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What is the solution set of $\frac{|x|}{x}<0 ?$ Justify your answer.

$\{x | x<0\}$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 8

Solving Rational Inequalities

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Campbell University

Oregon State University

McMaster University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

00:53

Determine the solution set…

01:03

Describe the solution set …

00:18

Solve.$$|x|=0$$

00:58

Which of the following has…

00:28

The solution set of the in…

01:22

00:59

The inequality $|x|>0$ …

00:22

Solve.$$(x-1)(x+4)<…

00:47

Explain why $|x|<-4$ ha…

00:27

Graph the solution set.

all right, guys. So when this problem, we're given the inequality. Absolute value of X divided by X, is less than zero. And the problem asks us to find the solution set for this inequality. And now, in order to solve this inequality, we need to split this up into two cases. One where what work is in a different color pen? One where X is positive. X is greater than 01 where X is less than zero and one where X is equal to zero. So let's start with our first case. X is positive if excess positive, it is a positive value. So let's say ax is equal to a A is this is this random positive into cheer positive rial number in the sense now X is equal to a We can plug in a for acts in our inequality and we have that absolute value A a over a less than zero absolute value of a is just a because we know that is already positive. So we have a divided by a is less than zero and we have one is less than zero. And right here we see that Oh, basically, we see that this is incorrect because one is not less than zero. So we know that values of X that are positive do not work for this solution Set. Now let's try our second. Ah, second option X is less than zero. Basically, ex less than zero means that access of the form acts is equal to negative A where a is just a just any not any positive number. It's just easy to write the right accent like this just to explicitly show you guys that it's a negative number. And now we plug in our negative. A four ex were in get his absolute value of negative, eh? Divided by negative, eh? And that's gonna be we're gonna say that's less than zero. Let's try to verify this. We know that the absolute value of negative A is just a So let's write that better, that the absolute value of negative A is just a and we have negative and the denominator and what we're seeing if we're saying that's less than zero, and we know that a divided by negative A is negative one and negative one is less than zero. So this is always true. Show this ex less than zero is a part of the solution set to this equation. And now let's try. Our lasts are Let's try our last about last. Basically condition for X X is equal to zero. And if X is equal to zero, we get absolute value of zero divided by zero. We were saying, that's less than zero automatically. This is a zero, by the way, we know that this is undefined does not exist, so X equals to zero is not a part of the solution set either. And as we can see, the only thing that's a part of this solution set. Basically, the answer to this inequality is when X is less than zero. That is the solution set to this to the equation. Absolute value of X divided by X, is less than zero. Huh? Thanks for listening, guys. And I hope this problem made sense, and I hope this helps solve it.

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