Welcome back everyone. We're going to continue with our suite on inequalities. Last time, we just showed you the basic ideas of what less than, greater than, less than or equal to than, greater than and equal to than looked like and have a picture of them on the number line. Now, we get a little bit more technical. Nothing too technical and show you how to do some algebraic manipulation with inequalities. We'll start by reminding you how to do it with equalities. First, what you do mechanically and why you would do it. And, when you do it is to solve for X. Then, we'll show you what is probably some new stuff. Which is, how to do it with inequalities, what works and then a very very big warning that there's one thing you can do with algebra equalities namely multiplying by negative numbers. That you have to be very very very careful about with inequalities to give away the punchline at the beginning. The only real difference is when you multiply by a negative number in equality you have to flip the sign. And, I will show you what that means. Let's get into it. OK first, what is probably some very very familiar material, was taught by algebra with inequalities. Suppose we start with something really simple like four equals four. Here's something that seems so obvious, that no one really appreciates what a powerful tool it is in an algebraic person. What am I saying? What I'm saying, is that if you take an equality and add the same thing to both sides. So take four and lets add three to the left side, lets add three to the right side and lets add the equal sign here. This is still true, if four equals four then four equals three equals four equals three. In fact, this is equal to seven, and this is equal to seven. So, let's put that as a general rule. If A equals b, then A plus C equals b plus C or A, b and C can be any numbers positive, negative, integers, fractions whatever. Who cares? Well, who cares is, this allows you to do algebra. For example, suppose someone tells you there's a mystery symbol called X. And X plus three equals ten and they demand that you solve for X. Meaning, what they're really asking you is, what is X? If X plus three equals ten. The simple one like this, you can sort of stare at it and say X probably has to be seven. But, you can also do it rigorously and technically using this algebraic rule. This part less tool you have in our arsenal. How could you do it? Well, you look here and you want to isolate x. If you knew that X was equal to something, you'd be done. If that's what it means to solve for X. So, let's Isolate it by getting rid of the thing next to it. Let's subtract three from both sides. In other words, if we write X plus three, minus three, this is equal to 10 minus three, X to three minus three is equal to X, 10 minus three is equal to seven. And hooray! We found out what we want to find out which we sort of knew already. Okay. So, that's nice. Let's start again with four equals four. And, let's multiply it by say two. Two times four equals two times four. This works. This is eight. And this is eight. So, I can multiply my number. I can also multiply by a negative number. So, if I start with four equals four and then multiply by negative three. Negative three times four, equals negative three times four. Over here this gives me negative 12, over here this gives me negative 12. And I'm happy. General rule, if A b and C are numbers, and C is not zero. And A equals b, then C times A equal C times b. Who cares? Well, again it allows us to do algebra. So, suppose someone comes along and says minus five X equals 15. Here, you solve for X. So, if we have to stare at this, with a little practice. You could probably figure out that X equals minus three. Another way to do it is to isolate the X. X is being multiplied by minus five. I need to somehow cancel that out. Let's multiply by negative one fifth. Right. So, i multiply by negative one fifth times minus five X that equal to negative one fifth times 15. Negative one fifth times minus five is one. This gives me that X is equal to minus 15 over five which is minus three. X equals minus three. I struck oil and I'm happy. OK fine. So, that tells us that out those two algebraic tools are in our arsenal. That you can add the same thing to both sides of the equation and you can multiply both sides of an equations still say true allow us to solve for things. OK. Let's go to inequalities. Suppose we start with four is less than seven. Suppose I add something to both sides. So, four is less than seven. Is it true that four plus two is less than seven plus two? Yup. Because, four plus two is six, seven plus two is nine. That works. Right. If I start with four is less than seven and I subtract something. So, say four minus one is less than seven minus one. That work. That says that three is less than six. And that works. So, general rule. If A is less than b then A plus C is less than b plus C. For any written numbers A, B and C we can even picture that on the number line. Right suppose here is A, here's b. That means, A is less than b is being left of it. And I add C, if C is positive that means I'm shifting A by C and I'm shifting b by C. There's no way that A plus C can jump over b plus C. Likewise, if C is negative. That's why that works. And, that's often very powerful. That allows us to do algebra inequalities just before. So when suppose someone tells us that X plus three is less than 10. And, they ask us to tell us more about what that means about X. So, if X plus three is less than 10, I can subtract three from both sides. And you all should practice with that. So, let's just see. I subtract three from the left I get X. I subtract three from the right I get 10 minus three. In other words X is less than seven. So, what does that mean? That means that. Whatever on earth X is. And it could be anything. It's not one particular thing. It has to be to the left of seven. In other words, X has to live in this blue zone here. Any number wants to be as long as that doesn't exceed seven. And that's what the information that X plus three is less than 10 looks like. By the way, in a later video we are going to talk about this being a half open array. This is going to be a symbol minus infinity to seven and X is in there. We'll tell you what that means. Don't get confused about it now I just want you to see that in advance. OK that's great. Now, here's a little warning. When you multiply inequalities by both side, it's little more dangerous. We start again with five is less than eight. And first let's multiply both sides by a positive number. So for example, three times five less than eight times five. Question mark. Yeah. That works because, three times five is 15, eight times five, oh dear arithmetic in public, eight times five, gears churning were eight times five is 40. I think that's right. So, 15 is less than 40 and that works. Here's a problem. Lets take five less than eight. And let's multiply both sides by a negative number say minus one. If I do minus one times five, and I ask are you less than a question mark minus one times eight. Turns out we get an electric shock through the keyboard. If, we ever try to write this because minus one times five is minus five less than question mark minus eight. No way. Right. Here's the real number line, here's zero, here's five, here's eight. Unfortunately, here's minus five, here's minus eight, minus five is not till after minus eight. This is completely awful. In fact, what is true is that minus five is greater than minus eight. Here's the general rule. Suppose, A is less than b. If C is greater than zero. A times C is less than b times C. But, if C is less than zero. Then A times C is greater than b times C. Just as we saw here. See you have to flip the inequalities. Let's see one example of that. So we do a little algebra. Suppose, someone tells us that minus two times X is less than 10. What does it mean about X? Well, lets multiply both sides by minus one half. So, if I multiply this side by and minus one half times minus two X, multiply this side by minus one half minus one half times 10. But now, I have to flip the inequality. Right. Because, I multiplied both sides of inequality by negative number. This gives me X is greater than minus five. Which pictured on the real number line here's zero, here's minus five. That means, X can be anything in here. So, if X and anything in here, and I feed X to this equation up here. This inequality up here, I make it true. That's really just the only pitfall you have to worry about. But, it can be a serious one. OK. That concludes all with you.