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When we are dealing with inequalities, we have so far learnt to illustrate these by either using a number line (if it is a single variable) or on a set of axes (if it’s double variable).


What we have to do now is illustrate the region on a graph that satisfies numerous inequalities.


A reminder about lines.

If the inequality you are drawing contains either > or <, you draw a dashed line.

If the inequality contains either ≥ or ≤, you draw a solid line.


Example: Illustrate the region that satisfies the inequalities.

x > 2, y ≥ x and x + y < 8

Step 1: Draw the limit of each inequality as a straight line.

If we draw these on individual graphs (you won’t need to do this in an exam), we can see the lines.

Notice that the middle graph has a solid line, this is because the inequality is greater than or equal to.


We would actually draw all of these onto one graph.


Step 2: Illustrate the region that satisfies all three inequalities.

There’s a short cut to this, in all my time teaching, I’ve never seen it not work at GCSE (it’s slightly different at A-Level)

Find the section that is defined by the inequalities. There is only one bit on here that works for all the inequalities, the bit in the middle.


Example: Are these points in the region defined by the inequalities x > 2, y ≥ x and x + y < 8

  1. (3,4)
  2. (2,6)
  3. (3,3)

If we plot these on the graph we previously created.

  1. (3,4) – This is in the middle of the region so it’s fine
  2. (2,6) – This is on a dashed line. Because a dashed line means > or <, points on the line don’t satisfy
  3. (3,3) – This point is on a solid line. A solid line means ≥ or ≤. Points on these lines do satisfy.

10 questions