So far we have dealt with algebraic expressions, which are a way of writing something down in symbolic form, and algebraic equations, which are a way of expressing a relationship between two things. But the relationship of equality is not the only kind of relationship that exists. Sometimes things are not equal to each other. In this situation we can use inequalities (or “crocodiles”) to describe the relationship.

An inequality tells us that one expression is larger than another expression, e.g. 2x+4 > 3x-5.

We solve inequalities by manipulating them until we have the unknown (or the letter) on its own (e.g. x<9). To achieve this goal, we follow exactly the same rules as when dealing with equations, that is we can do any arithmetic operation we like, as long as we do exactly the same to both sides of the equation.

There are 3 different ways that we can write our result, we should be aware of all of them:

• Using “crocodiles’” (Cambridge’s favourite way);
• Using set notation (My favourite way, and the way which prepares you for university);
• Using a number line (You need to know how to do this).

Sometimes we can have combined inequalities, like 3 < x+4 < 7. The way we deal with these is essentially the same, we are just doing it with more algebraic expressions

Example

Let’s try together to solve some of these. Each of our answers we will write in each of the three different ways mentioned above:

Exercise

Now let’s try questions 4 to 13 of the textbook exercise 8E on page 117:

The answers to these questions are below: