Plz explain all the properties of inequality.....required in increasing and decreasing functions.


Dear Student,
There are many properties of inequalities.
It is not possible to mention all unless asked for some specific ones.
Here are some basic properties of inequality.

There are four inequalities [<(less than)], [>(greater than)], [(less than or equal to)], [(greater than or equal to)]. The properties of inequalities are:

Let a, b, c be real numbers.
  • Transitive Property: If a<b and b<c then a<c. Similarly, if a>b and b>c then a>c.
  • Reversal Property: If a<b then b>a. Similarly, if a>b then b<a.
  • Law of Trichotomy: Only one of the following is true: a<b, a=b, a>b.
  • Addition Property: If a<b then a+c<b+c. Similarly, if a>b then a+c>b+c.
  • Subtraction Property: If a<b then a-c<b-c. Similarly, if a>b then a-c>b-c.
  • Multiplication Property: If a<b and c is positive then ac<bc and if a<b and c is negative then ac>bc. Similarly, If a>b and c is positive then ac>bc and a>b and c is negative then ac<bc.
  • Division Property: If a<b and c is positive then ac<bc and if a<b and c is negative then ac>bc. Similarly, If a>b and c is positive then ac>bc and a>b and c is negative then ac<bc.
  • Additive Inverse: If a<b then -a>-b. Similarly, if a>b then -a<-b.
  • Multiplicative Inverse: If a<b then 1a>1b. Similarly, if a>b then 1a<1b.
  • Non-negative Property of Squares: A square of a number is always greater than or equal to zero. a20
  • Square Root Property: If ab then ab if a, b0. Similarly, If ab then ab if a, b0
If you want some specific properties, kindly mention them in a separate thread.

Regards.

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