An inequality is expressed as any mathematical statement that uses one of the following symbols or inequalities.
To understand the concept of inequalities, we must keep in mind that equality means equal and inequality means there are five possibilities between terms/objects. Let us understand these types of possibilities through a table.
||Greater than or Equal to
||Less than or Equal to
Let us consider two variables X and Y.
Rules of Inequalities
|X > Y
||X Greater than Y
|X < Y
||X is Less than Y
|X ≥ Y
||X Greater than or Equal to Y
|X ≤ Y
||X is Less than or Equal to Y
|X = Y
||X is Equal to Y
The relation between two inequalities can be established if they have a common term. For example,
a) A > B, B > C
: By looking at this, we can easily define the relation that A > B > C which means A > C or C < A.
b) A < B, B < C
: By looking at this, we can easily define that A < B C or C < A.
c) A ≥ B, B ≥ C
: This implies that A ≥ B ≥ C, which means A ≥ C or C ≤ A.
A relation cannot be defined if they don't have a common term. For example,
a) A > B, C > B
: This implies that we cannot define a specific relation between A and C as both of them are greater than B.
b) B > A, D < B
: This implies that a relation cannot be established between A and D as both are lesser than B.
c) A ≥ B, B ≤ C
: This implies that a relation cannot be established between A and C as both are greater than or equal to B.
Determine the values of the variable that make the inequality true. The methodology used to settle an inequality is like that used to solve a condition/equation. That is, by utilizing fundamental operations, attempt to identify the variable on one side of the inequality.
There are two basic rules for solving inequalities and are similar to the rules for solving equations. They are namely:
(i) When the same constant is added to (or subtracted from) both sides of an inequality, the direction of inequality is preserved, and the new inequality is equivalent to the original.
(ii) When both sides of the inequality are multiplied (or divided) by the same constant, the direction of inequality is preserved if the constant is positive, but reversed if the constant is negative. In either case the new inequality is equivalent to the original.
: a) A > B b) B > C
: a) A > C b) C > A
On combining both the statements, we get: A > B > C
So, we can easily say that the conclusion a) follows i.e. A > C
: a) A > B b) B < C
: a) A > C b) A < C
Here, we can see that both A and C are greater than B but we cannot derive any relation between A and C.
So the answer to the above questions will be "nothing can be concluded".
: A > B > C < D≥E
: a) A > D b) D > B
: Here, nothing can be concluded because there is no definite relation between A & D and D & B. we cannot say that which one is greater, equal or lesser.