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In each of the following exercises, solve the given inequality.$$\frac{x^{2}-16}{x^{2}-25} \leq \frac{20}{11}$$

$x \leq-6$ or $-5<x<5$ or $x \geq 6$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:56

In each of the following e…

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01:23

03:27

01:16

03:13

02:25

03:26

03:31

03:43

03:51

04:32

01:43

In Exercises $17-26,$ solv…

03:07

07:30

03:18

02:16

04:02

04:23

so, whatever, I'm trying to solve a rational expression or rational inequality. What I like to do is just go ahead and move everything over toe one side. I don't like to try to, like, multiply and divide or do anything like that because a lot of times you're unsure if, like, if something, it's positive or negative, depending on the values for X, and I just normally find it a lot easier to move everything over. Um, so let's go ahead and do that first. So I'm going to subtract 31st on each side. Over here, they just cancel out. We get zero and then on the left side. Well, we could go ahead and get a common denominator. I didn't add everything up. So this negative 20/11 if we multiply by X squared minus 25 would give negative 20 X squared and then 20 times 25 or negative 20 times negative 25 I should say is plus 500. And then we multiply the next one by 11. That would be plus 11 x squared and then 16 times 11 is 1 76. That'll be negative. 1 76 and then this will be all over 11 X squared minus 25. Less than or equal to zero. Yeah. Now we go ahead and combined are like terms of the numerator s. So we could go ahead and combine these x squared. So that would be negative. Nine x squared that we could combine. 501 76 500 minus 1. 76 is 3 24. So plus 3 24 and then the denominator that still be all over 11 x squared itis 25. Listen very good to zero. All right, now the numerator We could go ahead and try to factor this, eh? So what I'm gonna do is I'm gonna factor out a negative nine from the numerator. And doing that would give some negative nine x squared and then 3 24 divided by 9. 36. So this would be minus 36 all over 11 x squared, minus 25. Less than or equal to zero. Yeah. Now let's go ahead and multiply each side by negative 11/9 just to kind of get thes thio. Cancel with each other. And remember, if we do this, then our sign here is going to flip, so that will be greater than or equal to zero. And now we would just be left with X squared, Uh, minus 36 over expert by 25. So I'm going to go ahead and, uh, factor those because it's just a difference of squares. And remember the difference of squares. Anytime you see something where it's like X squared minus a squared, this is just going to be equal to X y se x plus e. So in the numerator, 36 is six square, So this would be X by nus six x plus six all over X minus five x plus five like that. And in this form, we now can go ahead instead of a sign chart. Um, I set mine up a little bit different than I feel most people do. So all kind of walk through what I would have. So first we need to figure out where is the numerator and denominator from zero. So this first one is six. This is negative. Six. This is five and this is negative five. Then what I do just for kind of simplicity. I write down my factor terms going down this vertical charts with the X by six x plus six x minus five X plus five Um, and then I put going out like that and then up top here. So I'm gonna call this left side just f of X. So, like all of these kind of like multiplied and divide is ffx. And now let's put some of these doubts. So the number for this is gonna be negative. Six. Then we have negative five. Then we have five, and then we have six. So this top row is going to be once we've multiplied and divided all of these here. So what? All I'm going to do is first figure out is X minus six X plus six X minus five and X plus five women to be negative or positive for numbers less than negative six. So just pick some arbitrary number like, let's say number seven, um, plugged that in, so that would be a negative. Still, this would be negative. This would be negative. This would be negative. And now I take all of these, multiply them together and I put the result of top here, so four negatives will fly together, it would be a positive. Okay, um, and one thing I should probably also do, since we want this to be equal to zero as well. Remember, the only time it could be equal is in the numerator. So at six. And negative or six and negative six, we should have, like, closed circles for this. I'm just gonna kind of right that above Good. Um, Now we just go ahead, do the next one. But now for a number between negative six and negative five eso like negative 5.5. So this is still going to be negative. This is now positive. This is still negative, and that's still negative. And if we multiply all these together well, three negative and one positive make a negative. Now we go ahead and do the same thing. But for this middle row, eso we just fix the number between negative five and five. Easiest zero. So this is negative. Positive, negative. Positive. Multiply these together. Two negatives to positives is positive. Um, now for this next one, we have number 25 605.5 is a good one. So that's negative. And then the rest of these are going to be positive. Multiply all these together and then that would become negative. And then lastly, some number of larger than six like seven toe plug that in over here Positive, positive, positive, Positive. Multiply all these together and we get a positive. So the intervals we're going to be interested in, um are where it's positive or zero. So we said it was zero and negative six or six. So that would be going to the left of negative six between negative 5 to 5 and then to the right of six. Um, and then we can go ahead and actually right, This is like an interval notation. So I'll go ahead and write it using the interval notation off on this side here. So our solution is going to be well, this first one is going to be from negative infinity to negative six. But since we're including it, we use a bracket, so actually may expand the screen a little bit, and then we use our union, um then negative 5 to 5. But we exclude those will be used a parenthesis or negative 5 to 5, uh, union and then six and to the right, but we're including six will use the brackets of six to infinity. So this is one way we can write the answer. Another valid way is to just write in the inequality notation. So I just kind of write this below in green. But you could do it either way, and it doesn't really matter which way. You just kind of which way you prefer s Oh, this is just gonna be x. Not greater. Uh, X greater than or equal to negative six. Then here, this would be X. Listen, you go to five. Not less than strictly less than five less than negative five. I should be saying or in between these and then over here, we have six less than or equal to X like that. And so, actually, I got the inequality mixed up over here. That should be smaller than negative six. Not larger. Yeah, over here. Axes bigger. Yeah. So, um and I prefer doing the interval notation because sometimes I get the science kind of flipped around like I did there. But, I mean, you can put it in either way you want, and it will still be a ballot answer

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