The problem is when you divide both sides by a variable factor! you are losing information about the ineqality.
After you factor you get
from here take your two zero's (this is where the inequality can change signs)
so we get
Now we have three different parts of the number line
Now we check a point from each interval to see if it is a solution
In the original and we get so the above set is a solution.
Now we check with and we get so it is false and is not a solution.
finally we check with x=2 so this is a solution.
So the solution to the inequality is
For the rational ineqality get a common denomiantor and get the zero's from both the numerator and denomiator and do the same thing.