Question

2 $y < -\frac{1}{3}x + 4$

Explanation:

Step1: Identify the boundary line

The inequality is \( y < -\frac{1}{3}x + 4 \). First, consider the equation of the boundary line, which is \( y = -\frac{1}{3}x + 4 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m=-\frac{1}{3} \) and the y - intercept \( b = 4 \).

To graph the boundary line:

• Plot the y - intercept: When \( x = 0 \), \( y=4 \), so we have the point \( (0,4) \).
• Use the slope to find another point. The slope \( m = \frac{\text{rise}}{\text{run}}=-\frac{1}{3} \), which means from the point \( (0,4) \), we can go down 1 unit (since the rise is - 1) and to the right 3 units (since the run is 3). So we get the point \( (3,3) \) (because \( 4-1 = 3 \) and \( 0 + 3=3 \)). We can also go up 1 unit and to the left 3 units to get the point \( (-3,5) \) (since \( 4 + 1=5 \) and \( 0-3=-3 \)).

Since the inequality is \( y<-\frac{1}{3}x + 4 \) (not \( y\leq-\frac{1}{3}x + 4 \)), the boundary line should be a dashed line. This is because the points on the line \( y = -\frac{1}{3}x+4 \) do not satisfy the inequality \( y<-\frac{1}{3}x + 4 \).

Step2: Determine the region to shade

To determine which side of the line to shade, we can use a test point. A common test point is the origin \( (0,0) \) (as long as the line does not pass through it; in this case, the line \( y = -\frac{1}{3}x + 4 \) passes through \( (0,4) \), not \( (0,0) \)).

Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y<-\frac{1}{3}x + 4 \):
\( 0<-\frac{1}{3}(0)+4 \)
\( 0 < 4 \), which is a true statement.

Since the test point \( (0,0) \) satisfies the inequality, we shade the region that contains the origin.

Graphing Steps Summary:

1. Draw a dashed line through the points \( (0,4) \), \( (3,3) \), \( (-3,5) \) (or other points on the line \( y = -\frac{1}{3}x + 4 \)).
2. Shade the region below the dashed line (because the test point \( (0,0) \) is below the line \( y = -\frac{1}{3}x + 4 \) and satisfies the inequality \( y<-\frac{1}{3}x + 4 \)).

(Note: Since the problem is about graphing the inequality, the final answer is the graph with a dashed line \( y = -\frac{1}{3}x + 4 \) and the region below the line shaded.)

Answer:

The graph has a dashed line \( y = -\frac{1}{3}x + 4 \) (with points like \( (0,4) \), \( (3,3) \) on it) and the region below the line (containing the origin \( (0,0) \)) is shaded.