Solving a Linear Equation and its two Sibling Inequalities in Two Variables

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EXAMPLE: Discuss the inequality 3x – 7y < 21. Begin by analyzing the equation of the boundary line is 3x – 7y = 21. This is a linear equation in two variables so its graph is a line in the rectangular coordinate system. If x = 0, then y = –3, so the point (0, –3) is the y-intercept of the boundary line. If y = 0, then x = 7, so the point (7, 0) is the x-intercept of the boundary line. Plot those two points and draw the line through them to obtain the graph of the boundary line as shown in green in Fig 20.

The origin is not on the graph of the boundary line and is easy to test in either of the inequalities. Use (0, 0) as a test point in the inequality 3x – 7y < 21 to obtain 3(0) – 7(0) < 21 which is TRUE. Therefore (0, 0) is a solution of 3x – 7y< 21 and every other point in that half-plane is a solution of the inequality 3x – 7y < 21. Conclusion: The solution set for 3x – 7y < 21 is the half-plane containing the origin and bounded by the graph of 3x – 7y= 21. We graph the inequality by shading the half-plane which is its solution set as shown in blue in Fig. 21

Because of the Law of Trichotomy we know that the other half-plane (the un-shaded part) is the graph of 3x – 7y > 21. Conclusion: The solution set for 3x – 7y > 21 is the half-plane NOT containing the origin and bounded by the graph of 3x – 7y= 21. For consistency we show this graph in red in Fig. 22.

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