 This video is going to talk about absolute values and equations and inequalities. So here's an absolute value, real rough sketch here, last minute change. But what I want you to notice is that when I have a y value, then we're going to say that an absolute value is equal to, let's say this is at one. Okay, so the absolute value of something, we'll call it x, is equal to one. We also have the absolute value of negative x is also one. Okay, so if I'm going to solve an absolute value, I really have to look at it as I'm trying to find out what's in here. So either x is going to be equal to one, whatever's inside here is going to be equal to one, or inside here. And that's represented by negative x is equal to one. Okay, well we don't typically do it that way. We typically say yes, x is equal to one, the original problem. But it doesn't matter which side the negative's on, just one side has to be negative. Say x is equal to negative one. Same thing, we just chose to put the negative on the other side. So let's see if we can solve a problem. We have to get to this absolute value all by itself first before we can do anything else. So we have to peel the layers of the onion. So I have to take the 14 to the other side and I have to divide by three as well. Let's just do it one step at a time. Three times the absolute value of n minus five is going to be equal to 12 when I add 14 to negative two. And then the absolute value of n minus five is going to be equal to four when I divide by three. So that means that this n minus five could be a four, or n minus five could be equal to negative four. Then I take the absolute value of it and I still get a four. So let's see what we get. If we add five, n is equal to nine, and if I add five here, n is equal to one. Two answers. Both of those answers are what we need. Okay, so let's try again. Here we have the same thing. We need to take the 14 to the other side and then we need to divide by negative five. So the absolute value, negative five, sorry, times the absolute value of two y is equal to an adding 14. That'll be 20. And when we divide by negative five, we have the absolute value of two y is going to be equal to negative four. And right now we have a problem because an absolute value cannot be equal to a negative number. Absolute values are always equal to a positive or zero. It's not equal to a positive or negative, so we have no solution. Again, it's because of this wonderful little negative causes us to have this. If I have an absolute value that is less than some value, it implies that x is going to be between that negative k and positive k. And this is a nice little thing to keep to help you remember, less is nest, it's in between, it's going to be one interval. If x is greater than k, to have a value graph, notice it was all above the x-axis. If it's a greater than k, some value, we're going to have k is less than negative k, x is greater than k. We drop the absolute value and write the same one and then we take the x and multiply that k by negative one, so we have to flip the inequality. More is or. So here we have absolute value that is going to be less than or equal to some number. So it's going to be less is nest. We should end up with a nested interval. Same thing, got to add the seven, five, q minus two, less than or equal to 15 divided by five. So the absolute value of q minus two is going to be less than or equal to three. Drop the absolute value and write the inequality that you see and then we're going to take the dropping the absolute value and we're going to multiply negative one by three, but since we multiply by negative, we have to switch the inequality. So let's solve. Adding two, q is going to be less than or equal to five and adding two on this side, q is going to be greater than or equal to negative one. This is the smallest value. This is the largest value and it's it is bigger than the smallest, but it's less than the largest. So it really did nest. We would say q is, but we're writing it this way is between negative one and five and an interval notation that would say that we start at negative one. And since it said less than or equal to inequality, then I have a bracket because it equals and I go up to five. OK, so now we have to take this. It's already isolated, so we just have to take the two problems here. So M minus one is greater than five and then we're going to do M minus one is less than. So it's the inequality change the sign. So when we do this one, we add one to both sides. So M is going to be greater than six. And if I add one to this side or both sides over here, then I'm going to have M is less than negative four. If I think about that on a number line, just to show you the or idea. So I have my negative four and I have my six and it's greater than six. So that's going to go this way and it's less than negative four. So that's going to go this way. So you can see there's a gap in the middle. So we do have an or. It goes from negative infinity to negative four and it doesn't include either one because my inequality doesn't union and then it starts back up at six to infinity. Now before we go on, I want to show you how we could use this using a calculator and I want to solve the equation equal to five and I want to do the inequality, but it's all done with the same graph. So to do that, first you need to get into your Y equal screen, then get the math and arrow over to number and you see this is ABS. So absolute value is what that means. So you press enter and it takes us back to the Y equal screen and then I just put in what I have. I have X minus one inside my absolute value and then close it so that parentheses are like the bars and that we normally see. And then I'm going to break my second equation be the other side and that would be the five. So I'm going to do zoom six to make sure I'm in a standard window and I'm going to see this graph. Now if it were equal, then I would need to know what these two points right here were. That's where those two things are equal. So I want to find the intersection to second trace and then five and you can do enter, enter, enter. Now we'll give you the first one and that's negative four. We should have expected that. And then we do second trace five. But this time we have to get to the other side of our vertex or it will keep giving us the same answer. And I'm going to go up to look at Y two because that's a straight line and get me over there quicker. And I get over to the other side of my vertex and then I can just press enter, enter, enter and I find out that X is equal to six. Now if I'm talking about the inequality, now I want to know when my absolute value is greater than five. So I'm just going to sketch this on my paper here and then we can do some coloring. So I have a graph that looks something like this. This is X equal negative four and this is X equal six. And we want to know when the V is going to be greater than the five, which is in the blue. So when will this be greater? Well this is the five here. This line is X equal five and I want the red one to be greater. The Y values to be bigger, which means they're higher as I look at my graph. So I'm going to have this over here and I'm going to have this over here. And so now you can see that this over here is negative infinity up to the negative four as I go from left to right. And then I have a union because I have the blue space in the middle. But then my circle again over here starts back up at six and goes to infinity. Last problem here, so add eight to both sides so the absolute value of five U minus three is going to be greater than negative two. Well earlier we said that an absolute value couldn't be equal to a negative two. But an absolute value can be greater than negative two. In fact this absolute value is always a positive value so it will always be greater than negative two. So we would say that it goes from negative infinity to infinity.