Question

graph the inequality y ≤ (2/3)x - 3.

Explanation:

Step1: Graph the boundary line

The inequality is \( y \leq \frac{2}{3}x - 3 \). First, graph the boundary line \( y=\frac{2}{3}x - 3 \). The slope \( m=\frac{2}{3} \) and the y - intercept \( b=- 3 \). Plot the y - intercept at \( (0,-3) \). Then, use the slope to find another point. From \( (0,-3) \), move up 2 units and right 3 units to get \( (3,-1) \). Since the inequality is \( \leq \), the boundary line should be solid (because the points on the line are included in the solution set).

Step2: Shade the region

To determine which side of the line to shade, we can use a test point. A common test point is \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0\leq\frac{2}{3}(0)-3\), which simplifies to \( 0\leq - 3 \). This is false. So, we shade the side of the line that does not include the test point \( (0,0) \). In other words, we shade the region below the line \( y=\frac{2}{3}x - 3 \) (since the inequality is \( y\leq\frac{2}{3}x - 3 \), all the points with y - values less than or equal to the y - values on the line are in the solution set).

Answer:

The graph consists of a solid line with slope \( \frac{2}{3} \) and y - intercept \( - 3 \) (passing through \( (0,-3) \) and \( (3,-1) \) among other points) and the region below (including the line) this line is shaded.