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Calculus 1
Linear Inequalities and Compound Interest
101
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Nov 29, 2025
•
Understanding Linear Inequalities and Exploring Compound Interest
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Linear Inequalities and Simple Interest
Ever wondered how to graph areas instead of just lines on a coordinate plane? Linear inequalities use familiar symbols like ≥ (greater than or equal to) and ≤ (less than or equal to) to represent regions on a graph.
When graphing an inequality like y ≥ 2x + 3, first treat it like an equation to find the boundary line. Find the intercepts - in this case (0,3) and (-1.5,0) - and draw the line. Then shade the appropriate region .
Simple interest is a straightforward way to calculate growth on investments. The formula is I = PRT, where I is interest earned, P is principal (original amount), R is rate (as a decimal), and T is time in years. The total amount A = P + I.
Quick Tip: When solving simple interest problems, always convert percentages to decimals (6% = 0.06) and make sure your time is in years .
If you know three of the variables in the simple interest formula, you can rearrange it to find the fourth: P = I/RT, R = I/PT, or T = I/PR. This flexibility makes it useful for many financial problems!
Compound Interest and Sequences
Compound interest is what banks actually use - it's interest on your interest! The formula is A = P^(nt), where n is how many times interest is compounded each year.
Unlike simple interest, compound interest grows exponentially. For example, $10,000 invested at 3.5% for 3 years compounded monthly would grow to much more than with simple interest. The effective interest rate (EIR) tells you the actual percentage gain after compounding.
Sequences help us find patterns in numbers. An arithmetic sequence increases or decreases by a fixed amount each time (like 3, 11, 19, 27...). The general term is a + d, where a is the first term and d is the common difference.
Remember This: For reducible interest (like on loans), interest is calculated on the remaining balance after each payment. This means early payments save you more money in the long run!
A geometric sequence multiplies by a constant ratio (like 2, 6, 18, 54...). The formula for the general term is ar^, and the sum formula is a/. These patterns appear everywhere from salary increases to population growth!
Sequences and Statistical Measures
The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8...) is a famous pattern where each number equals the sum of the two before it. You'll find this sequence in nature, art, and architecture!
When analyzing data, we often look at univariate data - information about just one variable. For example, test scores from your class would be univariate data. We use several measures to understand this type of data:
- Mean: Add all values and divide by how many there are
- Median: The middle value when data is arranged in order
- Mode: The most frequently occurring value
- Range: The difference between the largest and smallest values
Quartiles divide ordered data into four equal parts. The median (Q2) splits the data in half, while Q1 (lower quartile) and Q3 (upper quartile) divide each half again. These values help us understand how data is distributed.
Pro Tip: When working with a large dataset, calculating quartiles helps you quickly identify unusual values (outliers) that fall far outside the typical range.
Finding these measures gives you a clearer picture of what your data actually means, which is essential for making good decisions based on information.
Bivariate Data and Correlation
Relationships between two variables can tell us powerful stories! Bivariate data examines connections between two different measurements, like height vs. weight or study time vs. test scores.
The relationship between variables can be positive (both increase together), negative (as one increases, the other decreases), or show no relationship at all. The strength of a relationship describes how closely the points follow a pattern.
A scatter plot helps visualize these relationships. Points that form a tight pattern along a line show a strong relationship, while scattered points indicate a weak or nonexistent relationship.
Visual Guide: Think of correlation as measuring how "well-behaved" your data points are. Perfect correlation means all points fall exactly on a straight line!
The correlation coefficient (r) gives a precise measure of relationship strength, ranging from -1 (perfect negative) through 0 (no relationship) to +1 (perfect positive). When analyzing real-world data, finding strong correlations can help you predict outcomes and make better decisions.
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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.
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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
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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
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Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Understanding Linear Inequalities and Exploring Compound Interest
_
_qpjx
@_qpjx
Mathematics gets more interesting when we can apply it to real-world situations! This guide covers essential topics from linear inequalities to statistical analysis that will help you solve practical problems and prepare for future math courses.
Linear Inequalities and Simple Interest
Ever wondered how to graph areas instead of just lines on a coordinate plane? Linear inequalities use familiar symbols like ≥ (greater than or equal to) and ≤ (less than or equal to) to represent regions on a graph.
When graphing an inequality like y ≥ 2x + 3, first treat it like an equation to find the boundary line. Find the intercepts - in this case (0,3) and (-1.5,0) - and draw the line. Then shade the appropriate region .
Simple interest is a straightforward way to calculate growth on investments. The formula is I = PRT, where I is interest earned, P is principal (original amount), R is rate (as a decimal), and T is time in years. The total amount A = P + I.
Quick Tip: When solving simple interest problems, always convert percentages to decimals (6% = 0.06) and make sure your time is in years .
If you know three of the variables in the simple interest formula, you can rearrange it to find the fourth: P = I/RT, R = I/PT, or T = I/PR. This flexibility makes it useful for many financial problems!
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Compound Interest and Sequences
Compound interest is what banks actually use - it's interest on your interest! The formula is A = P^(nt), where n is how many times interest is compounded each year.
Unlike simple interest, compound interest grows exponentially. For example, $10,000 invested at 3.5% for 3 years compounded monthly would grow to much more than with simple interest. The effective interest rate (EIR) tells you the actual percentage gain after compounding.
Sequences help us find patterns in numbers. An arithmetic sequence increases or decreases by a fixed amount each time (like 3, 11, 19, 27...). The general term is a + d, where a is the first term and d is the common difference.
Remember This: For reducible interest (like on loans), interest is calculated on the remaining balance after each payment. This means early payments save you more money in the long run!
A geometric sequence multiplies by a constant ratio (like 2, 6, 18, 54...). The formula for the general term is ar^, and the sum formula is a/. These patterns appear everywhere from salary increases to population growth!
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Sequences and Statistical Measures
The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8...) is a famous pattern where each number equals the sum of the two before it. You'll find this sequence in nature, art, and architecture!
When analyzing data, we often look at univariate data - information about just one variable. For example, test scores from your class would be univariate data. We use several measures to understand this type of data:
- Mean: Add all values and divide by how many there are
- Median: The middle value when data is arranged in order
- Mode: The most frequently occurring value
- Range: The difference between the largest and smallest values
Quartiles divide ordered data into four equal parts. The median (Q2) splits the data in half, while Q1 (lower quartile) and Q3 (upper quartile) divide each half again. These values help us understand how data is distributed.
Pro Tip: When working with a large dataset, calculating quartiles helps you quickly identify unusual values (outliers) that fall far outside the typical range.
Finding these measures gives you a clearer picture of what your data actually means, which is essential for making good decisions based on information.
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Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Bivariate Data and Correlation
Relationships between two variables can tell us powerful stories! Bivariate data examines connections between two different measurements, like height vs. weight or study time vs. test scores.
The relationship between variables can be positive (both increase together), negative (as one increases, the other decreases), or show no relationship at all. The strength of a relationship describes how closely the points follow a pattern.
A scatter plot helps visualize these relationships. Points that form a tight pattern along a line show a strong relationship, while scattered points indicate a weak or nonexistent relationship.
Visual Guide: Think of correlation as measuring how "well-behaved" your data points are. Perfect correlation means all points fall exactly on a straight line!
The correlation coefficient (r) gives a precise measure of relationship strength, ranging from -1 (perfect negative) through 0 (no relationship) to +1 (perfect positive). When analyzing real-world data, finding strong correlations can help you predict outcomes and make better decisions.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user