Solving absolute value equations

0:00- [Instructor] In this video, we're going to see that absolute value equations can have zero, one, or two solutions. And before we work some examples, let's just think about it graphically, why that is the case. So that's my x-axis, that's my y-axis, and let's say I had the graph of y is equal to something, I'll just call that A, times x minus something else, let's just call that B, plus something else.

0:29This is the general, or you could view this as the general absolute value equation. Well, this will have a graph something like this. I'm just gonna draw it very roughly. It's going to have this type of shape. It could also be downward pointing if A right over here were negative.

0:45But a graph like this, it obviously would keep going, if you said when does this expression on the right equal zero, or another way of thinking about it, for what x values does y equal zero? So y equals zero is right over here, and you could say, you could see from this graph that y is never equal to zero.

1:06So there's no x values that would make this left hand equal zero. So in that situation, you would have zero solutions if you had zero is equal to all of this stuff right over here. Now, let's say that this value right over here is when y is equal to two. So if you had two is equal to all of this stuff right over here, well you would have exactly one x value that makes that true.

1:32Now, let's say this over here is five. So if you had the equation five is equal to all of this business, well there would be two x values that make that true. So depending on the equation, you're going to have zero, one, or two solutions. Now, let's do some examples algebraically to see that happening.

1:57So let's say I had the equation negative four is equal to negative five times the absolute value of x minus three, and then all of that plus two. Pause this video and try to solve this absolute value equation. All right, now let's do this together. So what I try to do is isolate the absolute value part, the way that you would manipulate any algebraic equation.

2:22So we can start off by subtracting two from both sides, so that we get rid of that plus two over there. And we are going to get negative four minus two is negative six, is equal to negative five times the absolute value of x minus three, and then that's gone right over there.

2:42Now, let's divide both sides of this equation by negative five, so divide by negative five, divide by negative five. And of course we're going to get 6/5 is equal to the absolute value of x minus three. Now this is where it gets a little bit interesting. If the absolute value of something is equal to 6/5, that means that that something, that x minus three, could be equal to 6/5, but it also means that x minus three could be equal to negative 6/5.

3:17How do I know that? Well, if x minus three is equal to negative 6/5, you take the absolute value of that, you get 6/5. If it's equal 6/5 and you take the absolute value of that, well then you're going to get 6/5 again. So these are two possible solutions, so let's solve it.

3:32So if we add three to both sides of this equation, we're going to get x is equal to three plus 6/5. If you add three to both sides of this equation, you get x is equal to three minus 6/5. So three is the same thing as 15/5. 15/5 plus 6/5 is equal to 21/5 in this situation.

3:56And over here, this is equal to 15/5 minus 6/5, which is equal to 9/5. So here we clearly had two solutions. It was kind of like this scenario right over here. Two x values gave us that y value. Let's look at two more examples. So let's say that I had the equation six minus the absolute value of two x plus three is equal to six.

4:24Pause this video and see if you can solve this. All right, once again, let's try to isolate the absolute value. Let's subtract six from both sides. And we are going to get, on the left hand side we're just going to have negative absolute value of two x plus three is going to be equal to zero.

4:43Well, we can multiply both sides by negative one, or divide both sides by negative one, and you're going to get the absolute value of two x plus three is still going to be equal to zero. And so that means that two x plus three could be equal to zero, and that's it, because it's not like there's a positive version of zero and a negative version of zero.

5:06There's just zero. If the absolute value of something is zero, well that thing inside the absolute value sign must be equal to zero, and then you just solve from here. If you subtract three from both sides, we get two x is equal to negative three. Divide both sides by two, and we get x is equal to negative 3/2.

5:30And here we had one solution. And a good hint is is that when we isolated the absolute value sign here, on the right hand side you got a zero. If on the right hand side you got a positive number, you'd have two solutions. If on the right hand side you get a zero, you're going to have one solution.

5:45And as we're about to see, if on the right hand side when you isolate the absolute value, you get a negative value, well it's impossible to take the absolute value of something and get a negative value, and then you would have no solutions. But let's look at an example of that.

5:58I'm gonna do this in purple, just for the sake of variety. So let's say I have three times the absolute value of negative four x plus seven is equal to one. Well, let's subtract seven from both sides. And we are going to get three times the absolute value of negative four x is equal to negative six.

6:22Let's divide both sides by three. And we are going to get the absolute value of negative four x is equal to negative two. Well, how can I take the absolute value of anything and get a negative two? This is impossible. This will have no solution. So we've looked at all of the scenarios over here.

6:42Two solutions, one solution or no solutions.