Lesson Proving inequalities

Proving inequalities

Problem 1

The standard way to prove such statements is to start from the given inequality and to transform it to the simplest form.

using EQUIVALENT transformations.


If the final simplified inequality is TRUE, then the starting inequality is TRUE, too, since you can reverse the chain 
of equivalent transformations.


So, we start from  

     < .                   (1)


Multiply both sides by the positive number  (ax+y)*(bx+y).  Then inequality (1) is EQUIVALENT to

    a*(bx+y) < b*(ax+y).               (2)


Inequality (2) is EQUIVALENT to

    abx + ay < abx + by                (3)


which is equivalent to

    ay < by                            (4)


which with positive "y" is equivalent to

    a < b,                             (5)


which is given.


So, the reversed chain of VALID inequalities   (5) ---> (4) ---> (3) ---> (2) ---> (1)  proves the statement.

Problem 2

Let's consider this expression

    a^4 - a^3b + b^4 - ab^3.



Transform it this way  

    a^4 - a^3b + b^4 - ab^3 =  +  =  =  = .



So, our starting expression is the product of two quadratic polynomials

      and  .



They both are positively defined; in other words, they never take negative values.


Therefore,   >= 0  for all values of "a" and "b".



It implies that the original expression is never negative 

    a^4 - a^3b + b^4 - ab^3 >= 0.



It means that

    a^4 + b^4 >= a^3b + ab^3,


which is what has to be proved.


At this point, the proof is completed.

Problem 3

Your starting inequality is

    |x+3| < 1/2.     (1)


Taking off the absolute value symbol, it means that

    -1/2 < x+3 < 1/2.      (the compound inequality)


Multiply the last compound inequality by 4  (multiply all three its terms).
You will get an equivalent inequality

    -2 < 4x + 12 < 2.


Add 1 (one) to the last compound inequality (to all its three terms).
You will get an equivalent inequality

    -1 < 4x + 13 < 3.    (2)


But if (2) is valid, then also

    -3 < 4x + 13 < 3     (3)

is valid, too.  


    +--------------------------------------------------------------------------+
    |   In Math, they say "if (2) is valid, then (3) is valid even more so".   |
    +--------------------------------------------------------------------------+


The last inequality (3)  is the same as 

    |4x+13| < 3,

which we are requested to prove.


At this point, the proof is complete.