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We're going to go over how to solve absolute value equations.
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First here's the definition for absolute value.
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The absolute value of x represents the distance from x to zero on the number line.
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Alright, so x is some number, let's say.
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So let's say we wanted to get the absolute value of negative two.
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So here's negative two right here, and if we just count the spaces to zero, that's our
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answer.
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It's two spaces away, so the absolute value of negative two is two.
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Alright, what if we wanted to do the absolute value of positive two?
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Well here's positive two, and if we count how far away two is from zero, hey it's also
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two spaces away.
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So the absolute value of two is also two.
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Now since we're talking about distance, keep in mind that you can't get a negative number.
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Right?
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So if I count spaces, you can't count a negative, you can't say that's negative.
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So the absolute value cannot be negative, but could it be zero?
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What's the absolute value of zero?
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Well, here's zero.
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It's right on itself, and so how far is zero from zero?
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It's zero.
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So keep in mind the absolute value of something cannot be negative.
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Alright, and here's the next interesting thing.
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Notice that when I took the absolute value of negative two, I got two, and when I took
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the absolute value of positive two, I also got two.
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So what if an equation had been solved, the absolute value of x equals two?
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There's a couple ways you might think about this.
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One is you could just think of a number line, and you could say alright, I want x to be two
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spaces away from zero, because the absolute value of x means how far is x from zero?
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So if I go two spaces away from zero to the right, I'll get an answer, and two spaces
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to the left.
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Two spaces away.
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Give me two, also gives me a negative two.
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So that's one way of figuring out that there are two solutions here.
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So if I were going to solve this, I'm going to have two solutions by looking at the picture.
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x could be negative two, or x could be positive two.
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Another way you might remember, my note up here, the absolute value of negative two is
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two, and also the absolute value of two is two, you might just remember that.
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Well there's always two numbers that when you take the absolute value you get two.
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Another way you might think about it is this way.
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Whatever is inside here, which in case that could either be two, or it could be negative
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two, right?
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So the one that I take the absolute value I get two.
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So whatever is in here could be what's over on the right hand side, which is two, or
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it could be its opposite, which could be negative two.
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And this is how you would write that down, whatever is in there, which is x, it could either
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just equal the two, or it could be the opposite of two to make it true.
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Now that's the correct solution.
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It could be two, or it could be negative two, just like I had written over here by looking
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at the picture.
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But we want to check our answer in and make sure we're really right.
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This is an obvious one, but I'm going to check it because not everyone's going to be
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quite so obvious.
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So I'm going to check x equals two into the original problem, right?
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Solve absolute value of x equals two.
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And we're going to plug in two for x and simplify the left hand side, simplify the right
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hand side, there is nothing to do.
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And that's how I know for sure.
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That's a solution.
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And then I'm going to check the second solution into the original, and I'm going to plug in
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negative two in for x, simplify it, obviously by negative two is two.
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On the right hand side, nothing to simplify it's already two.
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And that's how I know x equals negative two is another solution.
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All right, try this problem, absolute value of x equals four.
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Solve for x.
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Again, you could do this by looking at a number line and figuring it out, right?
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If you're going to go four spaces to the right and four spaces to the left, you'll know
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that the initial will either end up on negative four or four.
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So those are going to be the solutions.
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Or you can take the equation as it is, and to get rid of the absolute value, you say,
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well, whatever is in that absolute value, x, that could either equal four or it could equal
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negative four.
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So the solution here, hopefully you got this, you could either have four as a solution,
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or negative four as a solution.
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Two ways of getting there.
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I'm not going to do the check on this one.
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It's pretty obvious if you put in four for x or you put negative four for x, the original
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problem will be true.
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So let's come up with a way of solving it if it's not quite so obvious.
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How about if instead of just to x in there, I had, hmm, maybe an x minus one was equal to
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five.
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And I wonder what x could be, well, you could do this on a number line.
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You could say, well, I know if I go five spaces to the right, that's what x minus one
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could be, right?
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Or if I go five spaces to the left, that's where x minus one could be.
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So if I go five spaces to the right, we know that number is really just the number five,
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right?
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And if I go five spaces to the left, I know it's really negative five.
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I really have two agents here.
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This x minus one could either equal negative five or could equal positive five.
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So x minus one, by looking at the picture, could either equal five or x minus one could
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equal negative five.
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So we could solve these by adding one to both sides.
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x is six.
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It's one possibility.
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And if I add one over here, my other equation, I get x equals negative four.
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So you know what?
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This original problem, it's got two different solutions, six or negative four.
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This is not just six and negative six.
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You don't get to just solve one equations.
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There's two completely different equations.
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Now you may not want to have to draw a picture to figure that out each time.
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So another way of thinking about is this.
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Well, whatever is in here could either equal this number five, right?
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Or whatever is in here could equal negative five.
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And that's exactly what I wrote.
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Whatever is in there, the x minus one could equal five or whatever is in there, it could
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be the opposite.
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Because if whatever was in there was equal to this could either equal five or negative
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five so that when I took the absolute value, I would get a positive five.
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So we could see graphically, those of you who are visual want to see it on a graph or
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you could sort of think of it algebraically, whatever is in here.
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If it was a five, we would get a solution or whatever was in here was a negative five
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when you took the absolute value, you would also get five.
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Now let's go ahead and check each of these in the original problem.
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So I'm going to do it.
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I'm going to check x equals six into the original problem, which is absolute value of x minus
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one equals five.
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So we're going to plug in six for x.
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And that gives me the absolute value of five and when you simplify that you get five.
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So definitely it looks like six is correct.
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And now we're going to check negative four in for x.
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So we're going to do, oops, we're going to plug in negative four for x, simplify inside
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the absolute value.
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And I got five on the left hand side, five on the right.
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So negative four is another solution to this problem.
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So those are the two solutions, six and negative four to the original problem, which was x minus
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one equals five, the solutions were six and negative four.
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And we show that those really are true.
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Now in general, this is the way we could do it algebraically always.
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If you have some stuff inside the absolute value and you have some number over here that
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is positive, you know this stuff inside could either
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equal k or it could be negative k to make it true.
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So anytime you have some stuff, whatever it is, take that stuff, whatever it is in there,
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and it gets to equal k or the stuff inside is equal to negative k.
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And we're going to solve problems like this on the next video.
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And I've only done problems where k is zero, but we'll discuss if k is zero or if k is
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negative as well.