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College Algebra: Solving Inequality Word Problems


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:20
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 68 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Equations & Inequalities (50 lessons, $65.34)
College Algebra: Inequalities in One Variable (3 lessons, $5.94)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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I thought we'd take a look at an application for using inequalities in the real world. We'll take a look at me and my second job as sales clerk, of course. I really pride myself on that sales clerk job. It means a lot to me. And let me tell you the two options that I have, the two pay options I have for weekly pay. One of them is that I get paid $100 a week. For me, that for educator as you may know, that's good money. And as a bonus for every sale I make, for every sale I get $8.00. So $8.00 per sale plus a hundred dollars a week. A pretty good deal. But even better deal is the following. They're willing to pay me, the same job by the way; they're willing to pay me $250 a week. That is a lot of money. Like a lot more than double this. But they only give me $3.50 per sale. And the question is which should I pick?
Well, the important thing to know is what kind of sales person am I and how much am I going to sell? So the question I want to pose right now is how many sales per week must I make so that Plan A would actually yield the higher paycheck. Now think about this. Even though I'm getting $250 a week here and here I'm only getting a hundred, since I'm getting $8.00 per sale here and only $3.50 per sale here, it's clear that if I make a ton of sales, this amount of money will start to accrue and I may actually grow over a salary over here. And I want to know where is that point. What's the smallest number of sales I have to make so that Plan A, the hundred a week plus $8 bonus per sale is actually the better deal? I make more money. So I want to know when is this thing bigger than this thing.
So actually that's an inequality I've got to set up. I've got to solve an inequality where this thing is greater than that thing and figure out how many sales we have. So let's take a look and see if we can actually do that. I'm going to take all this. I'm not going to throw this. I'll throw this $.50. This actually goes to one of our camera people. Very good. Caught them both, but the cash I will not throw. This I will pocket myself because we can't trust anyone. Now here we go. Let's take a look and see how you would play this out. So let's suppose that the number of sales that I actually have is s. So let's let s equal the number of sales I make per week. We're only looking at this weekly. Then with Plan A and, in fact, you can see they're all over here. At Plan A what happens? Plan A I get a hundred dollars for the week plus $8 for each sale. So how much would I make in one week? Well, I'd get that $100 lump check plus I'd get $8 for every sale and if I make s sales, then I'd be getting 8 x s dollars. So this is my total for Plan A. That's Salary Plan A.
What would be my total for the exact same week under Salary Plan B? Now look over there. There I get $250 in one lump sum plus the bonus is only the $3.50 per sale so that would be 3.5 or 3.50 times the number of sales I make. And the question that I posed to you is how many sales would I have to make so that Plan A is actually more attractive or more lucrative I should say than Plan B. So what I want to know is when does this number exceed that number? So now there's an inequality that I want to sell. So we set up the real world problem. I can take all these things off now, thank goodness. And now we can actually get down and see if we can solve this using that. So what I want to do is solve this inequality.
So let's see. What I'll do is I'll try to get all the s's and variables on one side and all the constants over on the other. If I bring this 3.5s over to the other side I would subtract it, the sign wouldn't change and I would see what? I would see 100 + and if I take 8s and subtract off 3.5s that would leave me with 4.5s, and that would be greater than 250. Do you see how I did that? I just subtracted 3.5s from both sides. When I do that on the right, they cancel. When I do that on the left, 8s - 3.5s gives me 4.5s. Now I'll take this hundred and bring it over to this side, so I subtract a hundred and I see 4.5s is greater than 150. If I take 250 and subtract off a hundred, I see a 150. And then to solve for s, I have to divide by 4.5. So if I divide both sides by 4.5, sort of running out of room here on my screen, 4.5, do I switch the signs? Absolutely not because 4.5 is positive. So what I would see is s > 150 4.5. And if we plug that into a calculator, what we'd see is this is 33 1/3. So s, which is the number of sales I make, has to exceed 33 1/3. So what's the smallest number of sales that I could make that would exceed 33 1/3? The answer is 34 sales. So if I make 34 sales or more, then I know Plan A is the more lucrative plan for me. I'll make more money. However if I make 33 sales or under, then I'd know I'd more money on Plan B. The question was how many sales should I make in order to have Plan A be the most lucrative? The answer is 34 sales or greater. So 34, 35, 36, all the way out to infinity. In fact, if I make infinite amount of sales I'll be so rich I wouldn't be talking to you right now.
Anyway, there's a great application of using inequalities to solve a real world problem. Enjoy.
Equations and Inequalities
Solving Inequalities
Solving Word Problems Involving Inequalities Page [1 of 1]

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