A first degree system of inequalities with two variables is a set that has at least two inequalities.
The solution set of an inequality is represented by a region of the Cartesian plane. The solution set of a system of inequalities corresponds graphically to the intersection of the solution set of each inequality of the system.
Consider the system formed by the following inequalities: |3x+6y\le12| and |x>3y-4.|
It can be represented as follows.
Any point belonging to region A only satisfies the inequality |3x+6y\le12.|
Any point belonging to region B only satisfies the inequality |x>3y-4.|
Any point belonging to region C simultaneously satisfies each of the inequalities.
Any point belonging to region D does not satisfy any inequality.
The solution set of this system of inequalities corresponds to region C, since it coincides with the intersection of the respective solution sets of each inequality.
When the border lines associated with a system of inequality are parallel, certain special cases may occur.
There may be an empty solution set (on the left) or the junction of the two border lines (if they are drawn in solid lines) may form the solution set (on the right).
The solution set of the two inequalities can correspond to a single inequality (on the left) or the region between the two border lines can form the solution set (on the right).
A polygon of constraints is the geometric shape delimited in a Cartesian plane by the meeting of the various solution sets of the given system of inequalities. This is the region that contains all the points that satisfy all the given inequalities.
Depending on the system of inequalities to be represented, the polygon of constraints can be unbounded when the region it defines is not confined on all sides (like the left polygon below). It can also be closed or bounded when the region it delimits is confined on all sides by segments (like the polygon below, on the right).
Given the following system of equations.
|y\leq x+2|
|y\leq -\frac{x}{2}+6|
|x\leq 6|
|y\ge-2x+4|
|y\ge0|
If each of these inequalities are sketched on a Cartesian plane, the following graphs are obtained.
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|y\leq x+2|
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|y\leq -\frac{x}{2}+6|
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|x\le6|
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|y\ge-2x+4|
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|y\ge0|
When all these regions are superimposed, the result is a region which is common to all the inequalities. This region, delimited by a black figure below, is called a polygon of constraints. The points inside this polygon are the solutions to all of the inequalities.
Here is a case where the polygon is bounded or closed.
In most real-life situations, the values of the variables cannot be less than |0.| Two inequalities called “non-negative constraints” are then added to the system. For example, if the problem’s situation refers to the number of hours worked, it is impossible to have negative values. So, the constraints |x\ge0| and |y\ge0| need to be added to the system of inequalities.
The points which make the vertices of the polygon of constraints are particularly noteworthy. Indeed, the points forming the vertices of the polygon are its extremes, or its maximum and minimum values. It is from these extremes that a system of inequalities can be optimized.
To determine the coordinates of the vertices of a polygon of constraints, first solve the appropriate system of equations for each vertex. It is possible to determine the coordinates of the meeting point of the two border lines algebraically, graphically, or using a table of values. It is important to note that a vertex is part of the solution set only if the border lines which form it are solid lines.
In the polygon of constraints below:
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the vertices F and G are not part of the solution set, since one of the lines which form them is a dotted line;
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the vertex H is part of the solution set, since the two border lines which form it are solid lines.
To learn more about the methods of solving systems of equations: