Question

graph the inequality.
$y \geq -\frac{1}{2}x - 3$

Explanation:

Step1: Identify the boundary line

The inequality is \( y \geq -\frac{1}{2}x - 3 \). First, graph the boundary line \( y = -\frac{1}{2}x - 3 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m=-\frac{1}{2} \) and the y - intercept \( b=-3 \). To graph the line, start by plotting the y - intercept: when \( x = 0 \), \( y=-3 \), so the point is \( (0,-3) \). Then, use the slope to find another point. The slope \( -\frac{1}{2} \) means for every 2 units we move to the right (increase in \( x \) by 2), we move down 1 unit (decrease in \( y \) by 1). So from \( (0,-3) \), moving 2 units right to \( x = 2 \) and 1 unit down gives the point \( (2,-4) \). Since the inequality is \( \geq \), the boundary line should be a solid line (because the points on the line are included in the solution set).

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 \( (0,0) \) (as long as it is not on the boundary line). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0\geq-\frac{1}{2}(0)-3 \), which simplifies to \( 0\geq - 3 \). This is a true statement. So we shade the region that contains the point \( (0,0) \), which is the region above the line \( y = -\frac{1}{2}x - 3 \).

Answer:

1. Draw a solid line for \( y = -\frac{1}{2}x - 3 \) (passing through \( (0,-3) \) and \( (2,-4) \) etc.).
2. Shade the region above the solid line (the region that includes the point \( (0,0) \)).