Testing solutions to absolute value inequalities

- [Voiceover] We have three inequalities here that involve absolute values and then below them we have potential values for x, and what I want to do in this video is see which of these potential values for x actually make the inequality true. So let's start with this first one in green, here on the left. It says the absolute value of x is less than the absolute value of negative seven. So let's think about which of these x values would make this true. And before I even try out the x values, let's see if we can simplify this inequality. Now the thing that might jump out at you and encourage you to try these on your own before I work through them, what might jump out at you is that we know what the absolute value of negative seven is. The absolute value of negative seven is how far from zero is negative seven? Well, it's seven to the left. The distance from negative seven to zero is seven, or another way to think about it is the absolute value of any number is always going to be the non-negative version of it. So this right over here, absolute value of negative seven is just going to be seven. So an equivalent inequality would be that the absolute value of x needs to be less than seven. So let's see if its true for x equals negative eight. So if x equals negative eight, then wherever we see the x we put a negative eight there. So let's see, is the absolute value of negative eight less than seven? Is that true? Well, the absolute value of negative eight is just going to be eight, so is eight less than seven? No, it's greater than seven. So that, x equals negative eight does not satisfy the inequality. Now x equals negative two? Well, wherever we see an x, let's put a negative two. So the absolute value of negative two needs to be less than seven. What's the absolute value of negative two? Well it's going to be positive two. Is positive two less than seven? Sure, two is less than seven. So this, this is x equals negative two, satisfies our inequality. The absolute value of negative two is going to be less than the absolute value of negative seven. Then, finally, x equals six. So the absolute value is the absolute value of six. Once again, everywhere I see the x I just put the six there. X equals six. We're going to say the absolute value of six, is that less than seven? Well the absolute value of six is once again, just six, Six is six to the right of zero. Is six less than seven? Yeah, sure, six is less than seven. So x equals six and x equals negative two both satisfy the inequality. Now let's do this one here in this magenta color. And once again, encourage you to pause the video and try to work through it out on your own. Let's try x equals negative four. So if x equals negative four, we're going to say the absolute value of negative four is greater than five. Absolute value of negative four, well that's just going to be four. Is four greater than five? No, four is less than five. So that doesn't work. Now, x equals three. Everywhere we see the x, replace it with a three. The absolute value of three, is that greater than five? Absolute value of three is just three, so is three greater than five? No, three is less than five. I think you're seeing, hopefully you're getting the hang of it. So finally, if x equals negative nine. The absolute value of negative nine, that would need to be greater than five. Absolute value of negative nine, well that's just going to be positive nine. So it's just going to be nine, and is that greater than five? Well, sure, nine is greater than five. So x equals negative nine satisfies the inequality. All right, now let's do these ones in this light purple color. The absolute value of x needs to be greater than negative 16. So there's something very interesting about this one. We don't even have to look at the choices here. Can you think of any x for which this would not be true? Well just think about it a little bit. The absolute value of a number, is it ever going to be negative? No, the absolute value of a number is going to be zero or positive. It's going to be non-negative, so this right over here is zero or positive, or we could call that non-negative. Zero or positive. So if this thing over here is zero or positive, something that is zero or greater, something that is zero or positive is always going to be greater than a negative number. So this is actually true for all x's. We don't even have to try them out. We could try them out, actually we will try them out, just so you see that, but it's going to be true for all x's, because when you take the absolute value, if you take, if x is zero it's going to be zero, but for any other value, any non-zero value of x, the absolute value of it is going to be positive. And let's just see that, we could put any x there, and this statement's going to be true. Absolute value of x equals negative 15. Well, the absolute value of negative 15, is that greater than negative 16? Well, the absolute value of negative 15 is positive 15, and of course positive 15 is going to be greater than negative 16. A positive is always going to be greater than a negative. So this is true. If x equals three, absolute value of three, is that greater than negative 16? Oh, whoops let me write it that way. Absolute value of three, is that greater than negative 16? Sure, absolute value of three is three. And three is positive, so it's going to be greater than a negative number. So that works. And as I said, any x would work there. And finally if x equals nine. Well, absolute value of nine, if x equals nine, is that greater than negative 16? Well sure, that's just nine, and that is greater than negative 16. Even if x was zero, then you would have zero is greater than negative 16, which is absolutely true. So any x here works.