🏠 Home › 📚 CSEC Mathematics › 📐 Linear Inequalities

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!

                \[ \text{Equations give POINTS} \quad \rightarrow \quad \text{Inequalities give REGIONS} \]
            

< less than

> greater than

≤ less than or equal

≥ greater than or equal

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)

CSEC Objective 2.2

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)

From 1D to 2D

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)

Core CSEC Skill

The 3-Step Method

For inequalities in the form \(ax + by ≤ c\) (or with any inequality sign), follow these three steps:

Draw the Boundary Line: Treat the inequality as an equation. For \(2x + 3y ≤ 6\), first graph \(2x + 3y = 6\). Find x-intercept (let y=0): \(x=3\). Find y-intercept (let x=0): \(y=2\). Connect these points.

Choose the Line Style: Check the inequality sign. Use SOLID line for ≤ or ≥. Use DASHED line for < or >.

The Test Point Method: Pick any point NOT on the line (the origin (0,0) is easiest). Substitute into the inequality. If TRUE, shade THAT side. If FALSE, shade the OPPOSITE side.

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.

Inequality Type

y < 2x + 1

Slope (m): 2

Y-Intercept (b): 1

👆 Click anywhere on the graph to test if that point satisfies the inequality!

6. Combined Inequalities (Systems)

Advanced Topic

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:

                \[ \begin{cases} 2x + y = 12 \\ x + y = 10 \end{cases} \quad \Rightarrow \quad \text{Subtract: } x = 2 \quad \Rightarrow \quad y = 8 \]
            

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:

Draw the first boundary: \(x + y = 5\)

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

Draw the second boundary: \(x = 1\)

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

Draw the third boundary: \(y = 0\)

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

Identify the Feasible Region

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:

Did I use the correct line style? (Dashed for <, >; Solid for ≤, ≥)

Did I use a test point (preferably the origin) to determine which side to shade?

Did I correctly calculate intercepts? (x-intercept: y=0, y-intercept: x=0)

Did I label the boundary line with its equation?

For combined inequalities, did I identify all vertices of the feasible region?

Did I clearly show which region is the solution (cross-hatching or shading)?

⚠️ Top 3 Mistakes Students Make

❌ Using a solid line for \(y < 2\)

✅ Fix: "Less than" means the line itself is NOT included. Use a DASHED line!

❌ Shading the wrong side

✅ Fix: Always use the (0,0) test point. If 0 ≤ statement is TRUE, shade the side with the origin!

❌ Mixing up x and y intercepts

✅ Fix: Remember: x-intercept has y=0; y-intercept has x=0. Check your answers!

9. Quick Quiz: "Solid, Dashed, or Number Line?"

Test Your Understanding

Which line style should be used for the inequality \(y > 3x + 2\)?

Solid line

Dashed line

Number line

No line needed

Solution: Dashed line! The ">" symbol means "greater than" but NOT equal to. Points on the line are NOT solutions, so we use a dashed boundary.

For \(2x + y ≤ 8\), which side should be shaded if (0,0) is tested?

The side containing (0,0)

The side NOT containing (0,0)

Above the line only

Below the line only

Solution: The side containing (0,0)! Test: 2(0) + 0 ≤ 8 → 0 ≤ 8 (TRUE). Since the statement is true, shade the side with the origin.

What represents \(x ≥ -2\) on a number line?

Open circle at -2, arrow left

Open circle at -2, arrow right

Closed circle at -2, arrow right

Closed circle at -2, arrow left

Solution: Closed circle at -2 with arrow to the right! Closed circle because ≥ includes the value, arrow points right because we want values greater than -2.

A feasible region is defined by \(x ≥ 0\), \(y ≥ 0\), and \(x + y ≤ 6\). What is a vertex of this region?

(6, 6)

(0, 6)

(6, 0)

(8, 0)

Solution: (6, 0)! Check: x = 6 ≥ 0 ✓, y = 0 ≥ 0 ✓, x + y = 6 ≤ 6 ✓. The vertices are (0,0), (6,0), and (0,6).

What is the correct boundary line style for \(3x - 2y ≥ 4\)?