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This is part three of how to solve absolute value equations. In parts one and two, we
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found out the way to solve the absolute value of something equals k, where k is greater
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than zero, is to solve these two equations, stuff equals k, and stuff equals negative k.
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Now, by the way, you never see a written as stuff when you are looking in a regular
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textbook, they will usually write it more formally and say, okay, absolute value equals k can
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be solved by writing x equals k or x equals negative k, but I like to just use stuff or
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junk or something like that, expression in the absolute value to get the idea to everybody.
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So these are basically saying the same thing, but now we are going to look at the example
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where k is not greater than zero. Alright, look at this example. Absolute value of 2x
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minus 3 is equal to negative 9. Now, notice this time, you have an absolute value equaling
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to a negative number. That's impossible, because the absolute value of something is greater
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than or equal to zero because we are talking about distance. So this is a big warning. You
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can't solve this. You cannot have an absolute value equaling to a negative number, so stop
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and just write no solution. It looks like a V. Okay, that is the answer. You don't have
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to do anything. Now, if you didn't notice this and you wrote 2x minus 3 equals 9 or 2x
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minus 3 equals negative 9 and you wouldn't solve this bad equation, when you checked it,
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you would hopefully find out it will never give you a solution. So the quick way is to
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stop right there. So what about if you had 3x squared minus 5x plus 2 plus 4 equals
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0? Well, notice that the absolute value is not isolated on one side of the equation. So
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first we are going to have to subtract 4 from both sides. So we have this and there it
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is again, you can't have the absolute value of something equaling to a negative number,
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so again you could stop right there and write no solution. Pretty easy, right? All right,
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let's look at the second kind of a problem. What if you had 2x minus 6 in the absolute
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value equaling 0? How would you solve this? Well, the absolute value of what will give
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you 0? There's only one thing, there's only one number that 0 spaces away from 0 and
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that's the number 0 itself. So the part in here has to equal 0 actually. So anytime you
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have an absolute value equaling to 0, you could just write whatever is in the absolute value
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equal to 0 and solve that. And that's easy to solve, we'll just add 6 to both sides and
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then divide by 2 and that's easy to check. Let's go ahead and just check it. We take the
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original problem and we're going to plug in 3 for x. So we have 2 times 3 minus 6 in the
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absolute value and we're going to simplify inside the absolute value until we get a simple
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number which is 0 and then the absolute value is 0 and we've got 0 on the other side and
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there you go. So for this problem, x equals 3 is the solution. So let's summarize. All right,
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so we've got 3 cases. How we would solve the absolute value of some stuff is equal to
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k. Either k is a negative number, that's when it's less than 0. Automatically, there will
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be no solution. The second case, k is actually equal to 0 in which case the stuff in the
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absolute value has to equal 0. Or the third case, which is the first case we actually worked
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on, is if k is greater than 0, you're going to solve these two equations. You're going
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to take the stuff in the absolute value, you take off the absolute value science, right?
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To write this. So the stuff equals k or the stuff equals negative k. All right, so notice
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that in all of these cases, you have the absolute value sign isolated on one side of the equation
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equaling to some number. All right, here's an example. Solve the absolute value of 2x minus
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5 plus 3 equals 9. Now I'm going to show you a classic mistake. And then I'm going to
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ask you what I did wrong, all right? So I'm going to write 2x minus 5 plus 3 equals 9,
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or 2x minus 5 plus 3 equals negative 9. Can you figure out the mistake here? The mistake
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is it wasn't in one of the forms I just gave you where the absolute value is isolated
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on one side of the equation. So you cannot just say everything on the left equals 9 or
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everything on the left equals negative 9. That's classic mistake number 1. Now let's just
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see why this isn't going to work. Let's go through and solve the one on the right. So
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we have 2x minus 2 equals negative 9. Then we're going to add 2 to both sides to get 2x equals
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negative 7, divide both sides by 2, and x equals negative 7 halves. So we're going to just check
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negative 7 halves into 2x minus 5 in the absolute value sign equals 9. So let's do that. We're
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going to put 2 times negative 7 halves plus 5. That all goes in the absolute value sign.
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And then inside the absolute value sign, we can cancel the 2's to get negative 7 plus
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5. And then in the absolute value sign, we get negative 2, right? And then what's the
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absolute value of negative 2? That's 2. And what's 2 plus 3? 5. And is that equal 9? No.
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So hopefully that convinces you that that didn't quite work out. Just to write 2x minus
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5 plus 3 equals 9 or 2x minus 5 plus 3 equals negative 9. Right. So that's classic example
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number 1. If you forget to isolate your absolute value sign, you're probably going to have
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some trouble. Let's look at another classic example. Actually not a classic example. Let's
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look at another classic mistake. So this is classic mistake number 2. Classic mistake
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number 2 is you might start off by isolating your absolute value sign, which is great by
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subtracting 3 from both sides. Okay. So in other words, go ahead and subtract 3 from both
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sides so that you would get this equation. Right. Now everything's fine. That's okay
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to hear. Okay. So far. And then you solve this one. 2x minus 5 equals 6. And add 5 both
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sides. Get 2x equals 11. Get x equals 11. How's when you divide by 2? And then a lot of
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people say, oh, so the answer is 11 halves and negative 11 halves. Okay. The mistake here
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is you only solved one equation and just put a minus sign. And for the answer, there's
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your problem. You can't do that. You have to solve two equations. You can't just solve
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one equation and then just put a minus sign in front of the other answer. Now we're going
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to show you why this does not work. Alright. Let's say you're trying to check this.
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We're going to just check that x equals negative 11 halves. So we have 2 times negative 11
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halves minus 5. And simplifying inside the absolute value, that gives you a negative 11
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minus 5. And negative 11 minus 5 is negative 16. And the absolute value of negative 16 is
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16. Put on the right hand side, you got 6. No. So this is not correct. That is not the
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correct answer. So those are two classic mistakes that you want to be careful of. Alright.
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Let's do one more example. How about we do this problem right? We're going to isolate
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the absolute value first to get absolute value of 2x minus 5 equals 6. And then we break
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it up into two equations to solve since the number on the right is positive. So 2x minus
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5 equals 6 or 2x minus 5 equals negative 6. Those are the two equations to solve. I don't
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have time on this video to solve them, but this is what you should get. X was 11 halves
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or X equals negative 1 half. And then you should be able to check both of these in the
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original absolute value equation above. Alright. Go on to the next video and we'll work
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on some more problems and examples.