Question

graph: $y < \frac{1}{3}x + \frac{1}{2}$
click or tap the graph to plot a point.

Explanation:

Step1: Identify the boundary line

The inequality is \( y < \frac{1}{3}x + \frac{1}{2} \). The boundary line is \( y = \frac{1}{3}x + \frac{1}{2} \), which is a straight line with slope \( \frac{1}{3} \) and y-intercept \( \frac{1}{2} \). Since the inequality is \( < \), the line should be dashed.

Step2: Choose a test point

A common test point is the origin \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality:
\( 0 < \frac{1}{3}(0) + \frac{1}{2} \)
\( 0 < \frac{1}{2} \), which is true. So we shade the region that includes the origin.

Step3: Plot the boundary line

• Find two points on the line \( y = \frac{1}{3}x + \frac{1}{2} \).
• When \( x = 0 \), \( y = \frac{1}{2} \), so the point is \( (0, \frac{1}{2}) \).
• When \( x = 3 \), \( y = \frac{1}{3}(3) + \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2} \), so the point is \( (3, \frac{3}{2}) \).
• Draw a dashed line through these two points.

Step4: Shade the region

Since the test point \( (0,0) \) satisfies the inequality, shade the region below the dashed line (because \( y < \) the line).

For the table, we can choose some x-values and find the corresponding y-values on the boundary line (even though the inequality is strict, the table can help plot the line):

• When \( x = -3 \), \( y = \frac{1}{3}(-3) + \frac{1}{2} = -1 + \frac{1}{2} = -\frac{1}{2} \)
• When \( x = 0 \), \( y = \frac{1}{2} \)
• When \( x = 3 \), \( y = \frac{3}{2} \)

So the table can be:

x	y
0	1/2
3	3/2

Answer:

To graph \( y < \frac{1}{3}x + \frac{1}{2} \):

1. Draw a dashed line for \( y = \frac{1}{3}x + \frac{1}{2} \) (using points like \( (0, \frac{1}{2}) \) and \( (3, \frac{3}{2}) \)).
2. Shade the region below the dashed line (since \( (0,0) \) satisfies \( y < \frac{1}{3}x + \frac{1}{2} \)).

The table (for the boundary line) is:

x	y
0	1/2
3	3/2