3.8 Regions
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Learning Objectives
- Understand how to represent linear and quadratic inequalities as regions on a graph
- Interpret and sketch the feasible region defined by a set of inequalities
- Identify solution regions for systems of inequalities
- Use test points to determine which region satisfies a given inequality
- Recognise when a region is bounded or unbounded
Key Concepts
What Are Regions?
In coordinate geometry, a **region** is the part of the graph that satisfies a given inequality or a set of inequalities.
When solving multiple inequalities, the **feasible region** is the area where all individual regions overlap.
Shading is used to represent the part of the graph where the inequality holds true.
Sketching Regions from Linear Inequalities
Plot each inequality one at a time using the method for linear inequalities.
Use a **dashed line** for strict inequalities (< or >) and a **solid line** for inclusive ones (≤ or ≥).
Shade the appropriate side of each line.
The region that satisfies all the inequalities is the **intersection** (overlap) of all shaded areas.
Example 1
Sketch the region defined by: y ≥ x + 2, y ≤ 4, x ≥ 1
Solution:
1. Plot the line y = x + 2 with a solid line and shade above it. 2. Plot the horizontal line y = 4 (solid) and shade below it. 3. Plot the vertical line x = 1 (solid) and shade to the right. The region is the overlap of all three shaded areas.
Using Test Points
If unsure which side of a line to shade, pick a **test point** (like (0,0)) and check if it satisfies the inequality.
If the test point makes the inequality true, shade the side **containing** that point.
If false, shade the **opposite** side.
Example 1
Determine the region for y < 3x - 1 using (0, 0) as a test point.
Solution:
Substitute into the inequality: 0 < 3(0) - 1 → 0 < -1 → False. So, shade the region **not containing (0,0)**.
Bounded and Unbounded Regions
A **bounded region** is enclosed by lines or curves — it has a finite area.
An **unbounded region** extends infinitely in at least one direction.
Bounded regions are especially useful in linear programming for optimization problems.
Interpreting Regions in Context
In applied problems, constraints such as 'x ≥ 0' and 'y ≥ 0' may represent real-world limits (e.g., no negative quantity).
Graphical methods can be used to solve real-life inequalities involving cost, profit, or capacity.
Test Your Knowledge
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