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Brianna said that $\frac{3}{x-2}=\frac{5}{x+2}$ is a rational equation but $\frac{x-2}{3}=\frac{x+2}{5}$ is not. Do you agree with Brianna? Explain why or why not.

Yes

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 7

Solving Rational Equations

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

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Hey, guys. So in this problem were given two sets off. Rational equations are 1st 1 is three over X minus two equals five over experts to and our second equation is X minus two over three equals expose two or five and other problems. States that the student named Briana considers this first set this for a set of equations of rational equation. She considers it a rational equation. However, she does not consider the second equation irrational equation and the textbook and success to, um, verified. Briana is correct, Incorrect and explain why. And in order to understand if she's correct or incorrect, we need to understand the definition of a rational equation and a rational function. Rational equations are basically any equations I'm gonna reviewed here. Rational equations, basically, or any equation when you have two rational functions said equal to each other. And in order to understand what rational functions are, I'm gonna explain that a bit rational functions functions are any functions such that there's a polynomial in either the numerator, the denominator or both. Meaning rational functions are of this four p of X over Q of X. And let's take a look at each of these ah functions in both of these equations. Three over X minus two. The reason. Constant function. It's a constant polynomial, so it is a polynomial X minus two. Linear. It's a linear function. It's a polynomial. Similarly, we see the F over five over exposed to sorry is also each each the numerator and denominator of both polynomial tze and weaken Tell that these two these two functions are rational function similarly over here X minus two. Now the function the polynomial is in the numerator still makes it about rational function are exposed to over five also a rational function because you have a polynomial in the numerator and the denominator. And so in both of these equations we have rational functions being set equal to each other. So in reality, both of these equations are indeed valid. Rational equations and Briana is incorrect in Mackey's of Briana is incorrect. Rihanna is incorrect. She's incorrect, that only the 1st 1 is a rational equation because in reality, taking a look at the definition of rational equations and rational functions, we see that both functions are rash boat by functions. I mean both equations, our rational equations. So I hope that answered the problem and thanks for listening us

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