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Chapter 7: Problem 9

Graph each linear inequality. \(2 y-3 x>6\)

Short Answer

Expert verified
The graph of the inequality \(2y - 3x > 6\) is a dashed line passing through the y-axis at y = 3 with a slope of 1.5, and the area above the line shaded.

Step by step solution

01

Rearrange the inequality

First, the inequality needs to be rearranged in the 'y' form, similar to the slope-intercept form of linear equations (y = mx + b). That means isolating 'y' on one side of the inequality. Doing this, we obtain: \(2y > 3x + 6\), then divide all terms by 2, leading to \(y > 1.5x + 3\).
02

Graph the boundary line

Next, knowing that 'm' is the slope and 'b' is the y-intercept in the form y = mx + b, we can graph the line y = 1.5x + 3. This line is the boundary of our inequality. Since the inequality symbol is 'greater than' (not 'greater than or equals'), the line should be represented as a dashed line.
03

Choose the side of the line

Finally, to determine which side of the line to shade, we can use a test point. The origin (0,0) is often the easiest point to use, unless it lies on the line. If the inequality is true when we substitute the test point, we shade that side of the line. If not, we shade the other side. In this case, substituting (0,0) into the original inequality, we get a false statement. Therefore, we shade the side of the line that does not contain the origin.

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