Please explain why x> 3 or x3 or x> 2?Consider the question: Solve for x in lR, X^2 -5x + 6 >0

For the inequality:

x^2 - 5x
+6 > 0

First we will
factor:

==> (x-2)(x-3) >
0

Now we have a product of two function. In order for the
product of two numbers to be positive ( > 0) then both numbers should be positive
or both numbers should be negative.

Then we have two
options:

(x-2) > 0  AND   x-3 >
0

We will solve.

==> x
> 2   AND  x > 3

==> x = (2, inf) n (
3, inf) = (3,in)

==> x = (3, inf )
............(1)

For the second
option:

(x-2) < 0  AND   (x-3) <
0

==> x < 2  AND   x <
3

==> x = (-inf , 2) n (-inf, 3) = (-inf,
2)

==> x = (-inf, 2)
.............(2)

Then, we have two possible
solutions: (1) and (2).

==> x = (-inf
, 2) U ( 3, inf)

OR:   x <2  OR  x
> 3