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

2 Answers
Oct 6, 2017

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.!

Oct 6, 2017

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]}