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

- (3,4)
- (2,6)
- (3,3)

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.