To graph the linear inequality [tex]\( y \geq -x - 3 \)[/tex], follow these steps:
1. Identify the boundary line:
The boundary line for the inequality is given by the equation [tex]\( y = -x - 3 \)[/tex]. This is a straight line with a slope of -1 and a y-intercept of -3.
2. Graph the boundary line:
- Start by plotting the y-intercept, which is the point (0, -3).
- From this point, use the slope to find another point on the line. The slope is -1, which means for every 1 unit you move to the right (in the positive x-direction), you move 1 unit down (in the negative y-direction). So, starting at (0, -3), moving right 1 unit to (1, -4), you'll have another point.
- Plot these points on the graph and draw a straight line through them. Since the inequality symbol is "≥" (greater than or equal to), the boundary line itself is included in the solution set. Therefore, draw the line as a solid line.
3. Shade the appropriate region:
- The inequality [tex]\( y \geq -x - 3 \)[/tex] indicates that y-values on the graph should be greater than or equal to the values of the line [tex]\( y = -x - 3 \)[/tex].
- To determine which side of the line to shade, you can test a point that is not on the line, such as (0, 0):
- Substitute (0, 0) into the inequality: [tex]\( 0 \geq -0 - 3 \)[/tex] → [tex]\( 0 \geq -3 \)[/tex].
- This is true. Therefore, the region that includes the point (0, 0) is the solution region.
- Shade the region above and including the boundary line because it satisfies the inequality.
4. Final graph:
The graph consists of the boundary line [tex]\( y = -x - 3 \)[/tex] drawn as a solid line and the region above it shaded.
This is the graph representing the linear inequality [tex]\( y \geq -x - 3 \)[/tex].
