Linear Inequalities
Inequalities can expressed by following four symbols:
< (less than), > (greater than), <= (less than or equal to) and >= (greater than or equal to)
(i) ax < b(ii) ax + b ≥ c(iii) ax + by > c(iv) ax + by ≤ c
Inequalities (i) and (ii) are in one variable while inequalities (iii) and (iv) are in two variables.
The following operations will not affect the order (or sense) of inequality while changing it to a simpler equivalent form:
(i) Adding or subtracting a constant to each side of it.
(ii) Multiplying or dividing each side of it by a positive constant.
Solution Region of Linear Inequalities in two variables
The graph of a system of inequalities consists of the set of all ordered pairs (x, y) in the xy-plane which simultaneously satisfy all the inequalities in the system.
To find the graph of such a system, we draw the graph of each inequality in the system on the same coordinates axes and then take intersection of all the graphs. The common region so obtained is called the solution region for the system of inequalities.
Vertex (Corner Points)
A point of a solution region where two of its boundary lines intersect, is called a corner point or vertex of the solution region.
Problem Constraits
The linear inequalities prescribe limitations and restrictions on allocation of available sources. While tackling a certain problem from every day life each linear inequality concerning the problem is named as problem constraint. The system of linear inequalities involved in the problem concerned are called problem constraints. The variables used in the system of linear inequalities relating to the problems of every day life are non-negative and are called non-negative constraits. These non-negative constraits play an important role for taking decisions. So these variables are called decision variables.
Feasible Region
A solution region which is restricted to the first quadrant is referred to as a feasible region for the solution of given constraits. Each point of the feasible region is called a feasible solution of the system of linear inequalities. A set consisting of all the feasible solutions of the system of linear inequalities is called a feasible solution set.
