# 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.

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.

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…

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:

Method 2:

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.

## Other Posts

How to fix a calculator that is in the wrong mode
If you’re studying GCSE or A Level maths, you will be spending …
Perfect numbers
“If you look for perfection, you’ll never be content”There are very few …
The Capture-Recapture Method – how to estimate the number of fish in a lake
In the previous blog, I explained the Difference of Two Squares, as …
The Difference of Two Squares
Mentioned specifically in the Edexcel Advance Information for the 2022 GCSE exams …
Advance Information for Summer 2022 GCSE Exams
Due to the disruption to schooling caused by the Coronavirus pandemic, exam …
Times Tables App Review – Math Ninja
For a child in primary school, there is nothing more important in …
How to use triangles to remember exact sin, cos and tan values
If you’re taking the higher maths GCSE, there are certain values of …
A more efficient method for listing the factors of a number
In this blog I will show you the most efficient way to …
GCSE and A-Level Results 2021
Unfortunately due to the pandemic, GCSE and A-Level exams were cancelled again …
The Look-and-Say Sequence
In a previous blog I looked in more detail into “the daddy …
How to represent a recurring decimal as a fraction
What is a recurring decimal? Some decimal numbers are easy to deal …
Divisibility rules for numbers 7-12
Following on from the previous blog, which showed you a series of …
Divisibility rules for numbers 2-6
In this blog I will show you how you can easily check …
The Fibonacci Sequence and the Golden Ratio
We have previously touched upon Fibonacci sequences when discussing different types of …
5 things that make a good maths tutor
Following on from our previous blog on why university students make excellent …
Triangles in GCSE Maths
Triangles come up a lot in GCSE mathematics. And there are a …
A guide to trigonometry (SOHCAHTOA) – Part 2
In the last blog, I introduced a method for using trigonometry to …
A guide to trigonometry (SOHCAHTOA) – Part 1
Following on from the previous blog on Pythagoras’ theorem, this blog will …
Easy as Py – a guide to Pythagoras’ Theorem
a2 + b2 = c2.That’s Pythagoras’ theorem. What it says is that …
5 more maths magic tricks
Following on from the previous blogs about the number 6174 and the …

This site uses Akismet to reduce spam. Learn how your comment data is processed.