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