Inequalities

Date: 08/01/97 at 02:44:30
From: Eytan Reiss
Subject: Greater than

How do I deal with an equation with a greater than > or less than < 
sign? I tried treating it like an equals sign but got a ridiculous 
answer.  

ex:  7+(X=7) > 3

Thanks and sincerely yours,
Eytan Reiss

Date: 08/07/97 at 14:13:51
From: Doctor Rob
Subject: Re: Greater than

Mostly you can work with inequalities in the same way you do with
equalities, but there are some important differences.  The rules are
more or less like this:

Let a, b, x, y, and z be any real numbers.

If x > y and y > z, then x > z.

If x > y and a = b, then x + a > y + b.
If x > y and a = b, then x - a > y - b.
If x > y and a = b > 0, then x*a > y*b.
If x > y and a = b > 0, then x/a > y/b.
If x > y and a = b < 0, then x*a < y*b.
If x > y and a = b < 0, then x/a < y/b.
If x > y and a > b, then x + a > y + b.
If x > y and a < b, then nothing can be said about x + a and y + b.
If x > y and a < b, then x - a > y - b.
If x > y and a > b, then nothing can be said about x - a and y - b.
If x > y and a > b > 0, then x*a > y*b.
If x > y and 0 > a > b, then nothing can be said about x*a and y*b.
If x > y and a < b < 0, then x*a < y*b.
If x > y and 0 < a < b, then nothing can be said about x*a and y*b.
If x > y and a > b > 0, then nothing can be said about x/a and y/b.
If x > y and 0 > a > b, then x/a < y/b.
If x > y and a < b < 0, then nothing can be said about x/a and y/b.
If x > y and 0 < a < b, then x/a > y/b.

If x^2 < a > 0, then -Sqrt[a] < x < Sqrt[a].
If x^2 > a > 0, then either x < -Sqrt[a] or else x > Sqrt[a].

All the above are still true if you replace > with >= or replace < 
with <=.

All the statements made above are true essentially because of the 
laws of signs for addition, subtraction, multiplication, and division.  
For example, x > y means x = y + p for some positive p (i.e. p > 0).  
If a = b > 0, then a and b are positive. Then a*x = b*(y + p) = 
b*y + b*p. Since p is positive and b is positive, p*b is positive (law 
of signs for multiplication), so a*x > b*y (since to get from b*y to 
a*x you add something positive).  The rest are similar.

I hope this helps.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/