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Foundation GCSE, Higher GCSE

The Problem with Solving Inequalities in GCSE Maths

In this blog, I will explain one of the most common mistakes I see made by GCSE maths students. We have already seen how to solve equations, but you also need to be able to solve inequalities in GCSE maths.

What is an inequality?

There are two inequality signs you need to be familiar with at GCSE level (both foundation and higher).

The first is the strict inequality:

                                                <                                             this means “less than”

In inequalities, the “mouth” of the inequality is always facing the larger number. Imagine the inequality is an alligator’s mouth – the alligator will always want to eat the larger number.

It means that if x < y, x must be less than y but cannot be y.

Or, if x > y, y must be less than x but cannot be x.

For example, these two inequalities are correct…

                                                1 < 3                                      clearly 1 is less than 3

                                                2 < 3                                      clearly 2 is less than 3

But these two inequalities are not correct…

                                                4 < 3                                      4 is not less than 3

                                                3 < 3                                      3 is equal to 3, but is not less than 3

The other inequality is not strict:

                                                ≤                                            this means “less than or equal to”

So this means that if x ≤ y, x can be less than y or equal to y.

Or, if x ≥ y, y can be less than x or equal to x.

For example, these two inequalities are correct:

                                                1 ≤ 2                                      1 is less than 2

                                                1 ≤ 1                                      1 is equal to 1, the inequality still holds            

Solving inequalities

Example 1
Solve 4x + 11 ≤ 27

Solving inequalities follow the same process as solving equations. The only difference is that instead of an equals sign, there is an inequality sign. For a recap of how to solve them, re-read that blog.

So solving this one is very easy.

inequalities example 1

So that is the answer. We leave it as x ≤ 4.

So example 1 was quite easy, but this next example is where it gets a bit more complicated.

Example 2
Solve 15 – 2x > 7

The difference with this example is the fact that the x has a minus sign in front of it. This causes a slight problem.

The problem with solving inequalities is that when you divide or multiply by a negative number, the sign needs to change. Here is the reason why:

We know that this inequality is true:

                                                                                               1 < 2

But look what happens when I multiply both sides by -1:

                                                                                                      1 < 2
                                                                                ×-1                                         ×-1
                                                                                                    -1 < -2

Now it reads “-1 is less than -2”. This is incorrect. So multiplying by a negative number has made the inequality incorrect. This is because the ordering of negative numbers is reversed. For negative numbers, the larger the number is, the smaller the negative number is. For example, 10 is larger than 8. But -10 is smaller than -8. So to remedy this problem, we have to switch the sign around. Fixing the example above with 1 and 2:

                                                                                                      1 < 2
                                                                                ×-1                                         ×-1
                                we need to switch the sign because we have multiplied by a negative number
                                                                                                    -1 > -2

Now let’s apply this to example 2.

inequalities example 2

That example included a lot of work with negative numbers. We had to do 7 – 15 = -8, -8 ÷ -2 = 4 and we had to remember to switch the sign around! If you, like many students, want to avoid negative numbers and remembering to switch the sign, there is another way to solve these questions.

We can get around both issues by moving the 2x over to the other side first. Like so…

alternative method for solving inequalities

There may have been one more step, but we have ended up with the same answer (albeit the 4 and the x have switched places and the signs switched) without needing to switch the sign ourselves or deal with any negative numbers.

This, in my opinion, is a much safer and easier way of solving inequalities where there is a “-x” involved however of course some students will still prefer the first way.

Here is one more example solved via both methods.

Example 3
Solve 20 – 3x ≥ 29

Method 1:

solving inequality example 3

Method 2:

alternative method for solving inequality example 3

Both methods are valid, it’s down to personal preference which one you choose to use.

If you would like to practice either of these methods some more, try our inequalities worksheet.

If you need some more help with these questions, book in a free taster session.

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