Intro to absolute value equations and graphs

Let's do some equations that deal with absolute values. And just as a bit of a review, when you take the absolute value of a number. Let's say I take the absolute value of negative 1. What you're really doing is you're saying, how far is that number from 0? And in the case of negative 1, if we draw a number line right there-- that's a very badly drawn number line. If we draw a number line right there, that's 0. You have a negative 1 right there. Well, it's 1 away from 0. So the absolute value of negative 1 is 1. And the absolute value of 1 is also 1 away from 0. It's also equal to 1. So on some level, absolute value is the distance from 0. But another, I guess simpler way to think of it, it always results in the positive version of the number. The absolute value of negative 7,346 is equal to 7,346. So with that in mind, let's try to solve some equations with absolute values in them. So let's say I have the equation the absolute value of x minus 5 is equal to 10. And one way you can interpret this, and I want you to think about this, this is actually saying that the distance between x and 5 is equal to 10. So how many numbers that are exactly 10 away from 5? And you can already think of the solution to this equation, but I'll show you how to solve it systematically. Now this is going to be true in two situations. Either x minus 5 is equal to positive 10. If this evaluates out to positive 10, then when you take the absolute value of it, you're going to get positive 10. Or x minus 5 might evaluate to negative 10. If x minus 5 evaluated to negative 10, when you take the absolute value of it, you would get 10 again. So x minus 5 could also be equal to negative 10. Both of these would satisfy this equation. Now, to solve this one, add 5 to both sides of this equation. You get x is equal to 15. To solve this one, add 5 to both sides of this equation. x is equal to negative 5. So our solution, there's two x's that satisfy this equation. x could be 15. 15 minus 5 is 10, take the absolute value, you're going to get 10, or x could be negative 5. Negative 5 minus 5 is negative 10. Take the absolute value, you get 10. And notice, both of these numbers are exactly 10 away from the number 5. Let's do another one of these. Let's do another one. Let's say we have the absolute value of x plus 2 is equal to 6. So what does that tell us? That tells us that either x plus 2, that the thing inside the absolute value sign, is equal to 6. Or the thing inside of the absolute value sign, the x plus 2, could also be negative 6. If this whole thing evaluated to negative 6, you take the absolute value, you'd get 6. So, or x plus 2 could equal negative 6. And then if you subtract 2 from both sides of this equation, you get x could be equal to 4. If you subtract 2 from both sides of this equation, you get x could be equal to negative 8. So these are the two solutions to the equation. And just to kind of have it gel in your mind, that absolute value, you can kind of view it as a distance, you could rewrite this problem as the absolute value of x minus negative 2 is equal to 6. And so this is asking me, what are the x's that are exactly 6 away from negative 2? Remember, up here we said, what are the x's that are exactly 10 away from positive 5? Whatever number you're subtracting from positive 5, these are both 10 away from positive 5. This is asking, what is exactly 6 away from negative 2? And it's going to be 4, or negative 8. You could try those numbers out for yourself. Let's do another one of these. Let's do another one, and we'll do it in purple. Let's say we have the absolute value of 4x-- I'm going to change this problem up a little bit. 4x minus 1. The absolute value of 4x minus 1, is equal to-- actually, I'll just keep it-- is equal to 19. So, just like the last few problems, 4x minus 1 could be equal to 19. Or 4x minus 1 might evaluate to negative 19. Because then when you take the absolute value, you're going to get 19 again. Or 4x minus 1 could be equal to negative 19. Then you just solve these two equations. Add 1 to both sides of this equation-- we could do them simultaneous, even. Add 1 to both sides of this, you get 4x is equal to 20. Add 1 to both sides of this equation, you get 4x is equal to negative 18. Divide both sides of this by 4, you get x is equal to 5. Divide both sides of this by 4, you get x is equal to negative 18/4, which is equal to negative 9/2. So both of these x values satisfy the equation. Try it out. Negative 9/2 times 4. This will become a negative 18. Negative 18 minus 1 is negative 19. Take the absolute value, you get 19. You put a 5 here, 4 times 5 is 20. Minus 1 is positive 19. So you take the absolute value. Once again, you'll get a 19. Let's try to graph one of these, just for fun. So let's say I have y is equal to the absolute value of x plus 3. So this is a function, or a graph, with an absolute value in it. So let's think about two scenarios. There's one scenario where the thing inside of the absolute value is positive. So you have the scenario where x plus 3-- I'll write it over here-- x plus 3 is greater than 0. And then you have the scenario where x plus 3 is less than 0. When x plus 3 is greater than 0, this graph, or this line-- or I guess we can't call it a line-- this function, is the same thing as y is equal to x plus 3. If this thing over here is greater than 0, then the absolute value sign is irrelevant. So then this thing is the same thing as y is equal to x plus 3. But when is x plus 3 greater than 0? Well, if you subtract 3 from both sides, you get x is greater than negative 3. So when x is greater than negative 3, this graph is going to look just like y is equal to x plus 3. Now, when x plus 3 is less than 0. When the situation where this-- the inside of our absolute value sign-- is negative, in that situation this equation is going to be y is equal to the negative of x plus 3. How can I say that? Well, look, if this is going to be a negative number, if x plus 3 is going to be a negative number-- that's what we're assuming here-- if it's going to be a negative number, then when you take the absolute value of a negative number, you're going to make it positive. That's just like multiplying it by negative 1. If you know you're taking the absolute value of a negative number, it's just like multiplying it by negative 1, because you're going to make it positive. And this is going to be the situation. x plus 3 is less than 0. If we subtract 3 from both sides, when x is less than negative 3. So when x is less than negative 3, the graph will look like this. When x is greater than negative 3, the graph will look like that. So let's see what that would make the entire graph look like. Let me draw my axes. That's my x-axis, that's my y-axis. So just let me multiply this out, just so we have it in mx plus b form. So this is equal to negative x minus 3. So let's just figure out what this graph would look like in general. Negative x minus 3. The y-intercept is negative 3, so 1, 2, 3. And negative x means it slopes downward, has a downward slope of 1. So it would look like this. The x-intercept would be at x is equal to--. So if you say y is equal to 0, that would happen when x is equal to negative 3. So it's going to go through that line, that point right there. And the graph, if we didn't have this constraint right here, would look something like this. That's if we didn't constrain it to a certain interval on the x-axis. Now this graph, what does it look like? Let's see. It has its y-intercept at positive 3. Just like that. And where's its x-intecept? When y is equal to 0, x is negative 3. So it also goes through that point right there, and it has a slope of 1. So it would look something like this. That's what this graph looks like. Now, what we figured out is that this absolute value function, it looks like this purple graph when x is less than negative 3. So when x is less than negative 3-- that's x is equal to negative 3 right there-- when x is less than negative 3, it looks like this purple graph. Right there. So that's when x is less than negative 3. But when x is greater than negative 3, it looks like the green graph. It looks like that. So this graph looks like this strange v. When x is greater than negative 3, this is positive. So we have the graph of-- we have a positive slope. But then when x is less than negative 3, we're essentially taking the negative of the function, if you want to view it that way, and so we have this negative slope. So you kind of have this v-shaped function, this v-shaped graph, which is indicative of an absolute value function.