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In $3-20,$ solve each equation and check.$$\frac{4}{y+2}=1-\frac{8}{y(y+2)}$$

$\{4\}$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 7

Solving Rational Equations

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Missouri State University

Campbell University

McMaster University

Harvey Mudd College

Lectures

01:32

In mathematics, the absolu…

01:11

02:08

Solve each equation.

01:14

Solve each equation.$$…

00:49

Solve each equation. Check…

00:43

Solve each equation and ch…

01:45

01:06

Solve the system of equati…

02:18

Solve each system using th…

00:51

02:26

01:12

Solve each system.$$\b…

in order to solve this problem out for. Why, while we're gonna need to do is we're gonna need to multiply the entire equation by whatever a common denominator would be for this problem. In order to cancel out all of our denominators. Looking at what I've got here, I have denominator of y plus two and I have a denominator of why times y plus two. While for our common denominator, it needs to be something that every problem can multiply up to meaning. I don't want to be missing anything. So I need to be multiplying by both y and y plus two because both dominators have a y plus two in common. But since one of them as a why I have to account for the most amount of stuff. So we need to multiply by a white times y plus two. Okay, if we're gonna do that, we have to multiply every single term by that. So if I take why times y plus two times my first fraction. A four over Why? Plus two? Well, the why plus twos would cancel out, leaving me with a four. And why, If I take one times why why plus two? Well, one times anything is just that number. So I get both the why and the why plus two. Not much going there for my last fraction. If I take why Times y plus two and multiply that times eight over why towns lifeless too Well, there's a y y plus two on top, and there's a y Y plus two on the bottom so they would cancel, leaving us with simply eight. I thought I changed. Now that we have cancelled out all of our denominators, we can actually go about trying to solve this problem out. I do see that I have a distributive property to do here. So we rewrite the four y and for why times, Why would be why squared why times To would be to Why? And we have the minus eight. So now I'm noticing that I have a y squared, which means we have a quadratic equation that we're trying to solve here. So we're no longer going to try to get all the wise on one side and all the constants on the other. Now that it's a quadratic, we want to get everything on the same side because if we get the equation equal to zero, then we can either try and factor this or we can try and use the quadratic formula, Whatever your personal preference would be. Meaning? I'm actually going to subtract this for why, To the other side, giving us zero equals Why squared two Y minus four. Why would be negative to why and minus eight. Now you have your choice of whether you would prefer to factor this or whether you would prefer to use the quadratic formula. It truly is up to you. I guess you could also complete the square if you wanted to, but I think most people prefer to either factories the quadratic formula cause those usually seem to take less time than completing the square, but you can choose what you want to do with that. Either way, you should end up with two answers off of this, and then we'll have to plug them back in to see which ones work. If you go to factor this two factor this you would be looking for two numbers that multiply too negative. Eight, but add to negative two. We talked about this in the sections previous to this, um, two numbers that multiply Negative eight, but add to negative two would be negative four. And positive, too, since the coefficient in front of the Y squared is just a one I don't have any more work to do. We simply have zero equals. Why, minus four as a factor and why, plus two as a factor. This has been fully factored. Now I can move on to the final step to solve of actually solving out for the number zero product, Property says. Since these two factors air being multiplied to each other and the equation is said equals zero, I am allowed to take each of these factors and set them equal to zero separately. So for the first equation, I'd add for to the other side and you get why equals positive four. For the second equation, I would subtract two to the other side and get why equals negative, too. So right now it looks like your answers are Y equals four and y equals negative, too. But before we say for certain that those are both our final answers, we need to take a look at the original problem, which was four over why plus two equals one minus eight over wide times y plus two. We need to plug both four and negative to infer wine at the same time. Some different times separate several times. If either of them plugs in and causes us to get zero in one of our denominators, that would cause the entire fraction to be undefined, which means that number is not actually a solution. We didn't do anything wrong, Algebraic Lee. We still solved out for the correct numbers. It's just they may not actually work when we actually plug back in. So let's start with 45 Plug in four. I'd have four over four plus two equals one minus eight over four times the quantity of four plus two. Well, four plus two is six that checks out fine. Four plus two over here is also six and four times six would give me 24 on this one, neither of which is zero. This problem checks out. Four is definitely one of our answers. Now let's try the exact same thing for if we plug in negative, too. That means we'd have four over negative Two plus two equals one minus eight over negative, too. Times negative, too. Plus two. Well, you may have already noticed it, but I can see for my very first fraction negative two plus two is zero four. Divided by zero is undefined. Which means it doesn't even matter what I have for the rest. This problem once one part of it's undefined. The whole thing is undefined. Which means while we did solve out for negative to correctly when he plugged back in, we realised that negative to does not actually work. Meaning are Onley Answer for this question is positive for

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