Common Core Algebra I.Unit 2.Lesson 13.Modeling with Inequalities.by eMathInstruction

That's going to allow you to take your smartphone or tablet, scan the code and come right to our videos. All right, let's begin. For the last few lessons, we've been learning all sorts of things about inequalities. Their properties had to solve them. How to think about compound inequalities, how to represent them on the number line, and how to use interval notation to simplify how we write them. Today, we're going to look at modeling real-world scenarios with inequalities. 

Now, as we go through this, I'm going to stress over and over again that when a problem asks you to write an inequality or use an inequality, you have to have an inequality. That is, you have to compare two expressions with a greater than symbol or a less than symbol. And of course, possibly a greater than or equal to symbol. Or a less than or equal to symbol. One of the major things about common core and especially the assessments of your common core knowledge is whether you understand the difference between an equation and an inequality. So make sure you're really paying attention to that as we work through the lesson today. But let's jump right into our first problem. 

A school is taking a field trip with 195 students and ten adults. Each bus can hold at most, and that's kind of an important phrase with inequalities. At most 40 students. We need to determine the smallest number of buses needed for the trip. Using a guess and check method determine the minimum number of buses needed, show evidence of your thinking. All right. Well, think about this for a second. Okay, no inequalities particularly needed right now, but we'll bring them in as we need to. Let's go with four buses. All right, let's see if this gets the job done. You know, we have four buses, and of course what we know is that we have 40 people. Per bus, and if we multiply that out and look at that, those buses cancel, we get 160. 160 people. All right? 

Now, is that right? Is that enough buses? I don't think so, right? And the reason why is that 160 is less than one 95 plus ten. In other words, 160 is less than two O 5. Okay? Well, let's try 5 buses. Right? If we take 5 buses and we multiply it by 40 people per bus. We get 200 people. Now that's still isn't correct because it's less than two O 5. It's close though. It's close. And I think you can probably see already that 6 buses will put us over the top. So we have 6 buses times 40 people per bus. You might wonder why I'm being so careful about my units, units are very, very important. So that will give us 240 people. And that is clearly greater. Than two O 5. All right? Great. So we need 6 buses. 

All right. Good enough. Let's take a look at letter B letter B says, let N be the number of buses taken on a trip. Okay? Right and solve inequality right and solve an inequality, not an equation, but in inequality, that models this problem based on N all right, that's not too hard. Watch what we're going to do. We're going to calculate the number of people based on the number of buses, right? If N is the number of buses and we multiply that by 40, this is the number of people. Or the number of possible people. The number of possible writers. Now, what we know is that when we take 40 and we multiply it by N, it must be greater than, or equal to 205. Now, technically, I suppose we could use greater than two O 5, but clearly, if somehow we could get exactly 205, that would also be okay. 

So in my mind, both the greater than or equal to symbol must must be taken care of. Now, that's really step one in this problem, right? They said right and inequality. They also said solvent inequality. Right? So our second thing that we have to do is we have to solve it. Now it's going to be very easy, right? It's going to be very, very easy because we're going to use the property of inequalities that just says that we can divide both sides by 40. Because that's not a negative, we don't have to switch the inequality symbol and what we see is that N must be greater than or equal to 5.125. Now, that's interesting. Because what that tells us is not that we can have 5 and one 8th buses. That's the .125. Because that wouldn't be viable. A fraction of a bus really isn't correct. 

So in my own mind, the correct answer here is N is greater than or equal to 6. This is really a non viable solution. Non viable. Viable means basically a solution that works. Solution that we can do something with. But N is greater than or equal to 6, that makes sense, right? And it's a good thing to just leave it that way. The minimum number of buses, right, is 6. So. Minimum number is 6. But any amount 6 or over will get the job done. It just might mean that we have a lot of extra space. Okay. So I'm going to clear out the text, pause the video now if you need to. All right, here we go. Let's move on to the next problem. More modeling. Now, exercise two says translate each of the following phrases into an inequality, but do not solve. Great. It's nice if I can just write the inequality, but I don't have to solve it. And that's not uncommon for you to be asked, and it's actually helpful to be able to just write inequalities. 

So letter a says, when three times a number N is increased by 12, the result is at least 32. Let's not worry about that part. Let's just worry first about three times the number N is increased by 12. Well, three times the number N that's easy enough. That's three N increased by 12. That means that we'll have to add 12 to it. And the result is at least 32. Now, if I said, because you want to be able to understand this English, this is more translation issue here. If I said I had at least $32 in my wallet, what would you now? Think about that for a moment. Well, you would know that I could have 32. I could have 40. I could have $67 in my wallet. What I couldn't have is 31. I couldn't have 31 50. I couldn't have 28. Because if I say I have at least 32, it means the amount of money in my wallet must be greater than or equal to 32. So at least translates. Into greater than or equal to. And that's it. We don't have to actually solve this. 

Let's take a look at letter B it says the sum of two consecutive even integers N and N plus two is at most 8. Well, some is easy enough. That's going to be N plus N plus two, right? At most 8. And again, I'd like to pause for a second here and talk about this wording. At most, let's use the money in my wallet thing again. If I said, I had at most $8 in my wallet, you would say, well, then the most amount you had was 8. You could have $3 in your wallet. You could have $6 in your wallet, right? This happens a lot in my own household. You know, we might be ordering food for dinner or something. You know, getting some pizza on a Friday night. And my wife might turn to me and she might say, hey, how much money do you have? And I say, well, I think I have at most $10. And what she should know at that point is, look, I can count on them, having something less than ten. I can't think that he's got 11 or 20 or 15. So the most you can have here is 8, but you could also have less than 8. So at most, translates into the less than or equal to. 

Now less than less than or equal to, right? If I say, hey, I have at most $8, it's okay for me to have exactly $8. All right, so at least greater than or equal to, at most, less than or equal to. That can be tricky tricky wording for a lot of students. So anytime you see it, I want you to always step back and say, hey, let me put some money in my pocket or anything you want. Money in your pocket, video games and your DS, whatever it is, right? Say a phrase involving those terms and think about what it means. All right? I'm going to clear out the text. So write down what you need to. Okay, here it goes. Let's move on. 

All right. Exercise number three says find all numbers for which 5 less than half the number is at least 7. Set up in inequality. Carefully define the expressions and solve the inequality. All right. We should probably carefully define the expressions first, but hey, why don't we use let N be equal to the number? All right. And let's first work on this phrase. 5 less than half the number. Let's not worry about this one yet. We'll get there. 5 less than half the number. Well, half the number is easy. That's one half times N or N divided by two, whichever one you prefer. So one half times N and now 5 less than it. Now that can be a very tricky for people too because they see the phrase less than and they immediately think this. Less than, unfortunately, has two distinct uses. It has the use less man, but it also has the use of subtraction. So 5 less than one half the number. Is at least 7. So again, think about that if you have a problem with the phrase at least 7, think to yourself, hey, I have at least 7 movies that I rented last week. Right? That means I could have rented 7 or 8 or 9 or ten, et cetera. So at least means greater than or equal to. Okay? So there's my inequality. I have to have an inequality here, okay? I do not want this. Even though that's a great start, we don't want it. It's not in inequalities. But now let's solve the inequality. Okay? 

Using a property of inequality says I can add whatever I want to both sides and it doesn't change the truth value of the inequality. Okay? I don't want to divide by one half. I'm going to multiply both sides by two. Two times one half is one. Now because two is a positive number. I have nothing to worry about. And I'll get N is greater than or equal to 24. And there's my solution. All right. By the way, in interval notation, it would look like this. We'd start at 24, and we'd go all the way to infinity like that. Now you don't have to include the interval notation. This is perfectly good. But just getting some practice from a previous lesson with interval notation. All right, not too bad. I think the trickiest thing about these things is that at most and at least business. That can be tough. So pause the video now and copy down what you need to. Okay, here it goes. Let's keep going. 

Find all numbers such that twice the sum of the number and 8 is at most four. All right, so we're doing exactly what we did last time. Find all numbers, such that twice the sum of the number and 8. Is at most four. All right, well, we're going to let N be the number again. Now what are we doing twice? Let's work on this. And then we'll work on the at most four. Twice the sum. So we're multiplying some sum by two, but what some are we multiplying by two? Well, the sum of the number and 8. Now it's very important. This wouldn't be correct. That's not correct because we're only multiplying the number by two. Remember your order of operations. If I want to multiply the sum by two, I have to have that in parentheses. 

All right, now let's deal with the at most four. Again, if I said to you, I have at most four apple trees, right? That could mean I have four. It means I could have three or two, et cetera. So at most, translates to less than or equal to. All right, so there's my inequality. Okay? I have to have an inequality. I can't just have a strict equation. Now how do I solve this while I have two choices? I could distribute the two through the parentheses, or I could just undo what has been done to N and I think I'm going to go with that approach. So in other words, I've added a, then I multiplied by two. So the first thing I'm going to do is divide both sides by two. Again, because two is a positive number. It does not change the truth value or it does not flip the inequality. And now I'm going to use a property of inequalities that says I can subtract anything I want from both sides it also doesn't change the inequality, watch out to -8 is negative 6. So every number less than or equal to negative 6 fits the bill.