 Hi, I'm Zor. Welcome to a new Zor education. Let's talk about inequalities. Well, this is an introduction to inequalities. Not too many problems will be solved. Just we will talk about what inequality is about. Well, let me just start with something relatively simple. X minus 5 greater than 0. This is an inequality. Let's just consider what actually we have written here. Well, on the left we have certain function. The argument is X. On the right we have certain constant. So, basically when we are talking about inequality, we are talking about some condition which we usually put on some function. It's a condition on a function that the value of this function are supposed to be, well, in this case, greater it might be actually less than or less than or equal or greater than equal. And some constant on the right. Well, first of all, let's think about what the function is. Function, as we know, is some kind of a rule which connects the elements of the domain with elements of core domain and somewhere inside the core domain we have range of all the values this particular function takes for all the different arguments from the domain. Now, core domain sometimes can be greater than the range. Sometimes it can be equal. For instance, function X square which is basically a function which converts let's say real argument to real value. Well, the real values, real numbers are domains and real numbers are the core domain. The range is, however, all the non-negative numbers. So it can be less, but it doesn't really matter. What does matter is that this particular constant A must be here. It can be either inside the range or it can be outside of the range. Again, X square we can consider this particular inequality when one belongs to the range or we can consider this particular inequality. One minus one actually doesn't belong to the range, but it still makes sense. So element A supposed to belong to the core domain that's fine. Now we are talking about the value of the function to be greater. In this case, it's greater, but again, you can always substitute any other inequality sign less than etc. So the value of the function f-attacks is supposed to be greater than the constant A which is an element of the core domain. Now, what does it actually mean? Well, it means that inside the core domain we have to have a relationship greater. So the core domain cannot be just some abstract set. It should be a set of ordered elements. So ordered means that forever to element I can always say which one is greater, which one is less. That's very, very important. And obvious example of this type of code domain is real numbers. Among real numbers we very easily establish, I mean it's already established basically, this order. We know which one is greater than which. So for two numbers from the set of real numbers, we can always say which one is greater. So if we will have a somewhere here, then we are interested in those values of the argument which are transformed into anything which is greater than a. So my direction of growth is this. So if this is a we are interested in those values from from the domain, which are reflected to this. So these are called solutions. So the values from the domain which satisfy this particular inequality are solutions to this inequality. And this is not a solution because it goes lower than a. So from the abstract standpoint, we are requiring that this particular set, the code domain, must be ordered. So for any two values from the code domain, we have to really find what exactly is the relationship between them, which one is greater than another. Now, how about the domain? Should domain be ordered? Well, not necessarily. And here, which is unusual, but that's actually true. Let's consider a completely different example of a function. Okay, the function is color, light, frequency. Now, you know that every color, red, blue, yellow has certain frequency of light expressed in some basically real number. Now, we can compare frequencies, but we cannot compare colors, which color is greater than another. But what we can have is, if I have a specific interval of frequencies, let's say we're talking about frequency from, I know, don't remember the frequencies of different colors, but from some value to some value, we are interested here. And it happens that red goes here. Red has a frequency which is in this interval. And let's say blue is outside of this. So what does it mean that if I want to solve the equation frequency of the light is supposed to be from this to this, then I can solve this equation by pointing to colors red and some other colors, which also have the frequency mapped into this particular interval and say these colors are solution to our inequality. And this color, blue, for instance, is not a solution. So I can basically from maybe finite, if you wish, set of different colors. I can always pick a subset which would correspond to an inequality, which I have basically conditioned on the frequency of this light. So the codomain must be ordered. Domain, not necessarily. However, in mathematics, usually these type of functions are not really considered. And what we do consider is obviously, usually, let's put it this way, the function of real argument, which has real values. And basically these are the functions we will that we will consider. Now, in this case, the expression, which is actually a solution to our inequality, also can be established in the terms of equal, not equal, greater, less than, etc. For instance, solution to something like this x square, let's say greater than 4. Solution to this are x less than minus 2 and x greater than 2. Because everything, which is less than minus 2, like minus 3, minus 4, etc. Being squared will give you correspondingly the value, which is greater than 4. And everything greater than 2, like 3, 4, etc. will also, being squared, will give you the value of greater than 4. So if this is an equality, this is the solution to this inequality. All right, so we understand what inequality is. It's a condition on some function which has values in a set, which is called codomain, where we can establish some kind of a condition, like greater than or less than, and some constant, which is an element of that codomain. That's the inequality. Now the solution to this inequality is a set of all x which, if applied the function f to them, will give the answer, which is greater than a. And the greater is established in the codomain, that's why we can't say. Now, obviously the relationship between greater than and less than, etc., are obvious. First of all, we all know what equal relationship is, right? So if we know the relationship, which is called greater than between two elements of the codomain, we can always establish all other equations, like what? For instance, less or equal is not greater. Less is not greater and not equal. What else we have? Greater or equal is greater or equal. So all possible relationships here, greater, less than equal, less and greater than equal, are defined using only one inequality sign. And obviously we also understand that the equality sign is also determined and is defined in any set. All right, so basically in my examples, I will use only one of those signs, like greater for instance. Everything else is basically assumed as a combination of some other logical conditions on this sign. So we have defined the inequality. We have defined a solution to this inequality. It's a set of such elements, which are greater than a. And all I would like to say is, well, basically the open question is how to solve equations. How to find these values of the argument, which would result in this particular condition on the value of the function. Well, there are different methods. And if you remember, if we were talking about equations, we had some transformations of the equations. We had invariant transformations, which do not actually change solutions. We had non-invarian transformations, which really had to be approached very carefully, because we might lose some solutions or we might gain some solutions. So we were talking about this. Well, similar approach can be basically used with with inequalities to solve inequalities. The only thing is, before doing this, which basically will be a subject of the next lecture, I would like to mention one other method, which is based on the graphics. And it's always related to a real function of real arguments, which is basically the most of the cases, which you will consider. So how can graphics be used to solve equations like this? And you know what, it's actually sometimes, whenever you can do it, it's really a good practice to do it. So you will see why. So let's consider a concrete example of the inequality, which I will solve using this graphics methodology. Inequality is x square minus 4x greater than minus 3. All right. Now, what we know is that this is a polynomial of the second degree, which has a graph of parabola. Now, let's think about what kind of parabola this is. Well, you remember that if you have roots of this parabola, so where the parabola is equal to 0 are the points where it intersects the x-axis, right? So if x is equal to this, y is equal to 0 and if x is equal to this, y is equal to 0. Now, looking at this, what are the roots of this? What are the solutions of the equation x square minus 4x equals to 0? Well, obviously x times x minus 4, right, is equal to 0. So one is x equals to 0, another is x equals to 4, right? So let me write, let me draw this particular parabola. So wipe out my example and draw a parabola, which is exactly corresponding to this one. So this is 4, this is 2, this is 1, this is 3 and my parabola in the middle point at 2 is equal to 2 times minus 2, which is minus 4, right? So it looks like this. It goes, now it's upwards, directed with its horns, as I call them, because the coefficient at x square is positive 1, in this case. So it goes up and the roots are 0 and 4, so that's why I draw it this way. So this is a parabola, which represents the function on the left. Now on the right, we have a constant minus 3 and we have to find those values of the argument, when the function would be greater than minus 3. Well, let's draw a line, y is equal to minus 3, which is here. Now, we see quite clearly that for x on this side, the parabola would be higher than minus 3 and x on this side would be higher than minus 3. Okay, what x are they? Well, I can basically draw this. Now, it looks like it's like 1 and 3. Well, let's do it exactly, not looks like, how to do it exactly. So my question is, when the parabola is exactly equal to minus 3? This is minus 3. So when the parabola is exactly equal to minus 3, at what x? Well, let's just solve the equation. x square minus 4x equals to minus 3 or x square minus 4x plus 3 equals to 0 and the solutions are x is equal to 1 and x is equal to 3. Product is 3, sum is 4 with minus sign equal to minus 4. So I'm just checking. And obviously, I chose the example with easy solutions. Right. So yes, exactly. It actually 1 and 3 exactly projecting down to these points of intersection between this line, y is equal to minus 3 and parabola. So what are the solutions to our inequality? It's all x's which are here and all x's which are here. So the solution is x less than 1 and x greater than 3. So these two areas are solutions to this particular inequality. Now, why did I choose to use the graphics in this particular introduction to inequalities? Because it actually helps you to see how exactly it looks like when something is greater or less than something else. Graphical representation always helps to feel better how this is all done. And yes, obviously, we can solve this particular inequality using certain methodology which will be presented in the next lecture, transformations of inequality. But still, even in that case, the graphics are sometimes utilized and very helpful. This is a simple case when all I have to do is draw initial graph of the function f of x, draw the graph of function y equals to this constant which is on the right, and basically graphically establish which areas are supposed to be the solution. And then another equation, now it's an equation which I have to solve would help to find exactly these two numbers one and three which are boundaries for these two areas. This is the right boundary for this area and this is the left boundary for this area of x. Well, that's it for introduction. As I was mentioning before, next lecture will be about transformations of the inequalities. And well, everything is, as you know, on Unisor.com and I do suggest you to register as a student, have somebody else as a supervisor or a parent maybe who will enroll you in certain classes that would allow you to go through exams and check yourself basically how you deal with all these issues. And that's it for today. Thank you very much.