Law of Trichotomy and A Sytem of Equations and Inequalities in Two Variables

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EXAMPLE: Discuss the system of inequalities

Begin by observing that this is a system of three linear inequalities in two variables. The solution set for each individual inequality is a half-plane. (The Law of Trichotomy). The solution set for the system is the set of points which are solutions of 2x + 5y ≥ 10 AND –3x + 2y ≥ 6 AND 2x – 7y > – 38. Therefore the solution set of the system is the intersection of the three individual solution sets. A good strategy is to graph each of the individual inequalities in different colors on the same coordinate system so the intersection is easily observed to be the region shaded in all three colors. For instructional purposes, each inequality will initially be graphed on its own coordinate system and then they will be combined.

The boundary equation for 2x + 5y ≥ 10 is 2x + 5y = 10.
If x = 0, then y = 2. Therefore (0, 2) is the y-intercept of the boundary equation.
If y = 0, then x = 5. Therefore (5, 0) is the x-intercept of the boundary equation.

The origin is not on the boundary line so we test it in 2x + 5y ≥ 10 to obtain 0 ≥ 10 which is FALSE. Therefore the graph of 2x + 5y ≥ 10 is the half-plane which does not contain the origin and is bounded by the boundary line 2x + 5y = 10. Because the symbol ≥ permits equality, the boundary line is part of the solution set and is therefore drawn as a solid line to indicate that it is part of the graph. The graph of 2x + 5y ≥ 10 appears in Fig. 23.
The boundary equation for –3x + 2y ≥ 6 is –3x + 2y = 6.
If x = 0, then y = 3. Therefore (0, 3) is the y-intercept of the boundary equation.
If y = 0, then x = –2. Therefore (–2, 0) is the x-intercept of the boundary equation.
The origin is not on the boundary line so we test it in –3x + 2y ≥ 6 to obtain 0 ≥ 6 which is FALSE. Therefore the graph of –3x + 2y ≥ 6 is the half-plane which does not contain the origin and is bounded by the boundary line –3x + 2y = 6. Because the symbol ≥ permits equality, the boundary line is part of the solution set and is therefore drawn as a solid line to indicate that it is part of the graph. The graph of –3x + 2y ≥ 6 appears in Fig. 24.

The boundary equation for 2x – 7y> – 38 is 2x – 7y = – 38.
If x = 0, then . Therefore  is the y-intercept of the boundary equation.
If y = 0, then x = –19. Therefore (–19, 0) is the x-intercept of the boundary equation.
The origin is not on the boundary line so we test it in 2x – 7y > – 38 to obtain 0 ≥ –38 which is TRUE. Therefore the graph of 2x – 7y > – 38 is the half-plane which contains the origin and is bounded by the boundary line 2x – 7y = – 38. Because the symbol > does not permit equality, the boundary line is not part of the solution set and is therefore drawn as a dashed line to indicate that it is not part of the graph.
The graph of 2x – 7y > – 38 appears in Fig. 25.

Recall that the graph of the system is the intersection of the graphs of the individual inequalities in the system. In Fig. 26 the three graphs are superimposed on the same coordinate system and their intersection becomes clear. The graph of the system of inequalities is shown in Fig. 26.

The solution of the system
is the triangular region containing red, blue, and green shading and bounded by the three boundary lines. Note the blue and green boundaries of the triangle are part of the graph and the red boundary is not part of the graph. This graph tells us that every point in the triangle or on the red or on the blue edge of the triangle is a solution of the system of inequalities. In terms of sets and unions we see that the solution set of the system is the set of points in the interior of the triangle UNION the set of points on the blue boundary of the triangle UNION the set of points on the green boundary of the triangle.

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