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O Level · Mathematics 0580

Solving & Graphing Inequalities

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Introduction to 2D Inequalities

What are 2D Inequalities?

A 2D inequality involves two variables and represents a region in the xy-plane, rather than a single line or point.

Examples of 2D Inequalities:

y < x
x + y ≥ 8
3x + 2y ≥ 12
y ≤ 2x + 5

1D vs 2D Inequalities

While 1D inequalities (like x > 3) represent a line or region on a number line, both 1D and 2D inequalities can be represented as regions in the xy-plane:

1D Inequality

Example: y ≥ 2

This represents all points on or above the horizontal line y = 2

2D Inequality

Example: y < x

This represents all points below the line y = x

Key Concepts

Region: A 2D inequality defines a region (area) in the xy-plane
Boundary Line: The line where the inequality becomes an equality
Included or Excluded: The boundary may or may not be part of the solution
Shading: Regions are shown using shading on a graph

How to Draw Inequalities on a Graph

The Four Steps

Replace the inequality sign with =
Draw the straight line as if it were an equation. For example, if you have y < 2x + 3, draw the line y = 2x + 3
Decide on line type: Solid or Dotted

Solid Line

Use solid lines for ≤ or ≥

This means the line IS included in the region

Dotted Line

Use dotted lines for < or >

This means the line is NOT included in the region
Identify which side of the line to shade
For y ≤ or y <: the wanted region is below the line

For y ≥ or y >: the wanted region is above the line

For x <: the wanted region is to the left of the line

For x >: the wanted region is to the right of the line
Test a point if unsure
Substitute the coordinates of a point into the inequality. If it's true, that point is in the wanted region.

Testing a Point

Example: For the inequality y < 2x + 3

Test the point (0, 0):

0 < 2(0) + 3 → 0 < 3 ✓ TRUE

Since this is true, the point (0, 0) is in the wanted region.

Visual Guide: Line Types

Shading Regions - Best Practice

The Smart Shading Method

Step-by-Step Process

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Also in the full note

Finding Inequalities from Regions
Worked Examples
Important Reminders
Interactive Inequality Tester
The Reverse Process
Step-by-Step Method
Determining Inequality Direction
Interactive Tool: Find the Inequalities

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