IGCSEMaths: ExtendedCIERevision NotesCoordinate Geometry & GraphsSolving & Graphing InequalitiesGraphical Inequalities

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First exams 2025

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Graphical Inequalities (CIE IGCSE Maths: Extended)

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Jamie W

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Jamie W

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Finding Regions using Inequalities

How do we draw inequalities on a graph?

  • First, see Linear Graphs 

To graph an inequality;

  1. DRAW the line (as if using “=”) for each inequality

    • Use a solid line for ≤ or ≥ (to indicate the line is included)

    • Use dotted line for < or > (to indicate the line is not included)

  2. DECIDE which side of line is wanted.
    • Below line if "y ≤ ..." or "y < ..."

    • Above line if "y ≥ ..." or "y > ..."

    • Use a point that's not on the line as a test if unsure; substitute its x and y value into the inequality to examine whether the inequality holds true on that side of the line

  3. Shade UNWANTED side of each line (unless the question says otherwise)
    • This is because it is easier, with pen/ pencil/ paper at least, to see which region has not been shaded than it is to look for a region that has been shaded 2-3 times or more

    • (Graphing software often shades the area that is required but this is easily overcome by reversing the inequality sign)

Worked example

On the axes given below, show the region that satisfies the three inequalities;

3 x plus 2 y greater or equal than 12      y less than 2 x      x less than 3

Label the region R.


First draw the three straight lines, 3 x plus 2 y equals 12, y equals 2 x and x equals 3, using your knowledge of Straight Line Graphs (y = mx + c). You may wish to rearrange 3 x plus 2 y equals 12 to the form y equals m x plus c first:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 y end cell equals cell negative 3 x plus 12 end cell row y equals cell negative 3 over 2 x plus 6 end cell end table

The line 3 x plus 2 y greater or equal than 12 takes a solid line because of the "≥" while the lines y less than 2 x and x less than 3 take dotted lines because of the "<"

0j84Yb-3_2-19-1-1-graphing-inequalities

Now we need to shade the unwanted regions

For 3 x plus 2 y greater or equal than 12 (or y greater or equal than negative 3 over 2 x plus 6), the unwanted region is below the line. We can check this with the point (0, 0);

table row cell " 3 open parentheses 0 close parentheses plus 2 open parentheses 0 close parentheses end cell greater or equal than cell 12 " end cell end table is false therefore (0, 0) does lie in the unwanted region for table row cell 3 x plus 2 y end cell greater or equal than 12 end table

For y less than 2 x, the unwanted region is above the line. If unsure, check with another point, for example (1, 0)

table row cell " 0 end cell less than cell 2 open parentheses 1 close parentheses " end cell end table is true, so (1, 0) lies in the wanted (i.e. unshaded) region for table row y less than cell 2 x end cell end table

For x less than 3, shade the unwanted region to the right of x equals 3. If unsure, check with a point

V90dxvma_2-19-1-2-graphing-inequalities

Finally, don't forget to label the region R

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Interpreting Graphical Inequalities

How do we interpret inequalities on a graph?

  • First, see Linear Graphs

To interpret inequalities/ to find a region defined by inequalities;

  1. Write down the EQUATION of each line on the graph
  2. REMEMBER that lines are drawn with:
    • A solid line for ≤ or ≥ (to indicate line included in region)

    • A dotted line for < or > (to indicate line not included)

  3. REPLACE = sign with:

    • ≤ or < if shading below line

    • ≥ or > if shading above line

    • (Use a point to test if not sure)

Worked example

Write down the three inequalities which define the shaded region on the axes below.

Kk~yY~ua_2-19-2-1-graphing-inequalities

First, using your knowledge of Straight Line Graphs (y = mx + c), define the three lines as equations, ignoring inequality signs;

2-19-2-2-graphing-inequalities

Now decide which inequality signs to use

For y equals x, the shaded region is above the line, and the line is dotted, so the inequality is

bold italic y bold greater than bold italic x

Check by substituting a point within the shaded region into this inequality. For example, using (2, 4) as marked on the graph above;

"4 greater than 2" is true, so the inequality y greater than x is correct

For y equals negative x plus 7, the shaded region is below the line, and the line is solid, so the inequality is 

bold italic y bold less or equal than bold minus bold italic x bold plus bold 7
or bold italic y bold plus bold italic x bold less or equal than bold 7

Again, check by substituting (2, 4) into the inequality;

"4 less or equal than negative 2 plus 7" is true, so the inequality y less or equal than negative x plus 7 is correct

For x equals 1, the shaded region is to the right of the solid line so the inequality is

bold italic x bold greater or equal than bold 1

(Vertical and horizontal inequality lines can also be checked by substituting in a point if you are unsure)

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    Jamie W

    Author: Jamie W

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.