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College Algebra: Solving Rational Inequalities Ex


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:49
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 95 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: Rationals, Radicals (3 lessons, $4.95)

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.

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So these rational inequalities really require a little bit of care and delicacy because you have all these little things to consider. You have to find out where things are 0, mark down where the denominators are 0, you have to mark as an undefined place, make the sign chart and read off the answer. So, it's a little bit involved and I wanted to show you a slightly more exotic, a little bit harder kind of problem just to put all these ideas together. So, like I said, this is a little bit tricky, but let's work through it together and see that it's nothing more than just the same idea with a little bit of care.
I want to find out all the points, all the values x, so that x² + x - 6 when divided by x - 4 is going to be greater than or equal to 0. Now, here I'm mixing all the mediums together. Do you see it? I've got a quadratic in here and I've got a division thing going. I have all this stuff happening at once and you're going to see the process isn't going to be any different than it was before. It's just a little bit more involved.
Step one, make sure I've got 0 on one side and everything else on the other. I've got that. Step two, make sure everything is just one fraction, if it's just the fraction stuff. Well, I have one fraction. Step three, if I see quadratics can I factor? Well, let's hope I can. Otherwise not going to be happy. So, let's try to factor the top of that thing. I'll put and x and an x. That makes a lot of sense. This tells me I'm going to have opposite signs. So, since these are both the same, I can write them any way I want. I need two numbers whose product is 6, but whose difference is going to be plus 1. That looks like 3 times 2 is going to work well. Where should I put the 3? Should I put the 3 by the negative? No, that would produce a negative x when I subtract off the plus 2. So, I'll put the 2 here and the 3 here. 3x - 2x is 1x and this times that is -6. Looks great.
Don't forget the denominator. You might be so excited to factor that correctly, you might forget about the denominator. That's a great mistake to make. Always make sure you have everything written down there just like it was. So, same exact inequality but now in a factored form. Now, I can sort of see all the factors. There's this as a factor, this as a factor, and actually the way I think about it, I consider that a factor too, even though it's division. So, I'm going to create one big sign chart. This is going to be a big one. This is like the mother of all sign charts. I'm going to mark down where every single factor vanishes.
So, let's see. Where does this thing equal 0? Well, wherever the top is 0. So, wherever this equals 0 or wherever this equals 0. So, when does x + 3 equal 0? Well, that equals 0 when x = -3. So, I want to mark down -3. I'll mark it way over here, -3. So, there's a point that makes this whole thing equal to 0. So, I'll mark it as a 0 here. If you want, you could actually maybe mark this like a little wall. You might want to put a little wall here, like a firewall. So, you can put this thing here. I'll make that green. So, you've got that. Now, when is this thing going to be 0? Well, that's 0, x - 2 is 0 when actually x = 2. So, I'll put 2 way over here. Again, you can put in all the points if you want, but I'm not going to. There's 2 and again, that thing will be 0 there. So, I put a 0 there and it's a firewall. So, I'll put this thing there.
But remember, I also want to mark down where the denominator of a fraction is 0, because those are the places where I know this thing is undefined. So, where's the bottom equal to 0? Well, x - 4 equals 0 when x equals 4. So, actually I'm going to put one more point here. It's sort of a scary point because it's undefined there. So, I don't put 0, I put undefined because you can't touch--don't touch that point. You cannot touch that point because if you touch that point with an x, this thing is undefined. So, you have to avoid that point at all times.
Well, now you'll notice I actually cut this line up, not into three regions but one, two, three, four regions. So, whatever's happening here, it's going to happen constantly throughout this region. The same thing here, here and here. So, I've got to go through each of these regions, pick a sample representative point, see what the sign of this is going to give, and then report the sign for that region. So, same exact process, but notice it's a little bit more complex, but not any harder.
So, I want to pick a point way off here to the right of 4. You can pick any number as long as it's bigger than 4. To really try to drive this home I'm going to pick a million. That seems like a big number. It is, but all I care about is the sign of this. Watch how I argue. I'm going to now plug in a million wherever I see an x. If I put a million in here, a million plus 3, that's positive. If I put a million in here, a million minus 2, that's still positive. A positive times a positive that's still positive. So, I've got a positive on the top. If I take a million and subtract 4, that's positive. So, I see positive divided by a positive. Well, that's positive.
You see how the factor that was a million really didn't make much of a difference at all? So, that whole region is a positive region. Am I going to include that? Absolutely, because I'm looking for where this thing is positive or 0. So, I'm going to include that. That's a happy answer. Okay, what about over here? I need something between 2 and 4. How about I pick 3? So, I'm going to pick 3 and plug it in. Let's see what happens and all I care about is the sign. If you want to you can actually compute the numbers, but I don't do that. 3 + 3, well that's positive. 3 - 2, that's positive. So, I see a positive times a positive that's positive on the top.
If I put a 3 in here, 3 - 4, well that's actually negative, -1. So, I actually have a positive divided by a negative. That actually is a net gain of a negative. So, in fact, this region here is a negative region. By the way, you might be saying, "Gee, you know, we're just wasting a lot of web time here. It's clear that whatever it does here, its going to be this opposite, and keep switching back and forth." Do you know what? That's a great guess and a lot of times that's true but you know what? There are times where it's not. So, you really have to check it because this could be positive, negative, negative, positive, or something like that. So, just because you switch doesn't mean that these things are going to flip flop. So, just a little word of warning - don't think you can sort of jump ahead. You really want to take it carefully.
Okay, now I need something between -3 and 2. How about 0? So, let's plug in 0. Here I see 3, that's positive. Here I see -2, so that's negative. So, positive times a negative is a negative on top. If I plug in 0 on the bottom, I see -4. So, I see a negative divided by a negative. Negative divided by a negative is positive. So, here it does switch back to positive. So, I see a positive here. Then what happens out here? Well, now we can pick anything we want. Let's pick negative a billion. How about that? Let's do negative a trillion. Just to really drive this home. Negative a trillion plus 3, negative. Negative a trillion minus 2, very negative. So, I've got negative times a negative that's a positive. If it take a negative a trillion and subtract 4, I get something else that's negative. So, now I have a positive divided by a negative that's negative. So, I see negative regions all here.
So, now I've created my sign chart here. I can read off my answer. I'm looking for where this thing is greater than or equal to 0. So, where is it positive? Well, it's positive in this land, but it's also positive in this land. Am I allowed to equal 0? The answer is, yes I am, because I see that little equal sign is there. So, wherever I see a 0 I'm allowed to capture that. So, I capture that with these brackets. So, those brackets mean I'm allowed to equal 0. I should put the bracket here too, because it's 0 there too. You don't, because that makes this undefined. Remember, you promised me you will never touch that point. You will never touch 4. So, I put and open parenthesis like that, but everything else is okay.
So, that's the graphical answer. Graphical representation or you could write it in the interval notation, by sort of copying down what you see. So, for -3 to 2 in brackets, because I'm allowed to include those end points. Then we have from 4 out to infinity, not including 4 and of course we never include infinity. How do we bring them together? Well, it's one or the other. Since it's an "or" we're taking a union. So, you could write it like this, with a union sign there. So you can see, even with a very complicated inequality like this, if you just take your time, make sure you have a 0, factor quadratics, make sure you have one fraction, mark the 0's at the top. Carefully mark the 0's of the bottom as undefined and then go through each little segment and see what the sign is, you can read off your answer. Easy as pie.
Equations and Inequalities
Inequalities - Rationals and Radicals
Solving Rational Inequalities: Another Example Page [2 of 2]

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