Skip to content
VirtusAcademy

Inequalities on Graphs / Regions

HigherHigher tier onlyAQAEdexcelOCR

Master inequalities on graphs / regions for GCSE Maths with structured, exam-style practice. This Higher resource covers representing inequalities as regions on a graph and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Use a dashed line for < or > and a solid line for ≤ or ≥.

Free downloads

These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An inequality can be shown on a graph as a region rather than a line. The line itself marks the boundary, and the region on one side of it contains all the points satisfying the inequality.

The type of line matters. A solid line means the boundary is included, used for \(\le\) and \(\ge\). A dashed line means it is excluded, used for the strict inequalities \(<\) and \(>\).

Deciding which side to shade is best done by testing a point. Substitute a convenient point such as the origin into the inequality: if it works, that point lies in the required region, and if not, the other side is correct.

Revision notes

Drawing the boundary

Replace the inequality sign with an equals sign and draw that line.

For \(y \le 2x + 1\), draw the line \(y = 2x + 1\), using a solid line because the inequality includes equality.

Solid or dashed

Use a solid line for \(\le\) or \(\ge\), and a dashed line for \(<\) or \(>\).

This is a mark in itself, and it communicates whether points on the boundary count as solutions.

Choosing the region

Test a point not on the line, usually the origin.

For \(y \le 2x + 1\) at \((0,0)\): \(0 \le 1\) is true, so shade the side containing the origin. Label the region clearly, and check whether the question wants the required region shaded or left unshaded.

Key points

  • The boundary line comes from replacing the sign with equals.
  • A solid line includes the boundary.
  • A dashed line excludes it.
  • Test a point to decide which side to shade.
  • The origin is usually the easiest test point.
  • Label the required region clearly.

Worked examples

Example 1

State the type of line for \(y > 3x - 2\).

Working

\[\text{The inequality is strict}\]the sign is > rather than ≥
\[\text{A dashed line}\]the boundary is not included

Example 2

For \(y \le x + 4\), decide which side to shade using the origin.

Working

\[0 \le 0 + 4\]substitute the origin into the inequality
\[0 \le 4 \text{ is true}\]the statement holds
\[\text{Shade the side containing the origin}\]state the conclusion

Example 3

For \(y > 2x\), test the point \((1, 1)\).

Working

\[1 > 2(1)\]substitute the point
\[1 > 2 \text{ is false}\]the statement does not hold
\[\text{The point is not in the region}\]shade the other side

Common mistakes

  • Using a solid line for a strict inequality.

    < and > need a dashed line, since the boundary is excluded.

  • Shading without testing a point.

    Guessing the side is unreliable. Substitute the origin and check.

  • Testing a point that lies on the line.

    Choose a point clearly off the boundary, such as the origin when the line does not pass through it.

  • Not labelling the region.

    The examiner needs to see which region you mean.

Exam tips

  • Draw the boundary line first, deciding solid or dashed immediately.
  • Always test a point rather than guessing the side.
  • Use the origin unless the line passes through it.
  • Label the region R or as the question directs.

Key terms

Boundary
The line marking the edge of a region.
Strict inequality
One using < or >, excluding the boundary.
Region
The area of the graph satisfying the inequality.
Test point
A point substituted to decide which side to shade.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.