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Graphical Inequalities for IGCSE Maths

Representing and identifying regions satisfying multiple inequalities. This subtopic is part of Inequalities in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding graphi

What You Need to Know

Representing and identifying regions satisfying multiple inequalities. This subtopic is part of Inequalities in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding graphical inequalities is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Graphical Inequalities

Graphical inequalities involve identifying and shading regions on a coordinate grid that satisfy one or more linear inequalities. The boundary line is drawn solid for ≤ or ≥ and dashed for < or >. The unwanted region is typically shaded (leaving the required region clear) — Cambridge IGCSE conventions ask for the 'wanted' region to be labelled R. Integer coordinate points inside the region can be listed as solutions.

Step-by-Step Method

  1. 1

    Draw the boundary line

    Replace the inequality sign with = and draw the line. Use a dashed line for < or > (not including the line), solid for ≤ or ≥.

  2. 2

    Test a point not on the line

    Substitute (0, 0) into the inequality (unless the line passes through the origin, in which case use (0, 1)). If the inequality holds, (0, 0) is in the required region.

  3. 3

    Shade the correct side

    IGCSE convention: shade the region you do NOT want. Leave the required region (R) unshaded.

  4. 4

    For multiple inequalities, shade progressively

    Add one inequality at a time. The required region is the unshaded area left when all unwanted regions have been shaded.

  5. 5

    List integer solutions if asked

    Identify all lattice points (whole number coordinates) within the unshaded region R.

Worked Example

Question

Shade the region satisfied by all three inequalities: x ≥ 1, y ≤ 4, x + y ≤ 6. Label the required region R.

Solution

Step 1: Draw x = 1 (vertical, solid line — ≥ included). Draw y = 4 (horizontal, solid — ≤ included). Draw x + y = 6, i.e. y = 6 − x (solid — ≤ included). Step 2: Test (0, 0): x=0 < 1 so this point fails x≥1. Test (2, 2): x=2≥1✓, y=2≤4✓, 2+2=4≤6✓. This point is in R. Step 3: Shade the regions that fail each inequality: - Shade left of x=1 (fails x≥1). - Shade above y=4 (fails y≤4). - Shade above x+y=6 line (fails x+y≤6). Step 4: Label the remaining triangular region R. Integer points in R: (1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(4,1),(4,2),(5,1).

Exam Tips for Graphical Inequalities

  • IGCSE convention: shade what you do NOT want. The required region R is left clear.
  • Dashed line for strict inequalities (< >), solid line for ≤ or ≥ — wrong line type loses marks.
  • Always test a point (usually the origin) to confirm which side of each line to shade.
  • When writing inequalities from a graph, check the slope and intercept of each boundary line carefully.

Practice Questions

Q1: Draw the region R that satisfies y > x, y < 4, and x > −2. List three integer coordinate points in R.

Show hint

y>x: dashed line y=x, shade below it (unwanted region). y<4: shade above y=4. x>−2: shade left of x=−2.

Q2: The region R is bounded by x ≥ 0, y ≥ 0, and 2x + y ≤ 8. What are the integer coordinates in R?

Show hint

Triangle with vertices (0,0),(0,8),(4,0). Integer points inside include (0,0),(1,0),(1,1),...,(4,0),(0,8).

Q3: Write down the three inequalities that define the region shown by: x ≤ 3, y ≥ −1, y ≤ x + 2.

Show hint

Read each boundary line from the graph. Check whether the shaded or unshaded side satisfies each.

Frequently Asked Questions

What is graphical inequalities in IGCSE Maths?

Representing and identifying regions satisfying multiple inequalities.

Is graphical inequalities in the Core or Extended syllabus?

Graphical Inequalities is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise graphical inequalities effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on graphical inequalities rather than trying to cover everything at once.

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