Sketching 2D Inequalities

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Cambridge (CIE) IGCSE International Maths — Extended

Sketching 2D Inequalities

Big idea: An inequality with two variables (like x and y) doesn't describe a single line — it describes an entire region of a graph, and you find that region by drawing the boundary line, then shading away everything you don't want until only the answer is left uncovered.

Quick Overvieweverything in this chapter, at a glance

A 2D inequality (e.g. y < x or x + y ≥ 8) involves two variables, and its solution is a whole region of the xy-plane, not just a line.
To draw a region: turn the inequality into an equation, draw that as a straight line, then decide which side is "wanted."
The line is solid for ≤ or ≥ (the line itself counts), and dotted for < or > (the line itself doesn't count).
Convention: we usually shade the unwanted side of each line, so the clean, unshaded patch left over is your answer region — normally labelled R.
Vertical lines (x = k) and horizontal lines (y = k) follow the same left/right and above/below logic.
You can also work backwards: given a shaded diagram, read off the equations of the lines and figure out which inequality signs produced that shading.
If you're ever unsure which side is which, test a point — plug its coordinates into the inequality and see if it holds true.

1. What Actually Is a 2D Inequality?building the mental picture first

From lines to regions

You already know that an equation like y = 2x + 3 draws a single, precise line on a graph — every point on that line satisfies the equation exactly, and no other point does.

Now swap the equals sign for an inequality sign: y > 2x + 3. Suddenly you're no longer asking "which points sit exactly on this line?" You're asking "which points have a y-value bigger than the line gives you at that x?" That's not one line anymore — it's an entire half of the plane, infinite in size, sitting above the line.

That's the whole shift in thinking this chapter asks of you: an equation is a boundary; an inequality is everything on one side of that boundary. The line itself just tells you where the boundary sits — the inequality sign tells you which side you actually want.

Even a 1-variable inequality becomes a region

Something like y ≥ 2 only mentions one variable, but on an xy-grid it still represents a whole region: every point on or above the horizontal line y = 2, stretching left and right forever. There's no restriction on x at all, so it fills the entire plane above that line.

Practice Question

Q1. In your own words, explain why the inequality x < 4 represents a region rather than a line, and describe exactly which region it is.

2. Drawing the Boundary Linesolid vs dotted — and why it matters

Step 1: turn the inequality into an equation

Before you can shade anything, you need the boundary itself. Take whatever inequality you're given and simply replace the inequality sign with an equals sign. So 3x + 2y ≥ 12 becomes 3x + 2y = 12. Now you draw that as an ordinary straight line — rearranging into y = mx + c form first if it helps you plot it.

For example, rearranging 3x + 2y = 12:

Rule

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Step 2: solid or dotted?
The quick rule for sloped and horizontal lines
Vertical lines are a special case
When in doubt: the point-test method (your safety net)
Shade what you don't want
Given a shaded region, find the inequalities
Fully worked example

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Sketching 2D Inequalities