 Hi, I'm Zor. Welcome to a new Zor education. I would like to talk about systems of inequalities and inequalities would be a general character, like not linear or not quadratic. It's whatever, whatever comes to mind. Now systems of inequalities, it means that we have more than one variable which we would like to have certain condition on to satisfy certain system of inequalities. Now, back to the inequality with a single argument, the solution to this inequality would be certain condition on one argument. Like for instance, if you have an inequality x where less than one, it's basically either x greater than minus one and it's supposed to be less than one, right? So that's what basically it means. This is an algebraic expression of the solution. At the same time, we can do it graphically. This is zero, this is one, this is minus one and we're saying that all axes which belong to this interval from minus one to one, two to one, are solutions to this problem. Now, so we have basically two ways of expressing the solution. If we are talking about more than one argument, if you have a system of inequalities, the algebraic solution of this type is practically possible because they're always x and y or whatever letters we're using for variables, they are interconnected and if geometrically in a single argument case, we have one or more intervals which we can describe each one with some algebraic condition on the argument. In case of a two-dimension, we have areas on the plane. This is one-dimensional, one line. Two-dimension is a plane. So we are talking about areas on the plane and areas on the plane are not easily expressed in this algebraic form. So we will use the geometric interpretation of the system of inequalities with more than one argument. That actually gives a very nice visual representation of the solutions. All right, so this is some kind of preamble. Now, let's go and let's just solve problems. I mentioned it before that with general inequalities, there is no recipe or whatever you call how to solve them. You just have to basically invent some new way of approaching the problem based on whatever your experience is. But in many cases, properties of monotonous functions do help and obviously, you have to be able to draw the graphs of basic functions from which inequalities are made. So number one. System of two inequalities with two variables. If you have an inequality, this one or this one, what you probably should start with is to draw a graph of a system of these equations. Draw the graphs when it's not inequality, but when it's inequality, when it's basically an equation. So to draw the area which is represented by the top inequality, you first have to draw a graph of a function, two arguments on the coordinate plane. And then you have to really take a look at what areas the plane is divided by this particular graph. Because if everything which has all the points which have coordinates satisfying this equation are on certain curve. So this is coordinate plane and this is your curve of this particular equation, whatever the equation is. So everything when it's equal is concentrated on this curve, which means everything which is not equal, which is less or greater, would be outside of this curve. And what's important is that this curve actually divide the plane in two halves. On one half, it's greater and another it's less, this particular sign. Now, I did explain actually why it happens, but let me just repeat this. It's very kind of interesting consideration. If you consider that there are two points here and here and in this point, it's let's say x square plus y square is less than 25 and in this case, it's greater than 25. Then I can always and they're both belong to one half, to the same half of this particular. So they can be connected with some curve, not necessarily this one, but whatever the curve is. They can be connected without crossing this line. If these points can be connected without crossing this line, it's, the expression is less than 25 in this particular case and greater than 25 in this particular case and considering that this function is really smooth in certain mathematical understanding of this and certain meaning, but you probably understand even without the details what smooth actual means. It means it's smoothly taking changes its value as argument is changing. So argument is changing from this point to this point. This is one pair of coordinate. This is another pair of coordinate. So the argument is changing smoothly, so the function should change smoothly, which means that somewhere in the middle, or not in the middle, but somewhere, it should actually be equal to 25. If this is less than 25 and this is greater than 25 and we are smoothly changing, it means that something is equal, which means that the function is equal to 25 somewhere outside of this curve, which we have already drawn, right? So that's impossible. That's why I'm saying that on one side of this curve, it's always greater than and another is always less than. So after you draw the line, you have to analyze which side corresponds to our particular inequality. So let's go back to our inequality. Now, what is this particular curve? How does it look like? Well, when we were talking about graphs of functions like f of x, y equals to zero. So if you have an equation like this, you were actually discussing this type of curve. If you have this and you have a point x, y, what is x square plus y square? Now, this is x. This is y. So this is a square of a distance and if I'm saying that this square of a distance from this point to the beginning of the coordinate is equal to 25, which is 5 square by the way, it means that the length of this segment is equal to 5. So all the points which have this particular property of having a distance from 0 equal to 5 are on some kind of a circle of the radius 5 around the center. So this is the graph of this particular curve. Now we are interested in less than 25. So it's either inside of the circle or outside of the circle. Okay. Now let's just check. I mean to know which side is which, we just have to take one point on one particular side. Let's say 0.00 inside the circle and check what's the value. Well, 0.00 would be less than 0.25. So 0.00 really is part of the solution of this equation, which means everything inside of the circle is a solution. So the first equation, inequality, sorry, the first inequality, results in cutting a circle out of the whole plane and heading inside of that circle. By the way, this is strict less than. It's not less or equal, which means that the border itself, the circle itself, is not included into the solution of this first equation. Okay. The second one. Why less than x square? Same thing. Let's start with y equal, equals to x square. This is parabola, as we know. It's something like this. Now, my question is, which side of parabola, this side, which is on the bottom, or the top, is this? Well, again, let's just check one particular point. Let's say this point. It has x is equal to 0 and y is equal to, well, 4, let's say. So it's on the top. It's above our parabola. So x is 0, y is 4. 4 less than 0. That's wrong, which means it's not the upper part of the plane, it's the bottom part. So this particular inequality results in the inside of the circle, not including the border, the circle itself, and this is everything below this parabola. So the solution is this area, again, not including the border inside of this kind of crescent, if you wish, you can call it. So inside of this crescent is the solution to the system of two equations with two different inequalities, or with two different variables. All right? So that's what it is. And the approach which we have taken right now is basically the same as we will use for other problems as well. So first you, instead of inequality, you try to draw the graph of corresponding equality and then you choose which side of the curve to take and then intersect whatever the sides, whatever the areas you have received from each individual in equality. You intersect them together. So next one. Load base 2 x minus 1 less than y and x minus 2 square plus y square less than y. All right. So first we start with a graph when y is equal to load base 2 at x minus 1. So what is the graph of this function? Well, as we know from the discussions about graphs, this graph is the same as this one shifted to the right by one unit. Right? So let's do it. So this is one. My load base 2, 2 is greater than 1, so the log is directed upwards. It's monotonously increasing function and it looks like this. Now, since I have to shift it by 1, the graph which we are interested in would look like shifted by 1, which means instead of 0.1 where it's crossing the x-axis at 0.2. And in this particular case this y-axis was asymptotic. Now, since we have shifted everything, this line would be, the vertical line would be an asymptotic thing. And this would be a graph. So this is our y is equal to log to x minus 1. Now, again, it divides the plane in two halves. The question is which half is ours? So where exactly y is less than log base 2 of x minus 1? Well, it's actually quite easy. If this is x, then this is, if this is x, then this is the y which is equal. So if it's smaller, which means it goes down. So any point which is below this particular curve is the one which we are interested in. So this is the area we are interested in. Now, second one. Again, as we discussed before when we were drawing the graphs of equations, this is different from x square plus y square is equal to 1 by shift to the right of this circle. It's a circle with the center 0 and the radius 1. Now we are shifting it to the right by two units. So the center will be 2 and the radius would be 1. So it would be in a circle like this. And we are talking about inside of this circle, right? So because the point x is equal to 2 and y is equal to 0, because this is point, the center obviously belongs. It's 0 plus 0. It's less than 1. So we have an intersection between the part of the plane below this logarithmic line and inside of the circle. So the result will be this particular area. And considering these are strict less than sign, the border is not included. So inside of this area is a solution. So all x and y, all pairs which are inside of this area are solutions to our system of equations. Next. Well, as you see, all these problems are about drawing the corresponding graphs of the function of two variables, well, actually equations of two variables. So the graph of this thing and then determining which part of the plane belongs and which one doesn't belong to the inequality in question. And finally, you just have to intercept all these areas. Next one. Load base 2 x less than y, 2 to the power of x greater than y, and x square plus y square less than 1. Okay, what's interesting about this equation is you probably used to have dealing with systems of equations, that the number of the equations is equal to the number of variables, because this basically assures that there is one and only one solution if it's more or less nicely done. I mean, if you have less equations than the number of variables, most likely we'll have an indeterminate number of solutions. All right, so in this particular case, we have two variables, but three inequalities. Well, that shouldn't really scare you because there is nothing wrong with this, since every inequality defines an area on the plane where x and y are supposed to belong to, to satisfy this particular inequality, and then the system means that we are intersecting these areas. Well, we can intercept not only two areas, or three, we can intersect three areas, or we can just have, you know, one particular inequality, and it's also an area. So basically the number of intersections is irrelevant in this case, which means we can have as many inequalities with as many variables and still get some meaningful solution. All right, so what is the meaningful solution in this case? y is equal to log x base 2, it's this, and we are looking for y, which is greater than, so it's above. So this is an area which we are interested in. Now, where is y equals to 2 to the power of x? Well, it's this one. So this is one, and this is one. But now we are looking for y, which is less than 2 to the power of x, so it's below this particular circle. So now we have above this, but below this. So we are talking about this area, some kind of funnel, if you wish. But then we have the third inequality, which is an insight of the circle of radius one. Which is this. So out of this funnel, we cut only the inside of this circle, and that's the result. That's if log base 0.5 x less than or equal to log 0.5 y 2 to the power of x less or equal 2 to the power of y. And again, our circle, which we used to have. Well, just notice that these are all less or equal signs. So the border of whatever area we will receive should be included. But now let's think about this differently. Log base 0.5 is a monotonously decreasing function, which means the greater the function, the smaller the argument. The greater the argument, the smaller the function. So the inequality between the values of this function means an opposite inequality between the arguments. So this particular inequality is equivalent to x being greater than y. However, we should really notice one interesting thing. Since this is the logarithm, we cannot allow x to be and y to be anything. They are only positive numbers. So together with this, I have to add this. Because otherwise logarithms would not be meaningful. So x is positive, y is positive, but x is greater or equal than y. These three are equivalent to this one. Now, how about this guy? Well, it does not restrict x and y. However, since 2 to the power of x is an increasing monotonously increasing function. So the first one is actually equivalent to this. The difference between functions is the same as difference between the arguments of this function. Finally, I have an inside of a circle. Inside of a circle means that I have to be somewhere here. So everything these three give us as a result. I mean these four. These four give us as a result should be positioned inside this circle. But let's just think what is the result of this? Well, x is positive, y is positive, but here we have a very interesting thing. We have x greater than y or equal and then we have x less or equal than y. This is quite an interesting thing because how can we intersect them? Well, very easily. The only thing which is really which really satisfies both x greater than y and x less than y or equal. Is x equal to y, right? Because if you have two numbers, one is always greater or equal to another and then at the same time it's less than or equal to another. Then the result is, so instead of these two, I can write x is equal to y. So this is what I have. And what is this? Well, y is equal to x is a straight line, but we are interested only in its positive range, which means only this one. And above and beyond everything, we have to cut from this the inside of the circle. So only inside of the circle is actually solution. So the result is this particular segment. Now you see it's not an area, it's a segment. So it's a segment of this line which lies inside the circle while including both ends. Since it's equal here, I include this border and and oh, sorry, not both ends. It's greater than, so it's only one end. So 0, 0 is not included. So this is basically the solution. This interval from 0 to 0.11, actually, right, because this is 0.11, everything including 0.11, but not including 0.00. And this completes this particular lecture about systems of inequalities. I will have another lecture probably, and that will probably be enough to get you acquainted with this particular area. So thanks very much for listening. I do recommend to, I do recommend you to start basically doing it yourself. The same problems. They are all in notes on unisor.com, obviously. And yeah, try basically to get the same results as I get here. I hope I was correct. And if anything is wrong, just send me an email. All right, good luck, and thank you very much.