Representing Inequalities as Regions (Cambridge (CIE) O Level Maths) : Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

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Finding Regions using Inequalities

What are 2D inequalities?

  • Recall that an inequality in one variable (1D inequality) represents a relationship that is not equal

    • An inequality of x < 7, represents all values smaller than 7

      • There are an infinite number of values than can satisfy this inequality

  • A 2D inequality represents a relationship between two expressions that is not equal

    • The inequality y > x represents all pairs of numbers x and y where the y value is greater than the x value

      • There are an infinite number of pairs of values that would satisfy this inequality

      • These pairs of numbers can be thought of as coordinates

      • On a graph, all coordinates above the line y = x would satisfy this inequality

  • If a 2D inequality includes either the symbol ≤ or ≥, then coordinates on the line itself also satisfies the inequality

    • E.g. y ≤ 2x represents all of the pairs of numbers where the value of y is less than two lots of the value of x

      • This is the region below the line y = 2x, but also being on the line y = 2x satisfies the inequality

How do we draw inequalities on a graph?

  • A set of 2D inequalities can be shown graphically using straight lines and shaded regions

  • To draw the correct lines:

    • Replace the inequality sign with “=” and draw that line

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

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

  • To decide which side of the line is the wanted side:

    • if "y ≤ ..." or "y < ..." then the wanted region is below the line

    • if "y ≥ ..." or "y > ..." then the wanted region is above the line

    • If you are unsure

      • substitute the coordinates from a point on one side of the line into the inequality

      • determine whether or not the inequality holds true on that side

    • For vertical lines:

      • the wanted region for x less than k is to the left of x equals k

      • the wanted region for x greater than k is to the right of x equals k

  • To do the shading:

    • Shade the unwanted sides of each line (unless the question says otherwise)

      • You are shading away any parts you don't want

      • This will leave behind a clear region that is the wanted region (rather than trying to look for the wanted region under multiple shades)

      • Label the wanted region R (unless the question says otherwise)

  • (Be careful if using graphing software, as some shade the wanted sides)

Worked Example

Show, graphically, the region that is satisfied by all three inequalities below:

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

Label this region R.

First draw the three straight lines: 3 x plus 2 y equals 12y equals 2 x and x equals 3

Use your knowledge of Straight Line Graphs, y equals m x plus 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 is a solid line because of the "≥"

The lines y less than 2 x and x less than 3 are dotted lines because of the "<"

Graphing showing a solid line for the equation 3x+2y=12 and dotted lines for the equations x=3 and y=2x.

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
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

Finally, don't forget to label the region R

V90dxvma_2-19-1-2-graphing-inequalities
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Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

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.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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