} { x - 2} - 3\geq 0`?

Answer 1

Update: x could be any real number but `x!=2`

Explanation:

First off, `x!=2` because then you would be dividing by 0.

So,

`(x-1)/(x-2)-3>=0`

Then add 3 on both sides,

`(x-1)/(x-2)>=3`

Multiply both sides by `(x-2)`

`color(blue)(x-1>=3(x-2))`

Distribute

`x-1>=3x-6`

Combine like terms

`5>=2x`

Divide by 2 on both sides

`2.5>=x`

So `2.5>=x` or `x<=2.5` but `x!=2`

However since you do not know if `x-2` is negative or positive, when you multiply both sides by `x-2` (`color(blue)(blue)`) it could switch the sign if that value is negative.

Therefore you could have `x-1<=3(x-2)` at that step which would lead to `x>=2.5`.

So really x could equal any number except for 2.

Thanks George C.!

Answer 2

`x in (2, 5/2]`, i.e. `2 < x <= 5/2`

Explanation:

Given:

`(x-1)/(x-2)-3 >= 0`

First combine the rational expression with the constant `-3` to find:

`((x-1)-3(x-2))/(x-2) >= 0`

That is:

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

In order that the left hand side be non-negative, we need one of the following:

• `-2x+5 >= 0` and `x-2 > 0`. Hence `x in (-oo, 5/2] nn (2, oo) = (2, 5/2]`

• `-2x+5 <= 0` and `x-2 < 0`. Hence `x in [5/2, oo) nn (-oo, 2) = O/`

graph{(-2x+5)/(x-2) [-10, 10, -5, 5]}