Question

In inequality log _1/2(x^{​​​​​​2}​​​​​-6x+12) ≥ -2 there is one step which says (x-2)(x-4) ≤ 0,. then how can we conclude that x<2 and x>4 ?

Answer 1

The first thing is
x<2 and x>4 cannot be the solutions.

Think of this examples:
If x<2, take a point x=1
then (x-2)(x-4)
=(1-2)(1-4)
=(-1)x(-3)=3
which is actually greater than 0 and not equal to 0. This contradicts the condition.

If x>4, take x=5
(5-2)(5-4)=3x1=3
which is again greater than 0 and not equal to 0
​​​​​​This too contradicts

The correct answer is
x≥2 and x≤4
therefore x belongs to [2,4]

Now if you check this:

We already know when x=2 or x=4 the product is 0 which is a solution.

Now take a point between 2 and 4, x=3
(3-2)(3-4)=1x(-1)=-1
which is less than 0

Hope you understood
Thank you