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When we are working with inequalities, we can illustrate these by either using a number line (if it has a single variable) or on a set of axes (if it has double variables).

We can also illustrate the region on a graph that satisfies numerous inequalities.

A reminder about which lines to use:

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

If the inequality contains either a or symbol, you should draw a solid line.

e.g. 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), we can see the lines more clearly.

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

Usually, we would actually draw all of these onto one graph, like this:

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

There’s a short cut we can use here.

There is only one section that works for all the inequalities - the bit in the middle of all three lines (shaded grey here):

Let's extend from this starting question now.

e.g. Are the points below in the region defined by the inequalities x > 2, y ≥ x and x + y < 8?

a) (3,4)

b) (2,6)

c) (3,3)

If we plot these on the graph we previously created, they will appear like this:

a) (3,4) – This is in the middle of the region so it is definitely within the region.

b) (2,6) – This is on a dashed line. As a dashed line means > or <, points on the line don’t count as being inside the region.

c) (3,3) – This point is on a solid line. A solid line means ≥ or ≤ so the line itself is included in the region.

Are you ready to investigate some regions independently now?

You may want to have some squared paper and a pencil handy in case you want to draw any of these graphs to compare with our maths teacher's versions.

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