Inequality Graphing Guide | IGCSE Maths Blog

Why Inequality Graphing Matters

Inequality graphing is a topic that bridges algebra and geometry. In IGCSE Maths, you will be asked to represent inequalities on a number line, shade regions on a coordinate grid, and identify inequalities from given graphs. These questions appear on both the Core and Extended papers and are typically worth 3 to 6 marks.

The good news is that the process is systematic. Once you learn the steps, you can apply them to any inequality graphing question the examiner throws at you.

Number Line Inequalities

The simplest form of inequality representation is on a number line. You need to understand the notation:

Filled circle (solid dot) means the value is included (≤ or ≥)
Open circle (hollow dot) means the value is not included (< or >)

For x > 3: draw an open circle at 3 and shade everything to the right. For x ≤ −1: draw a filled circle at −1 and shade everything to the left. For −2 < x ≤ 5: draw an open circle at −2, a filled circle at 5, and shade everything between them.

Always read the inequality carefully. The direction of the inequality sign tells you which way to shade.

Graphing Linear Inequalities in Two Variables

This is where it gets more interesting. A linear inequality like y < 2x + 1 represents not a line but a region of the coordinate plane.

The process has three steps:

Step 1: Draw the boundary line. Replace the inequality sign with an equals sign and draw the line. For y < 2x + 1, draw the line y = 2x + 1.

Use a solid line for ≤ or ≥ (the boundary is included)
Use a dashed line for < or > (the boundary is not included)

Step 2: Choose a test point. Pick any point not on the line. The origin (0, 0) is usually the easiest choice, unless the line passes through it. Substitute the test point into the inequality.

For y < 2x + 1, test (0, 0): is 0 < 2(0) + 1? Is 0 < 1? Yes. So (0, 0) is in the solution region.

Step 3: Shade the correct region. If the test point satisfies the inequality, shade the side of the line that contains the test point. If it does not, shade the other side.

Important: some exam questions ask you to shade the region that satisfies the inequality, while others ask you to shade the region that does NOT satisfy it (leaving the solution region clear). Read the instructions carefully.

Multiple Inequalities

Many IGCSE questions give you three or four inequalities and ask you to identify the region that satisfies all of them simultaneously. This is called the feasible region.

For example, find the region satisfying:

y ≥ 1
x ≤ 4
y ≤ x + 2

Step 1: Draw all three boundary lines on the same grid. Step 2: For each inequality, shade the unwanted region (the region that does NOT satisfy it). Step 3: The unshaded region that remains is the feasible region satisfying all three inequalities.

Label this region clearly with the letter R (or whatever the question specifies).

Finding Integer Points in a Region

A common follow-up question asks: “List all the integer coordinate pairs (x, y) that satisfy all the inequalities.”

To answer this:

Identify the feasible region on your graph
Look for all points where both x and y are integers
List them systematically — go column by column (x = 0, then x = 1, etc.)
Check each point satisfies every inequality

This is a pure accuracy exercise. Being methodical prevents you from missing points or including wrong ones.

Writing Inequalities from Graphs

Sometimes the question works in reverse: you are given a shaded region and must write the inequalities that define it.

For each boundary line:

Find the equation of the line (using y = mx + c or other methods)
Determine whether the line is solid (≤ or ≥) or dashed (< or >)
Test a point in the shaded region to determine the direction of the inequality

If the shaded region is above the line, the inequality is y ≥ mx + c (or y > mx + c for a dashed line). If below, it is y ≤ mx + c (or y < mx + c).

For vertical lines like x = 3, the inequality is x ≤ 3 or x ≥ 3 depending on which side is shaded.

Quadratic Inequalities

On the Extended paper, you may encounter quadratic inequalities such as x² − 5x + 6 < 0.

To solve:

Factorise: (x − 2)(x − 3) < 0
Find the critical values: x = 2 and x = 3
Determine where the expression is negative: between the roots, so 2 < x < 3

The key insight is that a quadratic expression changes sign at its roots. Sketch the parabola to visualise which regions are positive and which are negative.

For x² − 5x + 6 > 0, the solution would be x < 2 or x > 3 — the regions outside the roots.

Practical Tips for the Exam

Use a ruler for all straight lines — freehand lines lose marks
Label your lines with their equations so the examiner can follow your work
Use different shading patterns for different inequalities (horizontal lines, vertical lines, diagonal lines) so you can see where they overlap
Check your region by substituting a point from the unshaded area into all inequalities
Draw accurately — plot at least three points for each line to ensure accuracy
Use a sharp pencil for the dashed lines so the examiner can clearly see they are dashed

Common Mistakes

Using a solid line when it should be dashed, or vice versa
Shading the wrong side of a line
Not reading whether the question asks you to shade the wanted or unwanted region
Forgetting that the region must satisfy ALL inequalities simultaneously, not just one
Making errors when finding the equation of a boundary line
Not listing all integer points when asked

Quick Reference Table

Symbol	Line Type	Boundary Included?
<	Dashed	No
>	Dashed	No
≤	Solid	Yes
≥	Solid	Yes

Keep this distinction clear in your mind, and inequality graphing becomes much more straightforward.

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Inequality Graphing Guide for IGCSE Maths