Beg Algebra: Equations with 2 Absolute Values

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SOLVING EQUATIONS WITH TWO ABSOLUTE-VALUE EXPRESSIONS
So, let’s try a couple more examples of these equations with absolute values. Well, what I want to do is give you a chance to see what you do if you have an equation with two absolute values in it. For example, suppose I see 3 - 2x and that equals 5 - 4x. Now, how would you handle this? Well, all you have to remember is thinking about if two numbers are equal in absolute value, what does that mean? It means that they sort of must have the same size, but they just might be off by a negative sign. So, really again, even though they’re two absolute values they’re only two equations that are hiding, lurking in the background.
It’s either this thing equals that thing or this thing equals negative that thing. That’s all. So, even though they’re two absolute values, if you just think through what that means, right? If I say to you, “I’m thinking of two numbers and their absolute values are equal,” you know they’re either going to be the same number or they’re going to be one number and then its negative. So, again, just two equations to set up. One of them looks like this, 3 - 2x = 5 - 4x. Then the other one is a similar thing 3 - 2x = -(5 - 4x).
By the way, I think it’s kind of great and I hope that you can at least appreciate the fact that just by thinking through these things you can actually figure out sometimes what the solution is or how to proceed. Just by thinking. A lot of times people panic or just sort of thoroughly get disgusted. But if you can just sort of take it easy and think about what these things mean and not memorize the rules, you can actually make some progress. So, anyway, I like that feature about math, myself.
Let’s solve this now. I’m going bring the -4x over by making it a plus 4x. So, I’d have a plus 4x with this -2x, combined to give just the 2x and still have that plus 3 out there. I’ll write that on this side. Doesn’t make a difference how I write it. Since it’s a plus 3 I can just write it right here. Then on the other side I just have the 5. Remember I moved this over. If I bring the 3 over it becomes a -3 and I see 5 - 3 is 2. So, dividing through by 2, I see x = 1. So, there’s one answer.
Now, solving this thing. Let’s see, this looks to me like it would be 3 - 2x and then I have the -5 - 4x. Is that right? Why is it wrong? Well, again, I am making what I consider the number four on my top ten list of fantastic classic mistakes. Number four, the subtracting mistake. Don’t forget, when you’re subtracting a quantity you’ve got to subtract everybody. Remember folks share the negativity. So, that negative sign has to hit every single person. So, this is wrong. A negative times a negative. This should be a positive. Great mistake which I hope you’ll never make.
Anyway, now solving this I’m going to bring -4x and -2x is about -6x. Actually, would you permit me to bring the 3 over to this side now as well? Thank you. If I bring over that’s going to become a -3. So, I see -3 and -5 is around -8. Dividing both sides by -6, I see x equals a negative divided by a negative is a positive 8/6. Which if I cancel it becomes 4 over 3, so 4/3. So, I see this is another answer, x = 4/3. Again, you should go back, if you want, and check. For example, let’s just check the 1. If I put a 1 in here, I see 3 - 2, that’s 1, 1 is 1. If I put a 1 in here what do I see? I see 5 - 4(1) and 5 - 4 is 1, so 1 equals 1. If you were to plug the 4/3 back in here, what you would see is that these two numbers would be equal, but they would be off by a negative sign. So, when I take absolute values, I'm fine. You can check that if you want.
So, there’s a neat example of when you have two absolute values. The last example I want to look at together is one where you have fraction. So, suppose I have something like this, 3A - 4 divided by 2A + 3. Suppose that equals 1. In fact, I’m going to give you an opportunity to try this on your own. So, don’t get all upset or worried about the fact it’s a fraction. Just think about it as an absolute value equality. Think what that means. It means you’ve got to set up a couple of equations. See if you can set them up and see if you can solve them. I’ll give you a chance to do it right now.
Well, let’s see how you made out and let’s see how I make out. This just means that I have two equations. This thing here equals 1 or this thing here equals -1 and they both need to be solved separately. So, let’s try the first one. 3A - 4 over 2A + 3, that equals 1. Let’s solve that one first. The other one we have is right here so we don’t forget it, is going to be 3A - 4 divided by 2A + 3 and that equals -1. Well, let’s solve this one first. That’s going to actually require a number of little steps here. Let’s see if we can do some of them. So, what do I do first? © Thinkwell Corp.
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Well, I think what I’ll do here is cross multiply. That’s the best thing for me to do. This is 1 over 1. So, this top times 1 equals this bottom times 1. If you don’t like that, remember I could just multiply everything through by the denominator to a plus 3 and see the same thing. It would cancel here and appear here. In any case, I see that 3A - 4 = 2A + 3. Now, to solve this I’ll bring the 2A over, I subtract it. So, 3A - 2A becomes just an A. Actually, let me take the -4 and bring it over to this side as a plus 4 at the same time. So, plus 4 and 3 are 7. So, I see A = 7. So, A = 7 is the answer to that one.
Let’s now solve this one. How would this one look? Well, again, I’m just going to multiply everything through by 2A + 3. So, I see 3A - 4 equals negative times the quantity 2A + 3. Please remember, this is so important, to put those parentheses here. Because if you just put the negative sign in front of the 2A then you're going to forget to subtract that 3, which you really have to do. This is such a great mistake. I know I’m belaboring it, but I’m telling you everyone does this. So, really be careful and make sure you put everything in parentheses and remember to distribute that negative sign. It really will pay off.
Anyway, let’s see what happens here. I see a 3A - 4 = -2A and then when I distribute the negative sign, I get -3. Let’s solve this. I bring the -2A over. It becomes a positive 2A. 2A + 3A is 5A. If I take that minus 4 and bring it to this side as a positive 4, I see plus 4 - 3, which is 1. So, I see that A = 1/5. So, I see two answers, A = 7 and A = 1/5. Am I done? No, I’m not done. Because whenever I see an equation with a fraction, I always want to check my answer and make sure that for some reason I don’t have a 0 on the denominator anywhere.
So, let’s check our answers really fast. Let’s check this answer first. If I plug in a 7 here, 7 times 3 is 21 and 21 - 4 is what? Uh-oh, I have no idea. Is it like 17? Yeah, 17. So, 17 on top. What’s on the bottom? Well, on the bottom I have 7 times 2, which is 14, 14 plus 3 is happily 17. So, 17 over 17 is 1. The 1 is 1. So, this checks. This is okay. Now, 1/5 well, should I check 1/5? I guess I’ll check 1/5. Let’s plug it in and see what happens. I’ll have 3 times 1/5, which is 3/5 - 4 and I have to divide all that by 2/5 + 3. I hope this is worth it. This better be an answer.
So, what does that equal? Well, we have to get a common denominator. I’m going to do this really fast. The common denominator is going to be 5 or you could multiply top and bottom by 5. Maybe you’d rather do that. Let’s do that maybe. Multiply top and bottom by 5. It doesn’t change the value of anything because it’s 5 over 5, but when I distribute I see 3 - 5 times 4 is 30. On the bottom, I see just a 2 and when I distribute I get an extra 15. So what do I see here? I see a 3 - 20 is -17 divided by 17. Okay, well that equals -1, but remember I’m taking absolute values of this. So, I take absolute values of this, I take absolute values of this, I take absolute values of this, I take absolute values of this and it equals 1.
So, both answers are correct. No problem with it. We have the two answers. We have A = 1/5 and we have A = 7. We have our two answers and we checked them. So, that gives you a sense of how to proceed when you have absolute values or more than one absolute value. Just take it slowly. Think about what the absolute values mean remembering this equals one of two things. Either it equals the thing or it equals negative the thing. That’s the whole thing. See you soon.
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