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Chapter 7: Problem 56

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\). Discuss similarities and differences between the graphs.

Short Answer

Expert verified
The graphs of the inequalities \(3x - 2y > 6\) and \(3x - 2y \leq 6\) are quite similar. They both consist of the line and a shading region. However, the main difference lies in the shading direction. The first one shades above the line while the second one shades below the line and includes the line itself.

Step by step solution

01

Graph the first inequality

The first inequality is \(3x - 2y > 6\). To graph this inequality, first graph the equation \(3x - 2y = 6\) as if it was an equality. The line represents all the points (x, y) that satisfy the equation. Then, test a point not on the line (e.g., the origin (0, 0)) in the inequality. If it satisfies the inequality, shade that region. If it doesn't, shade the opposite region.
02

Graph the second inequality

The second inequality is \(3x - 2y \leq 6\). To graph this inequality, similarly graph the line for the equation \(3x - 2y = 6\). Then, test a point not on the line in the inequality. If it satisfied the inequality, shade the region. If it doesn't, shade the opposite region. In this case, unlike the first inequality, the line will be included in the area that satisfies the inequality.
03

Compare and analyze these graphs

Now, examine the two graphs for similarities and differences. The positions of the line graph in both inequalities are the same, but how they divide the plane is different due to the inequality symbols. In the first inequality, the points on the line does not satisfy the inequality, while in the second, they do. Also, the shading of the region is in opposite directions.

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