Examples of Quadratic expression on inequality

1. 

first consider ,
here  and 
means  has roots and real solution.

                   

consider, 
Since, 



now, inequality becomes,

for  we can cancel () , or since both the term of () will have the same sign so they will not affect the sign of inequality i.e. -2 is not a behaviour changing point.

Now, there are two behaviour changing points 1 and-1,
from wavy-curvy method,

therefore, 
-1 and 1 is in open bracket because expression is strictly less than zero... means cant be zero.


2.

Since, 
therefore, ...................................................(I)

i) consider, 
 and 
 (previous post) (http://iitmathseasy.blogspot.in/2013/03/quadratic-expression-concept-ii.html)

ii) and, , Modulus of x will always be positive and modulus of 0 will be 0
iii) , Square of any number will be positive and square of 0 will be 0

iv) , power of any positive number will always be positive since 'e' is a constant therefore,  cannot be 0

Above four expression cant change the sign of expression because they will always be positive
Therefore they will not give behaviour changing point

But since given expression is less than or equal to zero

therefore, solution set contains  since .............................(II)
and  , since ....................................................................(III)

since positive number multiplies with negative number gives negative number and if multiplies with positive number gives positive number. And in this inequality we find the values of x for which whole experssion will negative or zero.

So, in this inequality positive term expressions have no effects and we consider only those value of these expressions for which our question expression will get the value "0".

we can get rid of these positive expression and consider the fact that  and   from (II) and (III) and they can be solution of the inequality and from (I)  

so, the question becomes,

we are not concerned about odd powers, because they cant change behaviour of expression
there are two behaviour changing points -1 and -3,
by wavy curvy method,

here 
 and considering (I), (II) and (III)

SOLUTION OF INEQUALITY IS 


3. 

Since,, 
therefore, .................................(I)

consider,
i),
here,  and 
therefore,  why??? (http://iitmathseasy.blogspot.in/2013/03/quadratic-expression-concept-ii.html)

we can deal with it by two ways,

(1) since the quadratic expression will always be negative, if we multiply both side with -1 the quadratic expression will now always be positive and inequality get reversed.

because if 3>2
-3<-2

therefore, now left the part of quadratic expression( after multiply by -1 it will become always positive so expression can be left) the inequality becomes


we can also left  because  and  (from (I))
the inequality becomes 



since only behaviour changing point is -3
 solution is 

(2) since , therefore we can left the expression
the inequality becomes



where  is always negative
therefore only behaviour changing point is -3



understand,

clearly expression is +ve for less than 3
therefore solution is