Question

y < -\frac{2}{3}x - 2

Explanation:

Step1: Analyze the inequality type

The inequality is \( y < -\frac{2}{3}x - 2 \). For a linear inequality in two variables \( y < mx + b \), the boundary line is \( y = mx + b \), and we determine if it's dashed or solid and which side to shade. Since the inequality is "less than" (\(<\)), the boundary line should be dashed (but in the given graph, it seems to be a line, maybe a typo or for illustration, but the key is the shading direction). The slope \( m = -\frac{2}{3} \) and y - intercept \( b=-2 \).

Step2: Determine the boundary line

The equation of the boundary line is \( y = -\frac{2}{3}x - 2 \). To graph this, we can use the y - intercept (\( x = 0,y=-2 \)) and the slope (for every 3 units we move to the right along the x - axis, we move down 2 units, or for every 3 units left, we move up 2 units).

Step3: Determine the shading region

Since the inequality is \( y < -\frac{2}{3}x - 2 \), we shade the region below the line \( y = -\frac{2}{3}x - 2 \). To verify, we can pick a test point not on the line, say \( (0,0) \). Substitute into the inequality: \( 0<-\frac{2}{3}(0)-2\Rightarrow0 < - 2 \), which is false. So \( (0,0) \) is not in the solution region, so we shade the region that does not include \( (0,0) \), which is below the line.

Answer:

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

1. Draw the boundary line \( y = -\frac{2}{3}x - 2 \) (it should be a dashed line for \( < \), but in the given graph, if it's a solid line, it might be a mistake, but the key is the inequality). The line passes through \( (0,-2) \) and \( (3,-4) \) (since when \( x = 3,y=-\frac{2}{3}(3)-2=-2 - 2=-4 \)) or \( (-3,0) \) (when \( y = 0,0=-\frac{2}{3}x-2\Rightarrow\frac{2}{3}x=-2\Rightarrow x=-3 \)).
2. Shade the region below the line \( y = -\frac{2}{3}x - 2 \) (the region where \( y \) - values are less than the values on the line \( y = -\frac{2}{3}x - 2 \)).

(If the question was to graph the inequality, this is the process. If it was to check the graph, the boundary line is correct as \( y = -\frac{2}{3}x - 2 \) and the shading should be below the line.)