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Maths
5 Dec 2025
242
3 pages
Daniyal Ali @daniyalali
Quadratic inequalities might sound scary, but they're just equations that give you a range of answers instead of... Show more
Inequalities are mathematical statements that show when one value is greater than, less than, or equal to another. Unlike regular equations that give you one answer, inequalities give you multiple solutions within a range.
When you see symbols like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to), you're dealing with inequalities. These tell you exactly what relationship the numbers have with each other.
On a number line, you'll show these relationships using dots and arrows. If the inequality includes "equal to" (≥ or ≤), you fill in the dot completely. If it doesn't include "equal to" (> or <), you leave the dot empty to show that exact number isn't included.
Quick Tip Think of filled dots as "including" that number and empty dots as "excluding" it from your solution.
Solving linear inequalities works almost exactly like solving regular equations - you can add, subtract, multiply, and divide both sides to isolate your variable. The key difference is watching out for when you multiply or divide by negative numbers.
Let's look at the example 6 - b < 8 - 4b. You subtract 6 from both sides, then add 4b to both sides, giving you 3b < 2. Finally, divide by 3 to get b < 2/3 (or about 0.66).
Here's the crucial rule when you multiply or divide both sides by a negative number, you must flip the inequality sign. So if you had -2c < 2 and divided both sides by -2, the < would become >, giving you c > -1.
Remember Negative multiplication or division = flip the sign. This is the most common mistake students make!
Compound inequalities show you values trapped between two boundaries, like -3 < 3e ≤ 6. These are brilliant because they give you a clear range to work with.
To solve this, divide everything by 3 -1 < e ≤ 2. This means e is greater than -1 but less than or equal to 2. When you're asked for integer solutions (whole numbers), you'd list -2, -1, 0, 1.
Notice how -1 isn't included because e must be greater than -1, not equal to it. However, 2 would be included if it were an integer because of the ≤ symbol.
Pro Tip Usually you only need to list the first four integers that work - don't overthink it!
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
6
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore the fundamentals of inequalities in mathematics, including solving, graphing, and integer solutions. This comprehensive guide covers key concepts such as two-step and multi-step inequalities, set notation, and quadratic inequalities, providing clear examples and visual aids for better understanding. Ideal for GCSE Edexcel students.
MathsExplore the fundamentals of solving inequalities, including one-step, two-step, and multi-step inequalities. Learn the key rules, such as reversing the inequality sign when dividing by a negative number. This summary provides essential insights for mastering inequality equations.
MathsExplore the fundamentals of inequalities, including one-step, two-step, and multi-step solutions. Learn how to graph inequalities and understand compound inequalities with clear examples and step-by-step explanations. Ideal for students preparing for exams or seeking to strengthen their math skills.
MathsQuotes from every main character
English LiteratureApp Store
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Daniyal Ali
@daniyalali
Quadratic inequalities might sound scary, but they're just equations that give you a range of answers instead of one specific number. Think of them as showing you all the values that work within certain boundaries - like finding all the... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Inequalities are mathematical statements that show when one value is greater than, less than, or equal to another. Unlike regular equations that give you one answer, inequalities give you multiple solutions within a range.
When you see symbols like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to), you're dealing with inequalities. These tell you exactly what relationship the numbers have with each other.
On a number line, you'll show these relationships using dots and arrows. If the inequality includes "equal to" (≥ or ≤), you fill in the dot completely. If it doesn't include "equal to" (> or <), you leave the dot empty to show that exact number isn't included.
Quick Tip: Think of filled dots as "including" that number and empty dots as "excluding" it from your solution.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Solving linear inequalities works almost exactly like solving regular equations - you can add, subtract, multiply, and divide both sides to isolate your variable. The key difference is watching out for when you multiply or divide by negative numbers.
Let's look at the example: 6 - b < 8 - 4b. You subtract 6 from both sides, then add 4b to both sides, giving you 3b < 2. Finally, divide by 3 to get b < 2/3 (or about 0.66).
Here's the crucial rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign. So if you had -2c < 2 and divided both sides by -2, the < would become >, giving you c > -1.
Remember: Negative multiplication or division = flip the sign. This is the most common mistake students make!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Compound inequalities show you values trapped between two boundaries, like -3 < 3e ≤ 6. These are brilliant because they give you a clear range to work with.
To solve this, divide everything by 3: -1 < e ≤ 2. This means e is greater than -1 but less than or equal to 2. When you're asked for integer solutions (whole numbers), you'd list -2, -1, 0, 1.
Notice how -1 isn't included because e must be greater than -1, not equal to it. However, 2 would be included if it were an integer because of the ≤ symbol.
Pro Tip: Usually you only need to list the first four integers that work - don't overthink it!
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
6
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore the fundamentals of inequalities in mathematics, including solving, graphing, and integer solutions. This comprehensive guide covers key concepts such as two-step and multi-step inequalities, set notation, and quadratic inequalities, providing clear examples and visual aids for better understanding. Ideal for GCSE Edexcel students.
MathsExplore the fundamentals of solving inequalities, including one-step, two-step, and multi-step inequalities. Learn the key rules, such as reversing the inequality sign when dividing by a negative number. This summary provides essential insights for mastering inequality equations.
MathsExplore the fundamentals of inequalities, including one-step, two-step, and multi-step solutions. Learn how to graph inequalities and understand compound inequalities with clear examples and step-by-step explanations. Ideal for students preparing for exams or seeking to strengthen their math skills.
MathsQuotes from every main character
English LiteratureApp Store
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Sociology