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In each of the following exercises, solve the given inequality.$$(x+2)^{2}(x-3) \geq 0$$

$x=-2$ or $x \geq 3$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Campbell University

McMaster University

Baylor University

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In each of the following e…

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All right, We've got the inequality here. X plus two squared, multiplied by X plus, she's minus three, which is greater than or equal to zero. Our first thing we're gonna do is we're going to separate these into two different inequalities and just solve for X values as if this was an equal sense, what the experts to equal to heal. Then our X would be equal to negative two. And then for the second one, we have an expo minus three. And once again, um, the reason why I didn't put a square here because even if you took square root of this squared of zero, it would be equal to zero. So either way, it will be the same. Answer. Now, explain is three is equal to zero, so we could find our X to be three. All right, then what we're gonna do is we're gonna draw the number one, and we're gonna have our point negative to here. Uh, and we'll have a three here point right? And the objective is to find the region of X values to software inequality, and we would have a region here that goes from one of our X values to infinity the region in between our two X values and then the region from our lower X value to negative infinity. Okay, so what we're gonna do is we're gonna just choose a random value between three and infinity. And let's say we pick four and we'll put foreign into our total equation here. So we'll get four plus two square and a four minus three. And what we're looking to do is determine whether or not that answer comes out to be greater than or equal to zero. So we know a four plus 266 square to 36 36 multiplied by a one would be greater than or equal to zero. Therefore, we can confirm that this region here does work. So all X values in that region apply for this inequality to be true. Now, if we choose a region in between negative two and three, for example, if we choose, um, a let's let's pick a zero. Okay, If we pick a zero, then we have zero plus 22 squared and then we have a zero. Minus three is negative three. So we have four multiplied by negative. Three is equal to negative 12. This one was 36. So negative 12. We know it's not greater than equal to zero There for this region does not work. Alright, Finally, a number between negative two and negative infinity. We could pick negative three, maybe three plus two squared and a negative three minus three. So we have negative. Three plus two is negative. One negative one squared is a positive one Positive one multiplied by a negative six with negative six. Therefore, we can confirm that that region also does not work. Okay, However, we do want to make a note that this value does work cause we've already We've already tried plugging a negative too. And we know when we do that, we get a zero And since the inequality says greater than or equal to zero, we will include our negative too. So we'll say, Well, then, our total region of values for X would be X is gonna be greater than or equal to three. And if that's the case, that all then our inequality will be true. But also if X is the ant symbol here, X is equal to negative two. It also So this would be the only case where X is less than three. And it does work. We do have, like, one anomaly value here. Otherwise, the majority of your ex domain lies when X is greater than three. All right, well, I hope that clarifies the question there. Thank you so much for

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