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171
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30 Nov 2025
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Safwan Master
@safwanmaster
Inequalities are mathematical statements that compare two expressions using symbols... Show more
![INEQUALITIES
1.
2.
3.
4.
5.
6.
Solve the following inequalities.
(i) 2x + 3 < 1 - x [2]
(ii) 3(y-1) ≥ 5y - 8
Solve the following inequalitie](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FTPsArmhlstqvuwooQSCp_image_page_1.webp&w=2048&q=75)
Think of inequalities as mathematical "greater than" or "less than" statements that you solve just like equations, with one crucial difference. When you multiply or divide by a negative number, you must flip the inequality sign – this trips up loads of students, so remember it!
For linear inequalities like 2x + 3 < 1 - x, collect like terms and isolate x. You'll get x < -2/3, which means x can be any value smaller than -2/3. Simple inequalities with fractions like x/2 > 5 work the same way – multiply both sides by 2 to get x > 10.
Quick Tip: Always double-check your answer by picking a test value. If x < -2/3, try x = -1. Does it satisfy the original inequality?
The key is treating these like normal equations whilst being careful about that sign-flipping rule. Once you've mastered linear inequalities, you're ready for the trickier quadratic ones.
![INEQUALITIES
1.
2.
3.
4.
5.
6.
Solve the following inequalities.
(i) 2x + 3 < 1 - x [2]
(ii) 3(y-1) ≥ 5y - 8
Solve the following inequalitie](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FTPsArmhlstqvuwooQSCp_image_page_2.webp&w=2048&q=75)
Quadratic inequalities look scary but follow a clear pattern once you get the hang of them. Start by factorising the quadratic expression, find where it equals zero, then determine which regions satisfy your inequality.
Take x² + 2x - 15 ≤ 0. First factorise: = 0. This gives you critical points at x = -5 and x = 3. These points split the number line into three regions, and you need to test which regions make the inequality true.
For quadratics that are ≤ 0, you want the "dip" between the roots, so -5 ≤ x ≤ 3. For quadratics > 0, like 2p² - 7p + 3 > 0, you want the regions outside the roots: p < 1/2 or p > 3.
Memory Trick: Sketch a U-shaped parabola. If you want the expression < 0, choose where the parabola dips below the x-axis!
![INEQUALITIES
1.
2.
3.
4.
5.
6.
Solve the following inequalities.
(i) 2x + 3 < 1 - x [2]
(ii) 3(y-1) ≥ 5y - 8
Solve the following inequalitie](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FTPsArmhlstqvuwooQSCp_image_page_3.webp&w=2048&q=75)
When quadratic equations have real roots, their discriminant b² - 4ac must be ≥ 0. This connects algebra with inequalities in a really clever way that examiners love testing.
For the equation x² + kx + = 0 to have real roots, you set up k² - 4(1) ≥ 0. Simplifying gives k² - 4k - 12 ≥ 0. Now treat this like any quadratic inequality – factorise to get = 0, giving critical points at k = -2 and k = 6.
Since you want the quadratic ≥ 0, you need the regions outside the roots: k ≤ -2 or k ≥ 6. This means the original equation has real roots only when k falls in these ranges.
Exam Tip: Questions often ask you to "show that" an inequality holds first, then find the range. The showing part usually involves expanding and rearranging the discriminant.
![INEQUALITIES
1.
2.
3.
4.
5.
6.
Solve the following inequalities.
(i) 2x + 3 < 1 - x [2]
(ii) 3(y-1) ≥ 5y - 8
Solve the following inequalitie](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FTPsArmhlstqvuwooQSCp_image_page_4.webp&w=2048&q=75)
Set notation is just a fancy way of writing your answers that examiners expect in A-level work. Instead of writing x < 5/2, you write {x : x < 5/2}, which reads as "the set of all x such that x is less than 5/2".
When you have combined inequalities, solve each part separately first. For 6x - 7 < 2x + 3 AND 2x² - 11x + 5 < 0, you get x < 5/2 and 1/2 < x < 5 respectively. The combined solution is where these overlap: {x : 1/2 < x < 5/2}.
Real-world problems often create these combined inequalities naturally. A hotel room with length x might need x ≤ 88 for area constraints and 4x - 6 ≥ 30 for perimeter requirements.
Success Strategy: Always sketch number lines for combined inequalities – visual representation makes overlapping regions obvious and prevents silly mistakes.
![INEQUALITIES
1.
2.
3.
4.
5.
6.
Solve the following inequalities.
(i) 2x + 3 < 1 - x [2]
(ii) 3(y-1) ≥ 5y - 8
Solve the following inequalitie](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FTPsArmhlstqvuwooQSCp_image_page_5.webp&w=2048&q=75)
Graphical inequalities show up as shaded regions on coordinate planes, and they're actually quite straightforward once you know the rules. Solid lines mean ≤ or ≥, whilst dashed lines mean < or >.
For linear inequalities like y ≤ 2x, first draw the line y = 2x, then shade below it (since you want y values less than or equal to 2x). When you have multiple inequalities like y ≤ 2x, y ≥ 1, and y + x ≤ 3, the solution region is where all shaded areas overlap.
Quadratic inequalities work similarly. For y ≥ x² - 2x - 8, sketch the parabola first, then shade above it since you want y values greater than the quadratic expression.
Visual Check: Pick a point inside your shaded region and substitute into all inequalities – if they're all satisfied, you've got it right!
![INEQUALITIES
1.
2.
3.
4.
5.
6.
Solve the following inequalities.
(i) 2x + 3 < 1 - x [2]
(ii) 3(y-1) ≥ 5y - 8
Solve the following inequalitie](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FTPsArmhlstqvuwooQSCp_image_page_6.webp&w=2048&q=75)
Rational inequalities involving fractions need extra care because you can't simply multiply through by the denominator (it might be negative!). For 3/ < 1, rearrange to get everything on one side first.
Moving terms gives 3/ - 1 < 0, which becomes / < 0 or / < 0. Now you need the fraction to be negative, which happens when the numerator and denominator have opposite signs.
This creates two cases: either 2-x > 0 and x+1 < 0 (giving x < -1), or 2-x < 0 and x+1 > 0 (giving x > 2). So the solution is x < -1 or x > 2.
Advanced problems often combine rational expressions with discriminants. When 3/x + c = 4 - x has no real roots, rearrange to standard form and use b² - 4ac < 0.
Critical Warning: Never multiply inequalities by variables – you don't know if they're positive or negative, which affects the inequality direction!
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.
App 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
Safwan Master
@safwanmaster
Inequalities are mathematical statements that compare two expressions using symbols like <, >, ≤, or ≥. They're essential for solving real-world problems where you need to find ranges of values rather than exact answers, from calculating budgets to designing spaces.
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Think of inequalities as mathematical "greater than" or "less than" statements that you solve just like equations, with one crucial difference. When you multiply or divide by a negative number, you must flip the inequality sign – this trips up loads of students, so remember it!
For linear inequalities like 2x + 3 < 1 - x, collect like terms and isolate x. You'll get x < -2/3, which means x can be any value smaller than -2/3. Simple inequalities with fractions like x/2 > 5 work the same way – multiply both sides by 2 to get x > 10.
Quick Tip: Always double-check your answer by picking a test value. If x < -2/3, try x = -1. Does it satisfy the original inequality?
The key is treating these like normal equations whilst being careful about that sign-flipping rule. Once you've mastered linear inequalities, you're ready for the trickier quadratic ones.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Quadratic inequalities look scary but follow a clear pattern once you get the hang of them. Start by factorising the quadratic expression, find where it equals zero, then determine which regions satisfy your inequality.
Take x² + 2x - 15 ≤ 0. First factorise: = 0. This gives you critical points at x = -5 and x = 3. These points split the number line into three regions, and you need to test which regions make the inequality true.
For quadratics that are ≤ 0, you want the "dip" between the roots, so -5 ≤ x ≤ 3. For quadratics > 0, like 2p² - 7p + 3 > 0, you want the regions outside the roots: p < 1/2 or p > 3.
Memory Trick: Sketch a U-shaped parabola. If you want the expression < 0, choose where the parabola dips below the x-axis!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When quadratic equations have real roots, their discriminant b² - 4ac must be ≥ 0. This connects algebra with inequalities in a really clever way that examiners love testing.
For the equation x² + kx + = 0 to have real roots, you set up k² - 4(1) ≥ 0. Simplifying gives k² - 4k - 12 ≥ 0. Now treat this like any quadratic inequality – factorise to get = 0, giving critical points at k = -2 and k = 6.
Since you want the quadratic ≥ 0, you need the regions outside the roots: k ≤ -2 or k ≥ 6. This means the original equation has real roots only when k falls in these ranges.
Exam Tip: Questions often ask you to "show that" an inequality holds first, then find the range. The showing part usually involves expanding and rearranging the discriminant.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Set notation is just a fancy way of writing your answers that examiners expect in A-level work. Instead of writing x < 5/2, you write {x : x < 5/2}, which reads as "the set of all x such that x is less than 5/2".
When you have combined inequalities, solve each part separately first. For 6x - 7 < 2x + 3 AND 2x² - 11x + 5 < 0, you get x < 5/2 and 1/2 < x < 5 respectively. The combined solution is where these overlap: {x : 1/2 < x < 5/2}.
Real-world problems often create these combined inequalities naturally. A hotel room with length x might need x ≤ 88 for area constraints and 4x - 6 ≥ 30 for perimeter requirements.
Success Strategy: Always sketch number lines for combined inequalities – visual representation makes overlapping regions obvious and prevents silly mistakes.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Graphical inequalities show up as shaded regions on coordinate planes, and they're actually quite straightforward once you know the rules. Solid lines mean ≤ or ≥, whilst dashed lines mean < or >.
For linear inequalities like y ≤ 2x, first draw the line y = 2x, then shade below it (since you want y values less than or equal to 2x). When you have multiple inequalities like y ≤ 2x, y ≥ 1, and y + x ≤ 3, the solution region is where all shaded areas overlap.
Quadratic inequalities work similarly. For y ≥ x² - 2x - 8, sketch the parabola first, then shade above it since you want y values greater than the quadratic expression.
Visual Check: Pick a point inside your shaded region and substitute into all inequalities – if they're all satisfied, you've got it right!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Rational inequalities involving fractions need extra care because you can't simply multiply through by the denominator (it might be negative!). For 3/ < 1, rearrange to get everything on one side first.
Moving terms gives 3/ - 1 < 0, which becomes / < 0 or / < 0. Now you need the fraction to be negative, which happens when the numerator and denominator have opposite signs.
This creates two cases: either 2-x > 0 and x+1 < 0 (giving x < -1), or 2-x < 0 and x+1 > 0 (giving x > 2). So the solution is x < -1 or x > 2.
Advanced problems often combine rational expressions with discriminants. When 3/x + c = 4 - x has no real roots, rearrange to standard form and use b² - 4ac < 0.
Critical Warning: Never multiply inequalities by variables – you don't know if they're positive or negative, which affects the inequality direction!
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.
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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
