Properties of inequalities

There are some important properties of inequalities. That every student should know about.

1. Trichotomy Property:

   For any two real number a and b, one and only one statement of the following is always true. a<b, a=b or a>b.

   For example If $a=4$ and $b=5$ then only one of the above is true which is a<b\text{ or }4<5 and the other statements are not true. Such as 4 is not equal to 5 and 4 is not greater than 5.

2. Transitive Property:

   For any three real number a, b and c

   If a<b and b<c then a<c

   Means if a is less than b and b is less than c then a is also less than c. For example a=3, b=4 and c=5 then 3 is less than 4 and 4 is less than 5 thus 3 is also less than 5.

   also

   If a>b and b>c then a>c

   Means if a is greater than b and b is greater than c then a is also greater than c. For example a=5, b=4 and c=3 Then 5 is greater than 4 and 4 is greater than 3 then 5 is also greater than 3.

3. Additive Property:

   For any three real number a, b and c

   If a>b then a+c>b+c

   Means if a is greater than b then $a+c$ is also greater than $b+c$. For example a=5, b=4 and c=3 Since 5 is greater than 4 then 5+3=8 is also greater than 4+3=7.

   Also

   If a<b then a+c<b+c

   Means if a is less than b then $a+c$ is also less than $b+c$. For example a=3, b=4 and c=5 Since 3 is less than 4 then 3+5=8 is also less than 4+5=9.

4. Multiplicative Property:

   For any three real number a, b and c where c>0

   If a>b then ac>bc

   Means if a is greater than b then $ac$ is also greater than $bc$. For example a=5, b=4 and c=3 Since 5 is greater than 4 then 5x3=15 is also greater than 4x3=12.

   Also

   If a<b then ac<bc

   Means if a is less than b then $ac$ is also less than $bc$. For example a=3, b=4 and c=5 Since 3 is less than 4 then 3x5=15 is also less than 4x5=20.

   For any three real numbers a, b and c where c<0

   If a>b then ac<bc

   Means if a is greater than b then $ac$ will be less than $bc$. For example a=5, b=4 and c=-5 Since 5 is greater than 4 then 5x-5=-25 is less than 4x-5=-20.

   Also

   If a<b then ac>bc

   Means if a is less than b then $ac$ will be greater than $bc$. For example a=3, b=4 and c=-5 Since 3 is less than 4 then 3x-5=-15 is greater than 4x-5=-20.