# Modeling With Systems Of Inequalities Homework _BEST_

Students will be working through the whole activity before the debrief. For the most part, they won't have much trouble with this activity. As you walk around and monitoring the groups, make sure to be checking that their inequalities are correct. It can be really frustrating for a student to spend a lot of time graphing only to find out they graphed the wrong inequality.

## Modeling With Systems Of Inequalities Homework

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During the Check Your Understanding problem, students should write and graph inequalities that are in standard form. This is a great time to help students practice finding intercepts. Ask students, "What is the greatest number of minivans that they can get with this constraint? What's the greatest number of SUVs?" This will help them to graph much more efficiently. They also have to decide the scale on the axes so finding the intercepts first is very helpful.

Hello and welcome to another E math instruction common core algebra one lesson. My name is Kirk weiler and today we're going to be doing unit number 5 lesson number 8 modeling with systems of inequalities. I'd like to remind you that if you'd like to get a copy of the worksheet that goes with this lesson and in the company homework, just click on the video's description. As well, don't forget that at the top right hand corner of all of our worksheets we have a QR code that will allow you to use a mobile phone or a tablet to scan that QR code and bring you right to this video. Anyhow, let's get into a practical problem that involves modeling with systems of inequalities. Let's take a look at exercise one. John mose yards for his father's landscaping business for $10 per hour, and also works at a bakery for $15 per hour. He can work at most 52 hours per week during the summer, and he needs to make at least $600 per week to cover his living expenses. All right, let's make sure that we really have all of this down. Okay? He's going to work first landscaping business for $10 per hour, work at a bakery for $15 per hour. He can work at most 52 hours, 52 hours per week, apparently I don't care about the word week. And he needs to make at least $600 per week to cover his living expenses at least. Letter a if John works 14 hours mowing and 30 hours at the bakery, does this satisfy all the problems constraints, right? Now there's really two constraints. He can work at most 52 hours, and he needs to make at least $600 per week. So I'd like to get a pause the video and see if those working hours satisfy all the constraints. Let's take a look. Well, that's something that's pretty easy. Let's see if he's worked too many hours. So 30 plus 14, right? That gives me 44. He can work at most 52 hours, so that works. Let's see if you made enough money. Let's see. He's going to work. 14 hours mowing, and he's going to get paid $10 an hour to do that. So he's going to make a $140 mowing. Right? Then he's going to work 30 hours at the bakery. Times $15 an hour. That gives me $450. Okay. Add those together and we're going to have $590. And he needs to make at least 600. So no. One of the two conditions is one of the constraints is met, but not the other one. Well, let's try to model this, okay? Letter B says if X represents the hours John spends mowing. And why represents the hours he spends at the bakery? Write a system of inequalities that describes the scenario. Well, one thing that's pretty easy is that first condition. When I add X to Y, it's got to be less than or equal to 52. You can work at most 52 hours. So when I add the hours that he spends mowing to the hours he spends baking, right? Then that's got to be less than or equal to 52 hours. Okay, so that's kind of the equivalent of this condition. Now, how about this? Well, X represents the number of hours mowing, so ten times X will represent the amount of money he makes mowing. Then if I add to that, 30, no, not 30. One mistake. If I add to that 15, $15 per hour, times the number of hours he spends at the bakery, that's got to be greater than or equal to 600. This is what's known as a system. Of inequalities. Right? Letter C says, if John must work or walk, apparently. That's not right. If John must work a minimum of ten hours for his father, will he be able to make enough money to cover his living expenses. Show the work that leads to your answer. Well, I'd like you to play around with that a little bit. If John must work, not walk, but if John must work, a minimum of ten hours for his father will he be able to make enough money. Play around with that. All right, well let's see. Let's see, he's going to work ten hours for his father, right? Times $10 per hour. He's going to make a $100 just for his father. Right? Now how much time does that leave him? While he can work at most 52, so subtract off the ten hours, he's got to work for his father. He's going to be left 42 hours that he can work at the bakery, right? So if he takes 42 hours and we multiply that by $15 per hour, all right, what am I going to end up getting? I'm going to end up getting a value that I don't have written down for some reason. Let's do 42 times 15. And we're going to get 600 and $30. Well, obviously, that means that we're going to make $730 in this scenario. Now he's got to make at least 600. It's got to be greater than or equal to $600. And so yes, he does make enough. Okay? So he can work that minimum amount of time that his father needs him to and still make the $600 he needs to as long as he can work the other time at the bakery. All right, we're going to keep working with this problem because it's a very rich one. Let me clear out the text. All right, that's all gone. Let's take a look at the next part of the problem. Letter D says, graph the system of inequalities with the help of your calculator if needed on the axes below. Use the space below to think about how to graph these lines. So here's that system of inequalities that we had, right? So what we want to do is somehow graph them. Now, at this point, the expectation is you've played a little around a little bit with graphing, systems of inequalities. So one thing I would do is I would immediately put this X on the other side. It would become negative. And I would end up having this. Now, remember, when we graph an inequality, what we do is we first graph the equation that the inequality is based on, which is going to be Y equals negative X plus 52. Now the other equation, that one's going to take us a little more work. So let me put that thing down here. We're going to get 15 Y is greater than or equal to negative ten X plus 600. Divide by 15, divide by 15, divide by 15, all right? And we're going to get Y is greater than or equal to we could reduce the negative ten fifteenths. It does reduce nicely to negative two thirds. All right. And then 600 divided by 15 is 40. Okay? So I think I am going to use the calculator right now to help me out because I'm going to have it graph these two lines. Y equals negative two thirds X plus 40. And Y equals negative X plus 52. So let's open up the TI 84 plus. All right. Let's go into Y equals. Okay. And Y one, I think I'm going to put in the negative X plus 52. And in Y two, I'm going to put in negative two thirds X plus 40. All right. I have to think a little bit about the window. But remember, X represents the hours working at mowing. Why represents the hours of the bakery? Neither one of them can exceed 52, probably most of them are well short of it. So why don't we go into our window? Let's put X-Men in as zero. Let's put X max in at, I don't know, let's say 60. Let's do Y min also 50. Sorry, Y min also zero. And let's do Y max also 60. Okay? Once we have them all in, let's hit graph. See how they look. All right. Now it's kind of important to remember which ones which so here we've got the actually, let me use. The prefab line maker here. Do the red one and maybe label that. And why not? That'll help. Y equals negative X plus 52. And I think I'll go with a blue line for the other one. This, maybe not the best in terms of my spacing. Y equals negative two thirds X plus 40. All right, actually, let me go back to red. Now remember, what I'm really doing is I'm shading Y is less than or equal to negative X plus 52. So that's going to be. Everything down here. All right. Y is greater than or equal to negative two thirds X plus 40. That's going to be everything above it. Right? So where is the solution to a system of inequalities? Well, it is now this triangular region. Right here. It's the overlap, right? Wherever the shading overlaps is the solution to our inequality. So that is the solution to our inequality. Every point within here makes our system true. All right? Satisfies those conditions. I was kind of interesting. Put that guy back. Letter E John's father needs him to work a lot at the landscaping business. Show the point on the graph that corresponds to the greatest number of hours that he can work. While still covering his expenses. All right, so it's got to be a point somewhere in here. Points out here, right? He's mowing too much. Points in here are fine, right? If I'm in here, let's say, like a mowing in here, that's all right. So this has got to be it, right? Whatever point that is. Is the greatest amount that he can mow while still being in this region. All right? Um, letter F then says algebraically find the greatest number of hours, right? That he can mow, work for his father, and still cover his expenses. So how do we find that point algebraically? Pause the video for a second, think about that. All right, we can find that point algebraically. By solving the system of equations, equations, not inequalities, given by these two. So in other words, I can solve for when negative X plus 52 is equal to negative two thirds X plus 40. All right. Kind of up to you, but I think I'll add a positive X to both sides. That'll leave me with 52 over here. Now negative two thirds plus positive one would be a positive one third X think about that