Question

y ≤ -\frac{2}{3}x - 2

Explanation:

Step 1: Graph the boundary line

The equation of the boundary line is \( y=-\frac{2}{3}x - 2 \). For the y - intercept, when \( x = 0\), \( y=-2\), so the line passes through the point \((0,-2)\). Using the slope \( m =-\frac{2}{3}\) (rise over run, so from \((0,-2)\), we can go down 2 units and right 3 units to get another point \((3,-4)\) or up 2 units and left 3 units to get \((- 3,0)\)). Since the inequality is \( \leq \), we draw a solid line to represent the boundary.

Step 2: Test a point to find the shaded region

We choose the test point \((0,0)\). Substitute into the inequality \( y\leq-\frac{2}{3}x - 2\):

$$
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$$

This statement is false. So the region that satisfies the inequality is the region that does not contain the origin. We shade the area below the solid line \( y = -\frac{2}{3}x - 2 \).

Answer:

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

1. Graph the line: First, graph the boundary line \( y = -\frac{2}{3}x - 2 \). The slope is \( -\frac{2}{3} \) and the y - intercept is \( - 2 \). Since the inequality is \( \leq \), the line should be solid (to include the points on the line).
2. Determine the region: To find which side of the line to shade, we can test a point not on the line. A common test point is the origin \((0,0)\). Substitute \( x = 0\) and \( y = 0\) into the inequality: \( 0\leq-\frac{2}{3}(0)-2\), which simplifies to \( 0\leq - 2 \). This is false. So we shade the region that does not contain the origin (the region below the line \( y = -\frac{2}{3}x - 2 \)).