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

Finding regions using inequalities

What are 2D inequalities?

• A 2D inequality involves two variables

  • E.g. or

• The solution to a 2D inequality is a region in the plane

• The solution to a 1D inequality can also be represented as a region in the plane

  • E.g. represents the region of points that lie on or above the line

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 is to the left of

    • the wanted region for is to the right of

• 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)