Linear Inequalities in 2 Variables
From Points to Regions
Essential Understanding: Unlike equations that give us one or a few specific answers, inequalities give us infinite solutions that form a region on the graph. Learning to represent these regions is a core CSEC skill!
1. Introduction: The Concept of a "Solution Region"
When you solve an equation like \(2x + 3 = 11\), you find that \(x = 4\) is the only solution—a single point. But when you work with inequalities, you're looking for all the values that make the statement true.
🎯 The Big Picture
Think of a boundary line as a fence. The inequality tells you which side of the fence contains valid solutions. Every point in that shaded region is a solution to your inequality!
< and > (Strict)
Meaning: "Less than" or "Greater than" (but NOT equal)
Boundary: Dashed line (points on the line are NOT included)
Example: \(y > x\) means all points above the line
≤ and ≥ (Inclusive)
Meaning: "Less than or equal" / "Greater than or equal"
Boundary: Solid line (points on the line ARE included)
Example: \(y ≤ x\) means all points on and below the line
The Goal
By the end of this article, you'll be able to take any linear inequality and:
1. Draw the Boundary
Determine if line is solid or dashed
2. Shade Correctly
Use the test point to find the right side
3. Verify Points
Check that test points satisfy the inequality
Module A: Linear Inequalities in One Variable (The 1D View)
Before graphing on a 2D plane, let's understand inequalities using the number line—a one-dimensional view that builds intuition.
The Circle Rule
The type of circle tells us whether the boundary point is included in the solution:
Open Circle (∘)
Used for: < and >
Meaning: The number is NOT included
Example: \(x > 2\) — x can be 3, 4, 100... but NOT 2
Closed/Solid Circle (∙)
Used for: ≤ and ≥
Meaning: The number IS included
Example: \(x ≤ 2\) — x can be 2, 1, 0, -5...
Visual Guide: Number Line Examples
x > 2
Circle: Open (not included)
Arrow: Points right (greater values)
Solutions: 3, 4, 5, 100, ...
x ≤ 2
Circle: Closed (included)
Arrow: Points left (smaller values)
Solutions: ..., 0, 1, 2
The Arrow Direction Rule
| Inequality | Circle Type | Arrow Points | Reason |
|---|---|---|---|
| x > 3 | Open (○) | Right → | Greater than means larger values |
| x < 3 | Open (○) | ← Left | Less than means smaller values |
| x ≥ 3 | Closed (●) | Right → | At least 3 means 3 or more |
| x ≤ 3 | Closed (●) | ← Left | At most 3 means 3 or less |
Module B: Transitioning to the Cartesian Plane (The 2D View)
When we add a second variable, our boundary becomes a line instead of a point, and our solution becomes a region instead of a ray.
Vertical and Horizontal Boundaries
Key Insight
In one variable: \(x > 2\) means "all points to the right of 2"
In two variables: The same logic applies! The boundary is just a line instead of a point.
Vertical Lines: x > k or x < k
Example: \(x > 2\)
Boundary: Vertical line at x = 2
Shading: All points to the right of the line
Line Style: Dashed (strict inequality)
Horizontal Lines: y > k or y < k
Example: \(y ≤ 4\)
Boundary: Horizontal line at y = 4
Shading: All points below the line
Line Style: Solid (inclusive inequality)
Boundary Types: The Line Style Rule
| Inequality Sign | Line Style | Why? |
|---|---|---|
| < or > | ⚡ Dashed | Strict inequality — points on the line are NOT solutions |
| ≤ or ≥ | ━ Solid | Inclusive inequality — points on the line ARE solutions |
Module C: Linear Inequalities in Two Variables (Objective 2.1)
The 3-Step Method
For inequalities in the form \(ax + by ≤ c\) (or with any inequality sign), follow these three steps:
Worked Example: The (0,0) Test
Example: Graph \(x + 2y ≤ 6\)
Step 1: Boundary Line
Equation: \(x + 2y = 6\)
x-intercept: (6, 0) | y-intercept: (0, 3)
Step 2: Line Style
Sign is ≤, so use a SOLID line.
Step 3: Test Point (0,0)
Substitute: \(0 + 2(0) ≤ 6\) → \(0 ≤ 6\) → TRUE!
Since true, shade the side containing (0,0) — the bottom-left side.
Pro Tip: Why (0,0) Works Best
The origin (0,0) is almost always the best test point because:
- It's easy to remember and calculate with
- Substituting zero eliminates terms, making math simpler
- It's guaranteed to be clearly on one side or the other (rarely on the line)
Exception: If the line passes through (0,0), choose another point like (1,0) or (0,1).
5. Interactive "Shade and Check" Lab
Explore Linear Inequalities Graphically
Objective: Adjust the inequality parameters and click points to test if they satisfy the condition. Green = valid solution, Red = invalid solution.
👆 Click anywhere on the graph to test if that point satisfies the inequality!
6. Combined Inequalities (Systems)
CSEC often asks you to work with multiple inequalities at once. The solution is the region where ALL inequalities overlap.
The Feasible Region
When several inequalities are combined, they define a region called the feasible region—the set of all points that satisfy EVERY inequality simultaneously.
Real-World Example: Shop Constraints
Problem: A shop sells x cakes and y biscuits.
Constraints:
- \(x ≥ 0\) (Can't sell negative cakes)
- \(y ≥ 0\) (Can't sell negative biscuits)
- \(x + y ≤ 10\) (Maximum 10 items total)
- \(2x + y ≤ 12\) (Ingredient constraint)
The feasible region is the area where all four inequalities overlap!
The Vertex Finder
The vertices (corners) of the feasible region are critically important for later topics like Linear Programming. These points are found where two boundary lines intersect.
Finding Vertices
To find a vertex, solve the system of equations formed by two boundary lines:
Vertex at (2, 8)
7. Worked CSEC Exam Example
📝 CSEC Past Paper Style Question
Question: Represent the region defined by the inequalities \(x + y ≤ 5\), \(x ≥ 1\), and \(y ≥ 0\) on a graph.
Step-by-Step Solution:
x-intercept: (5, 0) | y-intercept: (0, 5)
Since sign is ≤, use a SOLID line
Test (0,0): 0 + 0 ≤ 5 → TRUE → Shade below the line
This is a vertical line at x = 1
Since sign is ≥, use a SOLID line
Test (0,0): 0 ≥ 1 → FALSE → Shade to the RIGHT of the line
This is the x-axis (horizontal line at y = 0)
Since sign is ≥, use a SOLID line
Test (0,1): 1 ≥ 0 → TRUE → Shade ABOVE the x-axis
The solution is where ALL three shaded regions overlap!
Vertices: (1, 0), (5, 0), and (1, 4)
Note: At (1, 4): 1 + 4 = 5 ≤ 5 ✓, x = 1 ≥ 1 ✓, y = 4 ≥ 0 ✓
8. Common Pitfalls Checklist
Before submitting your graph, run through this checklist to avoid losing marks:
9. Quick Quiz: "Solid, Dashed, or Number Line?"
Test Your Understanding
📝 Chapter Summary
Line Styles
- Dashed: < or > (strict, not included)
- Solid: ≤ or ≥ (inclusive, included)
Shading Method
- Use the (0,0) test point
- If TRUE, shade origin's side
- If FALSE, shade opposite side
Key Vocabulary
- Boundary Line: The edge of the region
- Feasible Region: Overlap of all inequalities
- Vertices: Corner points of feasible region