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Chapter 8: Problem 8

Graph each inequality. $$y>3 x+2$$

Short Answer

Expert verified
The graph of the inequality \(y > 3x + 2\) is a region above the line \(y = 3x + 2\), which is represented by a dashed line.

Step by step solution

01

Identify the boundary line

The inequality \(y > 3x + 2\) can be rewritten as \(y - 3x = 2\). The equation \(y = 3x + 2\) represents the boundary line. This line has a slope of 3 and y-intercept of 2.
02

Draw the boundary line

Plot the y-intercept at 2 on the y-axis. Then, use the slope (rise over run) to find the next point. Since the slope is 3, rise up 3 units and run (move to the right) 1 unit from the y-intercept. Connect these points to draw the line.
03

Identify the side for shading

Since the inequality is a 'greater than' (>) inequality, we will shade the region above the line. A quick way to decide the area for shading is to choose a point not on the line and substitute its coordinates into the inequality. If it satisfies the inequality, shade that region. If not, shade the opposite region. Here, let's choose (0,0), an easy test point.
04

Shade the appropriate region

Substitute (0,0) into the inequality. Since 0 is not greater than 2, we do not shade the region containing (0,0). So, shade the region above the boundary line. That represents all points (x, y) where \(y > 3x + 2\).
05

Notate the type of the line

As the inequality is a 'greater than' (>), and not 'greater than or equal to' (鈮�), the line representing the boundary line needs to be a dashed one. This means that the points on the line are not included in the solution.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed the solution set of \(y\) as \(x+2\) and \(x \geq 1\) without using test points.

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