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

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 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)

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