A linear inequality in two variables is an inequality statement containing two variables. A system of linear inequalities describes two or more inequalities with the same two variables.
For example, 3x + y $\geq$ 6 and x + y < 4 form a system of linear inequalities in variables x and y. A solution of such system of inequalities is an ordered pair (x, y), which is a solution of each of the inequality in the system.
For example, the ordered pair (1, 2) is a solution of the system given above. It can be clearly seen that (1, 2) is the solution for both the inequalities.
3 (1) + 2 = 3 + 2 = 5 $\leq$ 6 (True)
1 + 2 = 3 < 4 (True)
A system of inequalities in two variables may also contain inequalities containing only one of the two variables forming the system.
Example:
x + y $\leq$ 8
3x + 2y < 12
x $\geq$ 2 Inequality containing only variable x.
y $\geq$ 0 Inequality with only variable y.
Graphically, each of the inequality in the system is represented by a half - plane in the coordinate plane.
The solution of the system of inequalities is the intersecting or common region of all the plane regions representing the individual inequalities.
Let us look into an example to understand the graphical method of solving system of linear inequalities.
The steps applied for solving system of inequalities are as follows:
Example:
Graph the solution of the system of inequalities:
x - y $\geq$ - 4
3x + 2y < 8
x $\geq$ 0 and y $\geq$ 0
Solution:
The boundary lines for the above system are x - y = - 4, 3x + 2y = 8 and the two coordinate axes.
The points of intersection of the boundary lines form the vertices of the common region. Thus, the points O (0, 0), A ( 0, 4) and B($\frac{8}{3}$, 0) are the vertices of the solution of the given system of inequalities. For the above system, A (0, 4) is not regarded as a solution as it is point on the boundary line of the inequality 3x + 2y > 8 and not included in its solution set.
In Linear Programming problems, the constraints of an optimizing function form a system of inequalities. The ordered pairs corresponding to the vertices are tested for optimal function value (maximum or minimum).
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