Hello, jayshizwiz!
How about a graphic solution?
$\displaystyle x^2  3x + 2 \:<\: x+2$
The graph of $\displaystyle y \:=\:x^23x+2 \:=\:(x1)(x2)$
. . is an upopening parabola with $\displaystyle x$intercepts 1 and 2.
Its graph looks like this:
Code:

* *


* *

* *
 * *
  +  *       *    
 1 * * 2
 *

The graph of $\displaystyle y \:=\:x^23x+2$ looks like this:
Code:

o o


o o

o o o
 o o o o
   +  o       o    
 1 2

Everything below the $\displaystyle x$axis is reflected upward.
The graph of $\displaystyle y \:=\:x+2$ looks like this: Code:

 *
 *
 *
 *
2 *
*
* 
* 
* 
  *    +      
* 2 
* 

The graph of $\displaystyle y \:=\:x+2$ looks like this:
Code:

 ∆
 ∆
 ∆
 ∆
 ∆
∆
∆ 
∆ ∆ 
∆ ∆ 
  ∆    +      
2 
Sketch the two graphs on a set of coordinate axes.
Code:

o  ♥(4,6)
 ∆
 ∆
 ∆
 ∆
 ∆
o  ∆ o
 ∆
 ∆
 ∆
o ∆ o
 ∆
(0,2) ∆
♥ o
∆ 
∆ ∆ o o o
∆ ∆  o o o o
  ∆ . .  +  o       o    
2  1 2

When is the "parabola" below the "line"?
Answer: .$\displaystyle 0 \:<\:x\:<\:4$