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In each of the following exercises, solve the given inequality.$$\frac{5 x^{2}(4 x-7)(2 x+9)^{3}}{(x-2)^{5}(2 x+1)^{6}} \leq 0$$

$x<-9 / 2$ or $7 / 4 \leq x<2$ or $x=0$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Oregon State University

McMaster University

Idaho State University

Lectures

02:25

In each of the following e…

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03:51

04:23

03:18

So if we want to solve for this inequality, the first thing we need to do is determined. Where are numerous and denominator are both going to be zero. So I'll just go ahead and do that really quickly. So X squared would be zero at 04 x minus seven would be at seven horse two weeks plus nine would be a negative nine over to, uh, X minus two to the fifth. That would just be a two and then two x plus one. That would be a negative one half. Okay, so let's first go ahead and create a sign chart. With all of these eso, we'd have negative nine half negative half zero, uh, seven forces a little bit smaller than two. Yeah, So now what I'm going to do is I don't really set my sign tarps up a little bit differently. I feel than a lot of people do, because instead of just, like taking some number and then plugging it in, I go ahead and right all the factors going down the side here and then I figured pretty much do a sign chart for each of the factors. And then I multiply the result together and then put it along this, like top right here. And I find doing it that way, at least for me. Just makes things a little bit easier. Um, but, I mean, you could do it either way and you get the same answer, right? So let's see. So the first thing we're going to have, so it's five square. So go ahead and make a road there. Then we have four explains seven two x plus nine cubed. We have X minus two to the fifth. And then lastly, we will have two X plus one raise to sixth. Right? So the first thing I'm going to do since we want this to be equal to zero as well, I'm just gonna put a solid dot above where the numerator is zero. So that X is equal to 00 um, at seven. Force zero. And then at negative nine halfs, it's zero. So that way, I just don't happen to forget about it. Yeah. Now, the first thing I kind of wanna make note of is knows. No matter what number, we plug into two x plus one to the six and for raising it to an even power as well as five x squared. Since we're plugging these into even powers, those outputs will always be positive or zero. So we just go ahead and put positives going straight across both of these rows here and then that means we really only need to find the signs for these middle three. So let's go ahead and do that. So if we take some numbers smarter than negative nine have, say, like, negative 10 and blood this into these. So the forex my seven would be negative. Two x plus nine is going to be negative. And if I cubit, it's not really going to change anything s O that will still be negative. Um, explains to that will also be negative. And now we go ahead and multiply. All of these are so multiply these altogether and that would be so. Two positives and three negatives is a negative. So that's telling us to the left of negative nine halfs are functions negative? Now we come up here, repeat the same thing, but for some number between negative night house and negative one half, so like negative one will work. Eso forex. My sevens negative two X plus nine. That is going to be positive X minus two. That would be negative. So now we go ahead, multiply all these together. But the result appear. So three positives. Two negatives that multiplies for positive. Now, from negative one half to zero, we could do like 1/4. So if we plug in 1/4 here, so that would be negative. Still, that would be positive, and that would be negative. Then we go ahead. Multiple these together. Put them on top here. Eso three positives. Two negatives. That's still positive now. Some number between zero and seven force, so we could do like one. So if I plug one into forex, my seven, that's still negative. That would be positive. This would be negative. So multiply these together. So this is still going to give me something positive. Then we go ahead and plug it a number between seven Force and two so we could do like 1.9 eso. If I plug 1.9 into forex. My seven. It's slightly above, um, zero. So that's positive, um, to expose Stein that would be positive expires too. Well, that would be negative And so then if we multiply all these together, well, it's a bunch of positive and one negative, so that's negative. And then if I take to and plug it in now, all of these air going to be positive, so everything is positive and then we could go ahead by these together and then I'll give positive. So now we know what the sign is going to be for. Our function affects on all these intervals, and now we just need to find, well, where is this less than zero. So where is it? Negative end. So that's going to be to the left of negative night house. And then it's positive, positive, positive events, negative, beginning from seven, forced to to. But notice to we exclude, since it would be zero in the denominator which son to find. So we use an open circular so this would be our solution, which would be so Let's write that off on the side here so it would be negative infinity to negative nine house. But since it's a solid dot, we use a bracket to include a union, then we have zero here so you could write like bracket 0 to 0 bracket, but that's a little weird to right. So what I'm gonna do instead is just use thesis Aetna Tae Shin too, right? Zero. So it's just curly brackets and zero. And then lastly, we would have So we include seven force, and then we exclude too. So we use a parentheses there. And so then this here will be our solution. So again, it doesn't really matter. Um, if you just take the numbers and plug it into their directly once you get this sign chart set up or do it the way I do it here again, I just like doing it this way because I don't have to think it's hard I feel to do this might be a little bit more steps, but yeah, I just don't have to think it's hard to do it

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