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In each of the following exercises, solve the given inequality.$$(2 x+3)(3 x-5)(x+1) \leq 0$$

$$x \leq-3 / 2 \text { or }-1 \leq x \leq 5 / 3$$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Missouri State University

McMaster University

Idaho State University

Lectures

02:30

In each of the following e…

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03:41

05:39

04:23

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01:50

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03:31

02:42

04:15

01:54

Solve each inequality alge…

01:56

Solve each of the followin…

don't solve. But all right, we've got a question here. That is two x waas three multiplied by a three x minus five. And then I expressed one and all of that. It's less than or equal to you. Well, all right. In the way we do this, we saw first for X values. By setting this instead to be equal to zero, we take each parentheses and solve them individually. So we say two X plus three each factor. I should say we sent me cookies when we saw for our X here, and we'll get excess equal to a negative three over to you do the same thing for the next factor we have X is equal to 5/3. And finally, for our last factor, we have experts. One is equal to zero, so x is equal to negative more. All right, then we take a number line and we write out our negative one here. You've got a negative three halfs here, and then we'll have a 5. 30. Basically, we're trying to find the region of X values, the domain of exercise, for when this inequality is true. So we're gonna look at all of the regions, for example, five thirds infinity. And in the region between here in the region between here, eso fourth to determine whether or not those X values do exist in that region, that inequality is true, so we'll pick a value for five thirds and infinity, for example, which is to it's also too. Times two plus three, three times two minus five, the two plus one. Then we're just plugging into here we have a four plus three x seven seven multiplied by a one year and then seven, multiplied by a three would give us 20. We know 21 is certainly not less than or equal to zero. Therefore, we can confirm that the values in this region do not exist. They exist at five. Thursday, exist at negative one. Inequality is true at negative one. The inequalities true, five Thursday inequalities, Truth negative we have on. What we're doing is we're trying to determine whether it's true within the region's eso. It's not true in this region, so let's try to find a value in this region. See whether or not it's true. So let's pick zero in between these two values here and we'll get a three multiplied by negative five once applied by a positive one, and we'll get negative 15. We know negative 15 is certainly less than equal to zero before we can confirm that it does exist in this region. Okay, now, finally, for the region between negative one and negative one half, we can pick a negative 1.25 So we could say, um, let's take five ports. Or a matter of fact, I'll just keep it in decimals because I've got a calculator here. So all too negative, 1.25 multiplied by two was three in a three multiplied by a negative one point, uh, minus five. And then finally, a negative 1.25 What's more, I'm gonna use my handy dandy calculator here. So we've got two times negative. 1.25 was to be. It's just half okay? And then a three times a negative 1.25 Last five, it was a negative 8.75 and then finally, a negative 1.25 plus one is negative 0.25 and it really doesn't even matter what the answer comes out. We could just see, we have to. Negative. So it will be a positive value here. We know a positive value certainly will not be less than 20 So it doesn't exist in this region either. Finally, this truth value between negative 23 halfs and and negative infinity. So we'll choose negative to have two times negative too. Waas three a three times negative too minus five and a negative two plus one. So I have a negative four plus three, which is negative one. And then we'll have a negative six minus by which would be a negative 11 and then finally negative two plus one is negative one. We've got three negatives here, so we know we're gonna come out with a negative. Therefore, we can confirm that it does exist in this rich. All right? And then when we write out, our ex statement here will say that are inequality is true for one X is greater than or equal to negative one in less than or equal to five thirds where this region here and then it is also ah, valid when excess less than or equal to negative. Three house. Okay. Yeah. It exists in that region as well. All right. And we have negative 1 to 5 thirds, and then we have anything less than negative. Three house. All right, well, I hope that clarifies the question there. Thank you so much for watching.

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