Question

graph this inequality:
$y > \frac{1}{4}x + 5$
dot points on the boundary line. select the line to switch between solid and dotted. select a region to shade it.

Explanation:

Step1: Identify the boundary line

The inequality is \( y > \frac{1}{4}x + 5 \). First, we consider the corresponding equation \( y=\frac{1}{4}x + 5 \) to find the boundary line. The slope \( m=\frac{1}{4} \) and the y - intercept \( b = 5 \). Since the inequality is \( y>\frac{1}{4}x + 5 \) (not \( y\geq\frac{1}{4}x + 5 \)), the boundary line should be dashed (to indicate that points on the line are not included in the solution set).

Step2: Plot the boundary line

• Start at the y - intercept: When \( x = 0 \), \( y=5 \). So we plot the point \( (0,5) \).
• Use the slope to find another point. The slope \( \frac{1}{4} \) means for every 4 units we move to the right (increase in x by 4), we move up 1 unit (increase in y by 1). So from \( (0,5) \), if we move \( x = 4 \), then \( y=5 + 1=6 \). So we can plot the point \( (4,6) \). Then draw a dashed line through these two points (and extend it) to represent the boundary line \( y=\frac{1}{4}x + 5 \).

Step3: Determine the region to shade

To determine which side of the line to shade, we can test a point that is not on the line. A common test point is the origin \( (0,0) \), but let's check if it satisfies the inequality \( y>\frac{1}{4}x + 5 \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0>\frac{1}{4}(0)+5\), which simplifies to \( 0 > 5 \). This is false. So we need to shade the region that does not include the origin. Another test point: let's take \( (0,6) \). Substitute into the inequality: \( 6>\frac{1}{4}(0)+5\), which is \( 6 > 5 \), a true statement. So the region above the dashed line \( y=\frac{1}{4}x + 5 \) should be shaded.

(Note: Since this is a graphing problem, the final answer is the graph with a dashed line \( y = \frac{1}{4}x+5 \) and the region above the line shaded. If we were to describe the steps for graphing, the above steps outline how to get to the correct graph.)

Answer:

1. Draw a dashed line for \( y=\frac{1}{4}x + 5 \) (passing through \( (0,5) \) and \( (4,6) \) as key points).
2. Shade the region above the dashed line (the region where \( y\) - values are greater than \( \frac{1}{4}x + 5 \) for corresponding \( x\) - values).