Lesson 7.14
Shading Half-Planes
A line splits the infinite coordinate plane into two halves. Your job is to decide which half contains the "Truth".
Introduction
In Lesson 7.13, we learned to graph the line. Now we focus purely on the Shading Decision. The most reliable tool for this is the "Test Point".
Past Knowledge
Lesson 7.13 (Graphing Inequalities). You must know how to draw the boundary line (Solid vs. Dashed).
Today's Goal
Master the Test Point Method to determine shading direction with 100% accuracy.
Future Success
This logic is critical for Linear Programming, where you find the "Feasible Region" for business profits.
Key Concepts
The Test Point Method
When Y is not isolated (Standard Form), "Up/Down" logic fails. Use a spy!
Step 1: Pick a Point
Choose unless the line goes through it.
If the line hits , use .
Step 2: Plug it In
Worked Examples
Example 1: The Easy Test
BasicShade:
The Test
Pick .
Plug in:
Simplify:
is TRUE.
Conclusion: Shade the side that contains (0,0).
Example 2: The False Test
IntermediateShade:
The Test
Pick .
Plug in:
Simplify:
is FALSE.
Conclusion: (0,0) is a liar! Shade the OTHER side.
Example 3: Special Vertical Line
AdvancedShade:
The Logic
This is a vertical line at .
"X is less than 3."
Where are numbers smaller than 3? To the LEFT.
We don't really need a test point here, but if we used (0,0): (True). Shade left.
Common Pitfalls
Testing ON the Line
Never pick a test point that sits exactly on the line. The result will be equal (), which doesn't tell you which SIDE to shade.
Forgetting to Flip
If you convert to Slope-Intercept form by dividing by a negative, remember to FLIP the inequality sign! Or better yet... just use the Test Point Method on the original equation.
Real-Life Applications
Geofencing:
- GPS apps use inequalities to determine if you are "inside" a delivery zone.
- "If Latitude < X and Longitude > Y, then you are in the service area."
Practice Quiz
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