 Hi, I'm Zor. Welcome to InDesert Education. I would like to continue talking about inequalities and how to solve them. And right now we are talking about different transformations with equations which might help you to solve the problem, solve the inequality. Now, obviously in as much as transformations help to solve equations, transformations might help you to solve inequalities. So we're talking about different transformations and what actually can or cannot be done and when and how. And the rules are a little bit more complicated than with equations. Let me just remind you that if you have an equation something like effort X equals to A and you apply to both sides of the equation some function let's say function called T for transformation. The result of this function being applied to both equal basically arguments would also be an equation. So there is no rules so to speak. You just apply the transformation and that's it. It's a different question whether transformation is invariant or not invariant with invariant basically being of the nature that the solutions of this new equation of X basically are exactly the same as solutions of this original equation. But that's a different question. The transformation can be applied and there are no rules basically. It will be an equality between these two sides. With inequality it's slightly different and here is my first example. Let's consider an inequality AX plus B greater than zero. Now let's try to solve it in exactly the same fashion as we solve equality. The equation with correspondingly same left and right part. So how do we solve the equation AX plus B equals to zero? You add minus B to both sides you will get AX plus B minus B which is zero and zero doesn't change so it's AX and it will be minus B on this side. Then you divide by A both sides hopefully A is not equal to zero obviously and then you get a solution, right? Now let's do the same thing here. First we add minus B to both sides of the equation and we will have AX greater than minus B. Now we divide by A both sides we will have X greater than minus B over A. Now I'm in the red pen. Is it right? No. This is wrong. It's completely wrong. And let me just give you an example why it's wrong. Let's just have concrete numbers. Let's say minus 2X minus 2 greater than zero. Now we add minus minus 2 which is we add 2 to both sides we get minus 2X greater than 2. We divide by minus 2 we get X greater than 2 divided by minus 2 minus 1. Is it right? X greater than minus 1. Well let's put zero for instance. Zero is greater than minus 1. Put zero we will have minus 2 greater than zero which is wrong, right? So why did actually something like this happen? Well it's because some of these operations are actually completely wrong. I could not divide by minus 2 both sides of inequality and leave the inequality sign as is. And this is actually a subject of this lecture. So what can and cannot be done and why and under what circumstances with transformations of inequalities? So let's just do it step by step. I just wanted you to basically feel that simple transformations like you did with equalities. They might be applicable to inequalities but with very important rules which we have to really follow. So let's start from the beginning. Addition and subtraction. So if you have an inequality let's say P greater than Q. Can you add or for the same token subtract the same number? Let's say A and it will be P plus A greater than Q plus A. Now is this correct? So if this is correct, is this correct? And again formulated in the language of transformation the question is can we add the same number to both sides of equation? Regardless whether this number is positive or negative. Negative means it's actually a subtraction. Well the answer is yes. And here is why. What does it mean that P is greater than Q? Well if it's two real numbers and by the way I didn't mention it in the beginning but I should. We are talking about real numbers over right now. So everything whatever I'm talking about is about real numbers. Functions of real numbers with real arguments, real constants etc. Now P greater than Q means that P is equal to Q plus some difference between them. And what's important difference is positive. Right? It's a positive difference. We add it to a smaller number to get to the bigger number. Regardless by the way of the signs of P and Q. As long as the P is greater than Q then it means that the research number, the positive number, that being added to a smaller Q it will get P. Right? So if I'm saying that for instance 3 is greater than minus 3 what does it mean? It means that 3 is equal to minus 3 plus what's the difference between them? 6. And 6 is positive. That's what it means. Minus 4 is greater than minus 17. What does it mean? It means minus 4 is equal to minus 17 plus 13. Begin. 13 is positive. So regardless of the signs of P and Q the difference is always positive. Which we have to add to a smaller one to get to bigger number. Okay, so this is given. This is exactly the same as this. Now, we know this is a quality, right? So let's just add up A to both sides. We can do it without any problems. P plus A equals to Q plus D plus A. Or considering the commutative law of addition I can do this. So what do we have right now? What's the relationship between P plus A and Q plus A? Well, the relationship is such that for a Q plus A I have to add positive number D to get to the P plus A. Which actually is a definition of being this being smaller and this being greater. So that's why I retain the sign of the inequality after I add the constant A. Regardless, by the way, of the sign of A. Because the D is always positive and it remains positive. So the difference between P plus A and Q plus A is exactly the same positive D. And the Q plus A is a smaller number. So we can add the constant to a number. To both sides, to both numbers. The left and the right, the same constant. And our equation is supposed to be basically retained. And back to our example before, when I had AX plus D greater than 0, I added minus B. So AX greater than minus B. This is still correct. And it's equivalent to this one. Okay, that's all about adding and subtracting. Now, multiplying and dividing. We start from the same thing. P is equal to Q plus D. And D is positive. Now, let's multiply both sides by some constant. So, let's say K is a constant. I multiply P times K. And I multiply Q times K. And if I know that P is equal to Q plus D, when I multiply by K, I will have D. Right? Q plus D times K using distributive law of arithmetic. I've got this. So, what's the difference between P times K and Q times K? Well, this is the difference. If this difference is positive, it means PQ is greater than Q. PK is greater than QK. If this difference is negative, then the reverse is true. PK would be less than QK, right? So, it all depends on this particular sign of this particular number. Now, D is positive. So, what's the sign of D times K? That's exactly the sign of K. If K is positive, then this is positive. If K is negative, then this is negative. So, what it actually means, it means that if K is greater than zero, then PK is still greater than QK. So, multiplying by a positive number retains the equality. Multiplying by a negative number would change the sign of equality to the opposite. And the sign of this is the reason. Multiplying this by K gives me this. So, difference between PK and QK is this. And the sign of it, since G is positive, the sign of it depends on the K. So, this is basically the rule. Multiplying by a positive number, inequality is retained. Multiplying by a negative number and inequality is reversed to an opposite from a greater to a less. Well, obviously, we are not considering multiplying by zero, because then both sides would be equal to zero. That's not interesting. So, that's the rule. Again, you can remember it, obviously, that multiplying by a positive number retains the inequality, multiplying by a negative changes to the opposite. Or, you can think about it using this type of logic. So, if one number is greater than another, then this representation is true. And then we multiply both sides of the equation by K, which we can do, and the equation will be retained. And then we will have PK is equal to QK plus DK, and DK has a sign the same as the K, because the T is positive. Okay. That's all about multiplication. And, oh, by the way, I didn't really talk about division. It's exactly the same thing. I mean, if you will, instead of multiplying by K, you divide it by K, you will have P over K equals to Q over K plus D over K. The distributive law is still working. And again, these two are related using this. So, if this is positive, then P divided by K would be greater than Q divided by K. And the positiveness or negativeness of this depends only on the side of K, because D is positive. Right? So, it's exactly the same thing. All right. So, multiplication and division both depend on the sign of this factor or divisor. If the sign is positive, the inequality is retained. If the sign is negative, inequality is changing to the opposite. Next. Next is absolute variable. The short answer is we have absolutely no idea what happens with this particular inequality. And here is why. Absolute value is some function, right? So, if I would take something like 3 greater than minus 4, which is a true inequality, then I take the absolute value and I will get 3. I should put less. So, in this particular case, I should really reverse the equation, the inequality sign, right? From greater to less. However, in case 3 is greater than 2, the absolute value is the same as 3 and 2. So, the equation of inequality is actually retained. So, in some cases, inequality is supposed to be reversed. In some cases, inequality should be retained. So, this is incorrect. We cannot have it correct. Therefore, the short answer to absolute value function applied to both sides of the equation is you can do it easily. And we will return to this issue a little bit later in this lecture. But so far, the short answer is you cannot apply absolute value function to both sides of the inequality without anything. There is no general rule for absolute value. But there are some specific rules and I will address them later. And here is the beginning of this later. Let's consider a monotonically increasing function. What's an example of monotonically increasing function? Well, for instance, y is equal to x cubed. Remember, the graph of this would be something like this. It's monotonically increasing. Well, y is equal to x or y is equal to 2x. The x or 2x, they're also monotonically increasing functions. There are many monotonically increasing functions. For instance, y is equal to log 2 of x. For positive x, obviously, the graph would be something like this and it's monotonically increasing. And it doesn't really exist for a negative anyway. So, there are many monotonically increasing functions. So, my statement right now is that if function is monotonically increasing, then you can apply it to both sides of the equation without any change of the sign of inequality. So, if I have a monotonically increasing function f, then from p greater than q follows f of p greater than f of q. Why? Obvious reason, this is the definition of monotonically increasing function. Let's just remember, let's back to whatever the lecture was about monotonic functions. Monotonically increasing function if the function which is increasing with increase of its argument, right? So, and correspondingly, decreasing with decrease of the argument. Now, this is an increase from q to p and that's why it will be an increase of the function from fq to fp because that's the definition of the monotonic function. Which means that if p greater than q, then pq would be greater than qq or 7p would be greater than 7q. Now, this is because y is equal to xq is a monotonically increasing function. This is because y is equal to 7x is monotonically increasing function. So, basically, that's it. Now, if p and q are positive numbers, then log p base 2 would be greater than log q base 2. And all other monotonic functions, whatever we can think about, if they are monotonically increasing, then the relationship between p and q will be retained and transferred to f of p and f of q. Great. Now, obviously, without much talking about it, it's obviously if you're using monotonically decreasing function, then inequality should be reversed. And again, because this is a definition, with increase of the argument, monotonically decreasing function is decreasing. Increase argument decrees of the function. That's the definition. So, any decreasing monotonically decreasing function can be applied to this inequality and you will get another inequality with an opposite sign. An example, for instance, y is equal to, let's say, minus 2x. This is the monotonically decreasing function. Something like this goes through zero. So, I applied this and obviously it corresponds to whatever I was just talking about multiplication. If you multiply by a negative constant, then the inequality is reversed. So, it corresponds because multiplication by a negative constant is actually an application of monotonically decreasing function of this type, minus some constant x and the constant is negative. So, minus 2p would be less than minus 2q. Another example of monotonically decreasing function, let's consider function y is equal to 2 to the power of minus x. This is 1, this is 0, this is x, this is y. So, 2 to the power of minus p would be less than 2 to the power of minus q. Correct? So, monotonically increasing and decreasing functions can be applied to inequality with the result in correspondingly the same or opposite inequality for the result of this function. Now, we can re-examine our case of, let's say, absolute value. Now, is absolute value an increasing or decreasing function? Well, graph, 0, x, y is equal to absolute value of x. If x is positive, then y is equal to x, right? Because the positive values, absolute value is the same as the values themselves. So, the graph goes this way. If x is negative, then absolute value of x is minus x, right? What's the absolute value of minus 2? 2. What's the absolute value of minus 10? 10. So, whenever you have an absolute value of some number, like minus 2 or minus 10, you have to reverse it to get minus minus 2 to get to the 2 or minus minus 10 to get to the 10. So, on the negative side, it would be y is equal to minus x. That's the graph. So, this is the graph of y is equal to absolute value of x. It's this. Now, we see that this function is monotonically decreasing for the negative axis and this monotonically increasing for the positive axis, which means what? That if p and q both are greater than 0, then we can apply and we will retain the inequality. If both p and q are negative, then we can apply, but now function is monotonically decreasing and we can say that we will change the inequality to an opposite. From greater I got less. But, finally, if p and q are in different areas of the arguments, like one is positive and other is negative, then all bets are off because we cannot say anything about the function being monotonically increasing or decreasing in the entire area. So, only in this area it's that the increasing and in this area is decreasing. So, if both are in this area, we can consider it and we can apply it and we will have the retained inequality. If both are here, it would be a different inequality, but we still can say something. But if both belong to different sides of the x axis relative to 0, nothing can be said. So, that's a fuller answer to the question of whether we can apply the function absolute value to both sides. And, by the way, exactly the same thing can be said about a function of, let's say, x squared, y, same thing. This is parabola, right? That's the graph. So, when it's positive, it's increasing. When it's negative, it's decreasing. So, what I can say is, again, that if both P and Q are positive, then we can apply square function and we will get the same inequality between them. If both are negative, then equation inequality is changed to an opposite. But if they are none of these, one is positive, another is negative, nothing can be said about it. And, by the way, this is not only function x squared, but also x to the fourth and x to the fifth, sorry, x to the sixth and x to the eighth and x to the any even power. Because this is the same behavior of the graph. It will go down on a negative and it will go up to the positive values of argument, which is different from the graph of function with odd power. This is odd power, right? So, if n is equal to one, it's three. If n is equal to two, it's five, et cetera. Because the graph in this case would look like this. And the function is monotonous everywhere. So this function x to the odd number can be applied and then considering it's always monotonically increasing, the inequality would be retained. And incidentally function y is equal to minus x to the power 2n plus one. Because the graphs would be like this. It's monotonically decreasing. And then you can apply this function and you definitely know that inequality will change to the opposite. Because it's monotonically decreasing function. Well, the only one example I wanted to show is another type of transformation is function one over x. This is function y is equal to one over x. This is the graph. It's monotonically decreasing function on the negative side and also monotonically decreasing function on the positive side. However, what's important is that we really have to consider again both p and q belonging to the same side, either this one or that one. So if p and q both are positive or both p and q are negative, in both cases we should really, since it's monotonically decreasing here and there, we should really change the inequality to opposite. Just as an example, minus 2 is greater than minus 3. Right? Right. Now one over minus 2 would be less than one over than minus 3. Which is also true. If, however, p and q belong to different parts of the x-axis, let's say p is positive, q is negative, nothing can be said about it. Well, I mean, nothing can be said about the inequality. That's not exactly correct. I have to correct myself in this case. It depends. If p is positive and q is negative, because only that actually is possible if p is greater than q. Right? So they belong to different parts of the x-axis. So p must be positive and q must be negative. Otherwise, this particular inequality would not hold. But in this case, we definitely know that one over p would be also positive and one over q will be also negative as it was before. So the inequality will stand in this case. So if p is greater than 0 and q is less than 0, then one over p would be always greater than one over q, because this is positive and this is negative. But we don't really use the function and monotonic behavior of the function to derive this. It's just plain logic kind of thing. I'm urging you not to remember these rules, whatever I was just talking about. They're not supposed to be remembered. You are just supposed to understand that it all depends on what kind of function we apply. So that's something which I always wanted to inculcate into students' mind, that a generalized approach which you understand is much better than remembering few individual rules. What does it mean, for instance, that I was saying that, yes, you can add a constant to both sides. What does it mean you add a constant? It means you're applying a function x is equal to x plus a. That's the constant applied to both sides. It would be p plus a or q plus a. Now, is this function monotonically increasing or decreasing or something or nothing? Well, obviously, the graph of this function is y is equal to x, which is this, raised up or down, up or down by a, depending on whether a is positive or negative. But it's still monotonically increasing function. So from this monotonic behavior, we conclude that this function can always be applied because it's always monotonically increasing and it will retain the inequality. So it will have p plus a greater than q plus a. Now, multiplication. Same thing. It's an application of this function. Is it monotonic? Yes. It's monotonically increasing when k is positive. It's something like this. Or this doesn't really matter. It's still monotonically increasing, depending on k. But if k is negative, the function will go this way. And any of those functions are monotonically decreasing. So this is actually the reason that the multiplication by k will retain the inequality if k is positive and change it to the opposite if it's negative. Same thing with any other transformation because transformation is always a function. And based on behavior of this function, whether it's monotonic or not monotonic or monotonic where on this particular area, in that particular area, you can basically judge what kind of inequality you will get as a result. And that's what I would like actually to understand. It's always the result of the application of some transformation function which may be this, maybe this, maybe it's x square, maybe it's god knows what doesn't really matter. But behavior of the function is something which you always have to understand. And it's monotonically increasing retained inequality. If it's decreasing, inequality is supposed to be changed to the opposite. And all you have to understand is what's the behavior of the function? All right, that's it for this particular lecture. I will try to put as many practical examples in future lectures on inequalities as I can. And thanks very much for listening to me. Good luck.