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Chapter 7: Problem 55

Explain how to graph \(2 x-3 y<6\).

Short Answer

Expert verified
To graph the inequality \(2x - 3y < 6\), first rearrange it to \(y > \frac{2}{3}x - 2\). Plot this line on the graph by using the y-intercept and slope. As the inequality sign is 'greater than', shade the region above the line on the graph. This shaded region represents the solution to the inequality.

Step by step solution

01

Rearrange the Inequality

Rearrange the inequality to make \(y\) the subject. First, subtract \(2x\) from both sides to get \(-3y < -2x + 6\), then, divide every term by \(-3\), flip the inequality sign (because you're dividing by a negative value) to get \(y > \frac{2}{3}x - 2\).
02

Graph the Corresponding Line

Graph the line \(y = \frac{2}{3}x - 2\), which is the boundary of the inequality. This can be done by initially plotting the y-intercept, which is \(-2\), then using the slope \(\frac{2}{3}\) to find another point on the line. A line with a slope of \(\frac{2}{3}\) rises 2 units for every 3 units it runs.
03

Determine and Shade the Region

Since the inequality is \(y > \frac{2}{3}x - 2\), the solution will be all the points above the line. Therefore, shade the region above the boundary line \(y = \frac{2}{3}x - 2\). To check the shading direction, always take a test point that is not on the line, e.g., (0,0). If the test point satisfies the inequality, the side of the line on which it lies is the solution region.

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