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In each of the following exercises, solve the given inequality.$$x^{2}-4 x-41 \leq 0$$

$$2-3 \sqrt{5} \leq x \leq 2+3 \sqrt{5}$$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Missouri State University

McMaster University

University of Michigan - Ann Arbor

Lectures

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In each of the following e…

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Solve each inequality.…

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Solve each inequality.

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gives us the inequality. X squared minus four X minus 41 is less than or equal to zero. So first, excuse me, first thing we want to do is take this polynomial factor it. We could see it's not a simple factors. We're gonna plug it into the quadratic formula, and we will get Tu minus three, Route five and two plus three route. From there, we're gonna draw out a number line that goes from negative infinity to Infinity label our two X values. And we know that we're gonna have solid one cheer because it actually does include because it says less than or equal to so we can have solid lines there. Now, what we're gonna do is we're gonna pick values in between the regions aan den. We're gonna plug them into our equation and see if that that region of X values will satisfy the inequality. So let's pick a value between two plus 35 and eight. We could pick nine, So nine squared minus four times nine minus 41 gives us four, and we can see that four is certainly not less than zero or even equal to zero. So we can say Well, this this region here does not satisfy the inequality. Pick a number between two minus 35 and two plus 35 We can pick zero. So we do. Zero minus. We'll do zero minus zero minus 41. And we know negative 41 is less than zero. So we could say that the values do exist within this region. Finally, a value between two minus 25 and negative infinity. We could pick negative nine. So we'll get a negative nine squared multiplied by four minus of four, multiplied by negative nine in the minus 41. He was US 76. And we know that that's also gonna be larger than zero. Then this region is not going to satisfy the inequality. So then finally, when you write out your domain, your ex domain, you're gonna say, Well, X is greater than were equal to two minus three, Route five and less than or equal to two plus three, Route five. And whenever the values air within this region here you're Inequality is satisfied. All right, well, I hope that clarifies the question there. Thank you so much for watching

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