Explanation:
In order to graph the linear inequality, y ≥ -4x - 2, we must first transform this into its slope-intercept form, y = mx + b:
y ≥ -4x - 2
Replace the "≥" symbol into an equal sign, "="
y = -4x - 2 ⇒ This is the slope-intercept form where the slope, m = -4, and the y-intercept, (0, -2), where b = -2.
Graphing instructions:
- In order to graph this equation, simply plot the coordinates of the y-intercept, (0, -2).
- Then, use the slope, m = -4 (down 4 units, run 1 unit to the right; up 4 units from the y-intercept, then 1 unit to the left) to plot other points into the graph.
- Repeat this process until you have enough points to create a line with.
Important Note on Boundary Line:
Make sure to use a solid boundary line in your graph due to the "≥" symbol. This implies that the endpoints are included as part of the solution.
Shading a Half-plane Region:
In terms of determining which half-plane region to shade, we must first choose a test point to plug into the equation. The purpose of using a test point is to find out whether this test point would satisfy the given inequality statement. If it satisfies the inequality statement, then we must shade the half-plane region that contains the chosen test point.
The point of origin, (0, 0), is often used as a test point for this purpose. Let us substitute these values into the given inequality statement:
y ≥ -4x - 2
0 ≥ -4(0) - 2
0 ≥ 0 - 2
0 ≥ - 2 (True statement). Therefore, we must shade the half-plane region that contains the test point.
Attached is a screenshot of the graphed linear inequality, y ≥ -4x - 2, using the techniques discussed in this post.