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Solving Systems of Inequalities

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.
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:

  1. Graph the boundary lines of the inequalities. Use broken lines for inequalities with < or > symbols and solid lines for inequalities containing $\leq$ or $\geq$.
  2. The graph of the system is the region common to all the inequalities. If no such region is formed, then the system is termed as having no solution.


Graph the solution of the system of inequalities:
x - y $\geq$ - 4
3x + 2y < 8
x $\geq$ 0 and y $\geq$ 0


The boundary lines for the above system are x - y = - 4, 3x + 2y = 8 and the two coordinate axes.

  1. The half plane represented by the x - y $\geq$ - 4 lies below the boundary line including the points on the boundary line.
  2. The half plane represented by 3x + 2y < 8 is below the boundary line without including the points on the line.
  3. The two inequalities x $\geq$ 0 and y $\geq$ 0 are together represented by the first quadrant. 
The common region is thus the triangular region as shown shaded in the graph below:

Solving Systems of Inequalities

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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