Question

graph the inequality.
$y \geq \frac{2}{3}x + 2$

Explanation:

Step1: Graph the boundary line

The inequality is \( y \geq \frac{2}{3}x + 2 \). First, graph the line \( y=\frac{2}{3}x + 2 \). The slope \( m=\frac{2}{3} \) and the y - intercept \( b = 2 \). Plot the y - intercept at \( (0,2) \). Then, use the slope to find another point. From \( (0,2) \), move up 2 units and right 3 units to get the point \( (3,4) \) (since slope \(=\frac{\text{rise}}{\text{run}}=\frac{2}{3}\)). Since the inequality is \( \geq \), the boundary line should be a solid line (because the inequality includes equality).

Step2: Shade the region

To determine 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: \( 0\geq\frac{2}{3}(0)+2 \), which simplifies to \( 0\geq2 \). This is false. So we shade the region that does not include the origin. In other words, we shade the region above the line \( y = \frac{2}{3}x+2 \) (because the test point \( (0,0) \) is below the line and does not satisfy the inequality, so the solution set is above the line).

(Note: Since this is a graphing problem, the final answer is the graph with the solid line \( y=\frac{2}{3}x + 2 \) and the region above the line shaded. If we were to describe the steps for graphing, the above steps are the key parts. If we consider the visual graph, the solid line connects \( (0,2) \) and \( (3,4) \) (and other points following the slope) and the area above the line is shaded.)

Answer:

The graph has a solid line \( y = \frac{2}{3}x+2 \) (passing through \( (0,2) \) and \( (3,4) \) etc.) with the region above the line shaded.