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.

Did you follow those examples?

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.