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In each of the following exercises, solve the given inequality.$$\frac{x}{x-1} \geq \frac{3}{2}$$

$$1<x \leq 3$$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:07

In each of the following e…

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For the following exercise…

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short off wouldn't suffer our We've got a question here that gives us the inequality. X over X minus one is greater than or equal to three House. All right, so we're gonna solve for X dollars will start off with the denominator here. X minus one will set that equal to zero. And we know that case X would be equal to one. And then if we could solve for another X value by moving everything over to one side and, uh, setting equal to zero. So we have here, we'll start off by doing X is greater than equal to three halves multiplied by X minus one, which is the same thing as it is greater than or equal to three. Have X minus three halves. Subtract X from both sides of zero is greater than equal to half X minus negative minus three halves. All right, And then, if you divide everything by half, you'll get zero is greater than equal to X minus three. And if you set X minus 3 to 0, you would have x equal to three those of your two X values there. All right. And then you draw out a number line and you go from negative infinity too infinity. And you'll label your two X points as one and three the big values within this region to see whether or not it satisfies inequality. Uh, now we're gonna have a hollow one here because we know that this X equals one comes from the denominator where X minus one. We know if we actually plugged in a wondrous denominator, you would have a zero in the denominator which would lead to an undefined function. And then, if Extras three we know that that was that we could label as a solid circle. So first, to start off what's plugging a value between three and infinity? We could plug in four before over four minus one, which is the same thing is four thirds, and we know that four thirds is not greater than three halves. Therefore, we can confirm that this region does not qualify, does not satisfy the inequality pick a number between one and three, which is to so to over two minus one would just be equal to two. We know, too is greater than or equal to the rehabs. Therefore, this a region does satisfied inequality And finally we'll pick a number between one in negative infinity so we can choose zero on. We know zero. If we plugged in a zero for this extra then that would not be greater than equal to three house. And therefore, this does not. That region does not satisfied inequality. You picked your right out your extra mean by saying, whenever X is greater than one and less than or equal to three, inequality is satisfied. And the reason why we don't have a less greater than or equal to symbol is because here we have a hollow circle for for one. All right, well, I hope that clarifies the question. Thank you for watching.

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