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In each of the following exercises, solve the given inequality.$$x(2 x-5)<0$$

$$0<x<5 / 2$$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Missouri State University

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:23

In each of the following e…

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03:36

01:16

04:23

02:40

02:30

02:25

04:32

03:13

05:55

03:26

03:07

03:43

03:12

03:30

04:05

02:45

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All right, We've got some inequalities here. Where we have X multiplied by two X minus five is less than zero. Okay. And we're gonna use the number line system to help us find the regions of X. For this inequality be true. And the way we do that is we solve for R values of experts by setting this equal to zero and just looking at it, we know. Well, then this would be true when X is equal to zero and we know it would also equal to zero if we set to X minus five equal to zero. Right, Because if this is zero, the whole answer would be zero. And if this is zero, the whole answer zero. So you have X is equal to five over chips, right? These right here will be our X values. And what we'll do is we'll draw out a number line. We'll put zero here, and we'll put five over to here. We have infinity here in negative infinity. And then what we do is we way. Look at our three regions. We have a region here from five over to infinity. So one of our X values to infinity and then the region between our two X values and then finally the region from our lower X value to negative infinity. And what we do is we just choose any values between those regions and if it's true, than that region does exist. So let's start off by this region here. So we'll choose a value greater than five or two in less than infinity, so we can choose three. So we'll do. We'll plug that into this inequality here so we'll have three multiplied by two times three minus five and we're gonna solve for this and we'll get six minus five multiplied by three, which would just be three. Now it's three less than zero and we can see that it is most certainly not less than zero. Therefore, this region does not exist. Alright, Now we look at our second region which is any number from 0 to 5 or two and we'll pick two will do to multiplied by two times two minus five and we'll get four minus five is negative. One negative one times too negative too. So negative two is less than zero. Therefore this region does exist, so we know our X values will be anywhere between zero and five to now. Finally, let's check our our last region here, which is from zero to negative infinity. So we'll pick a negative one, We'll say. Well, negative one multiplied by two times a negative one minus five, and so have negative two minus five and negative seven. Negative seven multiplied by negative one, which gives us seven. That's a positive number, and that's not less than zero. Therefore, this region also does not exist, so our X values will then exist when X is greater than zero. Yeah, and when X is less than five, Richie. And that's how we use the number line system to determine our, uh, X values and to solve for inequality. All right, well, I hope that clarifies the question there. Thank you so much for watching

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