Modulus Inequalities Overview

IIT JWE mains and JWE advanced We are preparing for the mathematics This is free of cost, quality online education is being given to you From kindergarten to 12th standard Here we are talking about your coordinate geometry, calculus, vectors in 3D, algebra and trigonometry Means majorly your whole syllabus We have divided it into 5 chunks and covering all the chunks one by one First sector, first segment, coordinate geometry where we have straight line, pair of straight lines, circle, parabola, ellipse and hyperbola. Now we are on calculus where we are talking about relations and functions. Along with this, remember that we have to prepare thoroughly. Why thoroughly? Because we consolidate all the syllabus, all the concepts from the examples. Then when the chapter is finally completed, we revise all the concepts through practice questions. Where we have to write down the answers to all the questions of the categories. exposure and finally culminate the chapter with the previous years questions of GWE Advanced. This means a strong preparation and of course free of cost. Relations and functions which is the first chapter of Calculus. We have talked about inequalities, we have talked about 1 by x, x square and all such relations. And we are exploring different things about modulus, i.e. mod x. In which today we will talk about that sir, If we want to take any expression of modulus, then we will talk about how to solve the inequalities based on modulus. Very interesting, very interesting fact. Although we have taken a good discussion, introductory session of this, but we will continue this and cover it in a good detailed manner. So with that, let's see what things unfold for us today. First of all, listen carefully to this statement. If I ask you a very simple, simple and solved thing, then something like this is written in front of you that mod x is less than equal to a. What is your interpretation of this line? I will try to explain it to you. Listen carefully. As someone says, I write mod x is less than equal to 3. What do you think after reading this line? I will say, look carefully. When someone is saying that mod x is less than 3 or equal to it. So, can I say that here it will be minus infinity. Here, it will be somewhere infinity. Forget all these things, you tell me what does mod x mean? Sir, whatever the definition of mod x is, we have understood mathematically. But for our understanding, we have treated mod x as its value or its meaning is that x is a number whose origins, i.e. distance from 0. Mod x means the origin of a number, i.e. distance from 0. Now, is that distance always equal to 3 or less than that? Can you understand what I am saying? I mean, shall I say that he wants to say that somewhere here your point will be 0. This is to say, you have to find all those numbers, from 0, you have to find all those numbers which are within 3 distance or less than 3 distance. I repeat my statement. I want all those numbers, all those points here, which hold value from O, i.e. origin, i.e. 0, to 3 unit distance or less than 3 unit distance. For example, let me try to explain to you. Here we have taken a number. Look at this number. What is special about this number? The special thing about this number is that its distance is less than 3. But, if I had taken a number here, the special thing about that number or the problem would be that the distance from 0 to 3 units would be more than that number. Can you understand? The same holds true on the other side as well. If you observe things carefully, See, if I take a number from 0, then it is the distance and it is less than 3 units distance. Even if you take a number from minus 3, it is still less than 3 units distance. And 3 units distance will work, but not more than 3 units distance. I mean, you are able to understand the point. I want to say that if someone is saying that mod x is 3 or less than 3, I need All the values of x, its implication will be that all the values of x are allowed which are between minus 3 and plus 3. Did you understand this definition? I think it is a very simple, simple, sorted, direct definition. In this, we will do one more small thing. Listen carefully. Like if someone tells me, if someone tells me, listen carefully. Someone may say that mod x is greater than or equal to 2. What does this mean? See, for example, the same thing again. We take things on the number scale. And on the number scale, of course, here somewhere there is minus infinity. Here somewhere, what happens? Infinity. Right? Now somewhere in the middle, we will have 0 somewhere. Right? Now think about it. Mod x greater than or equal to 2. What does 2 mean? What does mod x mean? Mod x means a number whose distance is 0. We will assume that x is a number and its distance from 0 is mod x. Distance, mind you everyone, distance. So, if you go to the left hand side or right hand side, it does not make any difference and it doesn't matter at all. What matters for us is this, it is a very important thing to do, that sir, just remember this, that your distance should be more than 2 units. Now see, this distance is always greater than equal to 2. It means that the distance is either 2 or more than 2. Do you understand the requirement? Now, what does this mean? If this distance is more than 2 or 2, it means, sir, it is clear, where is the distance from 0? Sir, one distance is on this side, and one distance is on this side. Means, this distance is 2 units, and I want all those numbers which are equal or more than 2 units. So, shall I say, any number beyond 2 will be given to me at the equal or more than 2 units distance from the origin? If you take any number after 2, then it will be situated or placed at a distance of 2 units from 0. But will this be true only in any case? Look carefully. If I tell you that you take any number smaller than 2, then it will also be at a distance of 2 units from 0 or more. If someone is writing this and saying that mod x is greater than or equal to 2, mod x is greater than or equal to 2. So, what will I say? Listen carefully. I will say that x is either smaller than or equal to 2. Can you understand? We have another way to write this. If I want to write this, how will I write it? I will write that x belongs to minus infinity, Minus 2 k beech ka koi bhi number kiya fir. between 2 and infinity. And similarly, when we were talking about this here, we discussed another meaning of writing this. That another way is this, x belongs to any number between 3 and minus 3. In all these things, you must have noticed a small thing, that when I took these numbers, like 3 or 2, I took these positive numbers. Now, you can understand why I am taking positive. You can understand why I am taking non-negative numbers. Positive will be better than positive for now. More appropriate, non-negative. The reason for taking this is very simple. Try to understand it clearly. Look carefully. The reason for taking this number is very clear. What does clear mean? Look, if I tell you that mod x is greater than or equal to minus 2. First of all, there is no scope of being equal to minus 2. Because this value, the modulus of any number, is a distance. And distance is always positive. Or rather, at least non-negative. That is, there is no value greater than 0 or 0. But you have created a problem here that if you have increased it by minus 2, then this will always happen because distance is not negative anyway. So if you are saying that mod x, for which values will x be increased by minus 2, then this will always be true, right? Because there is no distance of minus 2. Or you think like this, tell me all those numbers whose mod is greater than minus 2. Sir, mod is always non-negative, means it is greater than 0 or 0. So, you can take any number, you can take minus 3.5, you can take 0 or 6. Mod of minus 3.5 is 3.5, mod of 0 is 0, mod of 6 is 6. So, take any number, its mod is always greater than minus 2. So if someone writes this, if ever, then I will say that this is true for all the real values of x. Whatever value of x you keep, this is justified because mod x is always like this. Not only this, but if someone had told me that mod x is greater than or equal to 0, I would say that this is also correct. Oh, like x square, how is x square? Always greater than or equal to 0. I hope you are remembering. So in the same way, mod x will always be greater than or equal to 0 and this will be true for every value of x. If someone asks me what values of x are true, then sir, can any number's distance from the origin be greater than 0 or 0? Sir, it is of all numbers. Are you able to understand the definition? I hope you are able to digest this. But with this, I want to tell you one more small thing. For example, if someone tells me yesterday that mod x is less than minus 5. Sir, please look carefully, you are talking nonsense. I want to explain to you that this is a very stupid thing, I mean, it is absolutely impossible. Because, understand, this is distance, this is minus 5. You think and tell yourself, can this distance ever be smaller than minus 5? How can distance be smaller than negative? Distance itself is a non-negative value. That means it is greater than 0 or 0, that means it is a positive value. And you are saying that distance should be smaller than a negative value. How is that possible? It is not negative at all. So, the question of being smaller than it is... Can you understand what I am saying? Let me explain it a little more clearly. You understand this, tell me any number whose mod is less than minus 5. Sir, all the numbers in the world have a mod greater than 0 or 0. So, there is no question of being less than minus 5. So, this will be a solution in which I will say that for this, no value of x will exist, that is, you will say 5. 5 means empty, null. 5 does not mean 0. I did not say that the value of x will be 0. I am saying that no value of x is acceptable because you try anything, will not give the mod of x equal to or less than minus 5. Are you understanding the requirement, condition and things? These are the things that are being written here and being tried to say that whenever I will get inequalities based on modulars, then how will we think about them? Read it carefully. First thing, someone may say that mod x is equal to or less than a, but the condition is that a should be positive, we have already talked about this. So, we will always remember that if a is equal to or less than a, then x will be between a and minus a. This is the first conclusion which you note down. That means if someone says that mod x is equal to 2 or smaller than that, then I will say that x is between 2 and 2. If mod x is smaller than 3, then I will say that x is between 3 and 3. Of course, here it was equal, so it will be equal here also. Here it is not equal, so it will not be equal here. This is a very basic, simple thing which you understand as I believe. There is no problem with this. I mean, we have gone towards our standard conclusion. Then what? Then the second important thing that comes out is this. Listen carefully. This is another version of this which we just saw. Sir, look carefully. First talk about this. Whenever I get a question, that f is less than or equal to a or f is greater than or equal to minus a. So what will we do in that case? I am coming on this. First remember this. Whenever someone says that mod of x is less than or equal to a, So, we are talking about this, but first listen carefully, you will conclude x between minus a to a. We have already done this, right? Apart from this, one more important thing, sometimes someone says to yesterday, sometimes someone says that your mod x is between 0 to a. Sometimes someone says that your mod x is between 0 to a. So, then you will have to keep in mind that a, sorry, here a will not come, basically x should have come, so x will be your minus a to a, Okay. and it will be excluding 0. This is something wrong as far as I can think. I will explain this in a different way. There is a little correction scope in this which I feel. Look carefully. Let's understand. As I told you yesterday that mod x is between 0 and 3. Mod x is between 0 and 3. So what do I need for the technique? I want all the values of x that are between 0 and 3. So the first thing is that any number mod is anyway greater than 0. So there is no need to write this anyway. And when it says that x should be greater than 3 or 3, then the question arises that x will be between minus 3 and 3. The only condition that I find wrong is that if the mod x is bigger than 0 and smaller than a, 1 is smaller than 1, so if it is 0, it doesn't seem equal. So what happens? So you say that you don't want those numbers which are at 0 unit distance from the origin. I mean, I don't want 0 because 0 is 0 in the distance. So, remove 0 and take all the other values. If there was a question that x is greater than 0, mod x is greater than 0 and 3k is equal to or smaller than that. So, you will say x is between minus 3 and 3, but there is a problem in this. The problem is that you don't take 0. You will say x is the values from minus 3 to 0 or you will say all the values from 0 to 3. But you will not include 0. Why? I don't want points with zero distance. And who is at zero distance from zero? Zero. So zero will not be included. I hope you are understanding this. And you are building a good understanding of this. That how and why he is writing things. With that, let's move a little further. Let's see some more things. In the same way, we will talk about one more thing. Which we have done. That sir, if someone says, Is mod x greater than or equal to a? Of course, this will also be true when our a is positive. We have already done this. So, it's a simple thing, sir. Either you take a value smaller than or equal to a, because now we are looking for points which are situated at a distance or more than a distance. And I will say that whenever someone says that mod x is equal to or greater than a, then x will be either equal to or less than or equal to or greater than a. It is a very simple thing. And this is being explained to you by different examples. Like if mod x is equal to 3 or bigger than that, then x minus 3 is equal to 1 or smaller than that or bigger than 3. If mod x is greater than 2, then x minus 2 is smaller or greater than 2. I hope you understood this. In the same way, sometimes a function I get a function. If I ever get a function, like suppose someone asks me in a question, Sir, the mod of e to the power x is, let's say, smaller than 2. So what will you say? If I remove the mod, I will say, e to the power x is, although this doesn't happen, but still, I will write it down for now, that it will be between minus 2 and 2. Are you able to understand what I am saying? Like if someone writes down, Like, if someone writes to you yesterday that log x is a function of yours, right? And this function of yours is greater than or equal to 5. So, you will say that if the value of your log x function is greater than or equal to 5 of its mod, then log x is either... It will be either smaller than minus 5 or this log x will be greater than plus 5. In this way, you must be thinking. Are you understanding these expressions? I hope this is not just valid on x, but also valid on functions. There is one more small thing. Listen to this carefully. This is a matter of work. Is parham We can also draw a direct conclusion, but I am teaching you indirectly that things have to be done internally. And how it is happening in reality, I will explain in both ways. First, look, if someone tells me that mod x is bigger than a and smaller than b. In between, I know that both a and b are positive. This is important when you apply inequalities on mod q. We have already discussed the problem of q regarding negative. Now if someone is asking me that mod x is between a and b, then what does this mean? First, I will try to explain the graphical meaning to you. If I bring things on the number scale, then this will remain minus infinity and this will remain infinity. Here it will remain 0 somewhere. I am assuming that a is small and b is big, of course. In mod x, the left hand side of a is this side and the right hand side is this side. Both are positive, so it is a small and b is big. So, shall I say a will be here. Instead of a and b, why not take some good numbers so that you can understand better. Like I write here 3 and here I write 5. So, I hope you are able to realize that here 3 will be, here 5 will be, here 0 will be. Similarly, Here minus 3 and here minus 5. Now when someone is telling you, first understand what this means. Now forget this for a while. Now just look at this part. Now just look at this part. What does this part mean? Sir, I will say that this part means I need all those values of x. whose origin is always greater than 3 or equal to 3. So, equal to 3 or greater than 3 means either go here or go there. But I have to satisfy this too. I mean, this and this. should satisfy. Means, both have to be satisfied. Do you understand? So, it is saying that I want all the values of x which are less than 0 to 5 units or less than 5 units distance. Less than 5 units or less than 5 units distance means either this, I hope you are understanding, or this. Because these are all the numbers which on the number scale, are situated at a distance less than 5 units. Now, when I want to satisfy both, what will I do? I will find the intersection of both. And if I find the intersection of both, then the points that are obtained are from minus 5 to minus 3, and from 3 to 5. Are you able to understand my point? Think like this. It is saying that we need a distance more than 3. So, 3 or more than 3, we get it from 3 forward or from minus 3 backward. It is saying that less than 5, equal to 5 or less than 5, so, equal to 5 or less than 5, I get it from 5 backward. or beyond minus 5, I get values here. And to satisfy both, the common intersection area of both is this. Do you understand what I am saying? Look, if you are not able to digest it like this, I am telling you this way, don't apply it, I am just explaining. If someone says that your mod x is between 3 and 5, what I will say is that I will split it into two parts. I will say that your mod x is greater than 3. And along with this, what do I want? I want that mod x is smaller than 5. When someone is saying that mod x is bigger than 3, what does this mean? We have just taken out that mod x is bigger than 3. Meaning that x will be smaller than minus 3 or x will be bigger than 3. Along with this, what do I want? I also want that mod x is equal to 5. When mod x is smaller than 5, then x will be between minus 5 and 5. Are you understanding these things? Now, in all these things, I want to tell you a small thing, that if you look at things on the number scale, if you look at things on the number scale, then this is minus infinity and this is plus infinity, right? Somewhere 3, 5, somewhere minus 3, minus 5, you will get, right? Now, you tell me what he wants to say. Look, first let's read this part. He says, minus 3 to the power of minus 3, so minus 3 to the power of minus 3 means this. It says that it is greater than 3. So, it said either take this or take that. This is the solution. And, what did it say? It said between minus 5 and 5. Now, do you understand? We are going there by roaming around. So, the common conclusion of both is and means Intersection, both are common. So, where is the common of both? One is this zone and the other is this zone. I will repeat the same thing, that if someone says that mod x is between 3 and 5, then I will give a conclusion that this is true when x is from minus 5 to minus 3 or from 3 to 5. Do you understand this? Do you understand the final conclusion? The final conclusion is that if someone says that mod x is is between a and b. We know that both a and b are positive and we also know that a is greater than 0 and b is greater than a. Meaning both are technically positive. And in between, I will say that I don't need to think much. I have understood this and learnt it. If someone asks me, who does x belong to? So if mod x is between a and b, then from where will x be? Sir, it will be from minus b to minus a. Or it could be A to B. Did you understand the conclusion? This is what is being explained here. I hope you have understood things very well with this example and clarity. No confusion, no doubt. I hope this is clear. With that, let's move ahead and shift our focus to some new things. Will you remember all these things that have been taught? If you remember all these things, then let's try this question. See what it is writing. With this, you will be able to build a good understanding of modulus. And until you don't understand this, whenever you encounter a mod in maths, you will be scared and scared. So generally, these are very small and simple things. But we are discussing them in great detail so that I want you to up comod. 1 by x or x square, based on these inequalities, you should not make mistakes. You should encounter them properly and reach the right answer logically. You know the reason why we are doing this. So, like here if I am talking, leave all the other things. If you have any confusion, then I will assume for a little while, mod over x minus 1 minus 5, for a little while, let me write k. I don't have to write this, just to explain to you. So, what is this question written? This question is written, mod of k, mod of k greater than equal to 2. Now, you understood what I said. And if someone says mod of k is greater than equal to 2, then we say that in this case, k will be either less than minus 2, or k will be greater than plus 2. Now, you understood what I said. Now, I have written all these things to explain to you. I mean, I want to tell you that if any expression's mod is greater than 2, I mean, I want all those points which are more than 2 unit distance. The value of mod is either less than or equal to minus 2. And the value of mod is always greater than or equal to plus 2. Do you understand both these reasons? Let's simplify this a little more. Godson will say, why is this conjunction added more? You can understand, right? Either this or this. And it won't be added. You can understand the difference. Now what, sir? Now look, take minus 5 there. Then how much will it be? This will be mod over x minus 1. When minus 5 goes there, what happens? Try to understand the thing carefully. When minus 5 goes there, then plus 5? Minus 2 plus 5 is 5 minus 2, that's 3. So mod over x minus 1 is what? 3 to the power of 1. Let's think about the same thing here. Here it is written and here it is written mod over x minus 1. If minus 5 is there, then how much? Sir, this is 7. Are you all able to understand this? Are you all understanding this or not? Again, I will try to explain one more thing. For example, let's assume that x-1 is lambda. So, what is written here? It is written that lambda is less than or equal to 3. And when someone says that mod lambda is less than or equal to 3, then what do we call lambda? Sir, lambda is between minus 3 and 3. Are you listening to me? I hope you are able to understand after listening to me. So, this is what is written here. mod over x minus 1. This mod of x-1 is equal to 3 or smaller than this. So, shall I say, I will conclude that x-1 will be between 3 and 3. And along with this, I have to give one more thing here that if... someone says mod over x minus 1 is greater than 7 or 7. Again, let's assume that I call x minus 1 lambda. So, he is saying that the mod of lambda is greater than 7. So, what conclusion do you draw from this? You say, sir, that the lambda will be either less than 7 or greater than 7. I hope you have not forgotten this. So, I am trying to apply this here and trying to explain you things. If you are able to understand things properly. So, what will we do here? What will this mean, sir? This means that x minus 1 is either less than or equal to minus 7 or x-1 will be greater than plus 7. Are these things getting clear? There is no doubt or problem. I hope you are understanding every solution clearly. Then we will talk about this expression. If I want to sort out this, I try to remove minus 1, so I do plus 1 on all three sides. one So, minus 3 plus 1 is what? Minus 2. This is what? X. And this is what? 4. Right? Either this will happen, or what will happen? Either. If we work out this part, then shift 1 to that side. Shift 1 to that side, then what is it saying? It is saying, look carefully, this is what? Minus 7 plus 1, that is minus 6. So, what is this? It is saying, X is smaller than minus 6. Is there any problem or problem with this? I hope you are agreeing with that. And similarly, if I find here, then what is it saying? If we take minus 1, then it is 8. So it is saying that your x is greater than 8 or greater than 1. Are you understanding all three things? Are you understanding all three things very clearly? Now I will try to plot all three things on the number scale without thinking too much. Just don't use your brain. Now work out the things that you have been told. For example, what will happen somewhere, sir? Minus 8. Minus infinity. Somewhere infinity. Now look at the numbers. What are these, sir? Minus 6, minus 2, 4 and 8. Somewhere it is minus 6, somewhere it is minus 2, somewhere it is 4 and somewhere it is 8. We don't need any other number. Now because there is more between all of them, more means union. Means we don't have to take intersection because if there is AND anywhere, then I would say take out the common in this and in this. But there is more between these three, means we have to take this too, we have to take this too and we have to take this too. And means union, means take whatever is given, that will be your answer. So what will be your answer first? I will start from here. Take all the small values from minus 6, sir. Sir, do you want to include minus 6 or yes? So, we include minus 6 and take all the smaller values from it. Absolutely, we took it. Then, what is he saying? Sir, take all the values between 4 and 2 and include them. So, from minus 2, all the values between 4 were taken and they were also included. Did you understand this? And what else? And if I see, sir, including 8, take all the bigger values than 8. This way, your solution will be made. Did you understand this? Do you see this solution clearly? That this is how you will get the answer. If I want to write the final answer of this question in a sorted manner, then I will write it on the next page. Or if there is a scope here, then will it work if I write here? Look carefully. I am writing your answer here. So if I am asked, to whom does x finally belong? So x finally belongs to, minus infinity to minus 6, include minus 6. Union, minus 2 to 4, minus 2 and plus 4, include both. Union includes 8 to infinity and infinity is not included anyway. So this will be your final answer. I just want to say that if you keep any value that lies in this domain, your inequality will be satisfied. But if you keep any value between minus 6 and minus 2, or any value between 4 and 8, then it will not be satisfied. This whole question, this whole solution, each step, has been understood in a very clear manner. if there is no problem in any step. So should we move forward and shift our focus on the next things? If I move forward from here, what comes out of the next part? Let's see. See, what comes out of the next question? Listen carefully. If we move towards the next question, then here is your next question. What will we do, sir? Sir, first of all, no intelligent person should think that we have to cross multiply mod x minus 2 here. This is a foolish approach which we don't do with inequalities. Do you remember? So what I will do is, the best idea is bring minus... this one on the left-hand side. So, let's bring it here. So, see what happens. It becomes minus 1 upon mod over x minus 2. And when this 1 comes, then minus 1 is greater than or equal to 0. Did you understand all this point? Now, if you keep it on this side, cross multiply, or repeat your L-seam process, then there is no problem. So, what happens? Sir, this becomes minus 1. This becomes minus mod x and plus 2. Divided by what? Divided by mod over x minus 2. Do you understand this point? I hope you are seeing it clearly. Sir, if we simplify it a little, will it be good? Absolutely, it would be a good thing if you do this. It is being tried to simplify it a little. So look carefully. Minus mod x as it is, 2 minus 1, 1. So, how much does this make? Sir, this makes 1 minus mod x divided by what? Divided by mod x minus 2. It's greater than or equal to 0. Now, to simplify our problem for a little while, to sort out our problem for a little while, I'll do a small work. Can I replace mod x with any variable, k, lambda, whatever you like, can I replace it with that? So, for a little while, I'll assume that your mod x, That is for the sake of the list. I hope you are not getting into any trouble with this. Okay, one more thing. If I had taken the negative sign common from here, if I had taken the negative sign common from 1 minus x, then what would that become? Sir, that would become mod x minus 1. But when you take the negative sign common and shift it to that side, then technically you have to multiply the negative signs on both sides. So what will be greater than or equal to? Sir, that will be less than or equal to. Are you able to understand the point? We are following the basic standard math rules and doing nothing. Now if I rewrite this expression, what will be the expression? You will see it a little more carefully. So the expression is, k minus 1 upon k minus 2 less than equal to 0. Sir, we have read this somewhere. Oh, we have seen this somewhere. We have learnt this in the method of sine scheme or the wavy curve method. And if we think about it, we have a standard method for this, minus infinity. Even if you don't write this, minus infinity plus infinity is by default understood. For numbers, when the leftmost goes to the extreme, then minus infinity and the rightmost will approach plus infinity. And which critical points are there in between? 1 and 2. So, you will get two terminating points, whatever you want to call it, plus 1 and plus 2. And let's know the simple thing, sir, plus, minus, plus, if there is still any doubt, then think about it. Look, try any value greater than 2. Any value greater than 2, let's say 4. 4 minus 1 is positive, 4 minus 2 is positive. So, what is this? Positive. So, definitely, this also has power 1, this also has power 1. So, there is no particular change. Just remember that you should not include 2. Sir, why don't we include 2? Because if the value of k is 2, then the denominator is 0, and the problem will arise, we cannot afford to take the denominator as 0. So, if it is plus, and here the power is 1 and 1, which means it is odd, then it will be minus and it will be plus. I hope you remember this sine scheme method, this Wavy Curve method, which we have discussed in detail. From here, what did I want? I wanted the expression less than equal to 0. What does less than equal to 0 mean? Can I include 1? I can include 1, because 1 has 0, and 0 is allowed, but, Denominator ka 0 hona allowed nahi hai, isli humne 2 ko nahi liya. Matlab agar mutse koi pooch raha hai, ki mujhe expression negative chahiye. Toh mai kahunga, k jo hai apka sir. K jo hai apka, wo kaha se kaa tak hona chahiye, wo 1 se 2 ke beech mein hona chahiye. Sir, aap ye keh rahe ho, aap ye keh rahe ho, ki k jo hai apka, wo 1 se 2 ke beech hona chahiye. Lekin question k par toh tha nahi. Question kis par tha sir? Question mein k ki jaga humare paas mod x tha. You can click mod x if you want, but you... unnecessarily confused. That's why we tried to write it this way. So, can I write from here that mod x will be smaller than 2 and bigger than 1? Did you understand this point? Is it coming to your mind after seeing this? Is it clicking in your mind after seeing this? I hope it is. That if your mod x is between a and b, that means I need such points, I need such numbers on the number scale which are 0 to 1 unit distance or is situated at less than 2 units of distance. Is it clicking? Where do I want to take it? I will say, sir, your answer is clear, it is straight, where do you want to go? You come from minus 2 to minus 1, I hope you remember these things. Or you come from 1 to Or you can go from 1 to 2. Why did we put open bracket on 2? Hope you understand. Why did we put close bracket on 1? You understand. I have written the answer directly from here. If you don't understand, think like this. I will tell you both the ways. Like, think on number scale. So, on number scale, first, The points for 1 and 2 will be 1 and 2 and on this side, minus 1 and minus 2. Now, he was saying that, please understand this, first tell me all those points, if I just consider this, then tell me all those numbers which are more than 1 unit distance. So, what does it mean to be more than 1 unit distance? Either I am ahead of 1 and 1 or minus 1 and behind minus 1. But he said that it is less than 2 unit distance. and the distance is less than 2 units. And what does it mean to be less than 2 units? It means that it is between 2 and minus 2. And if I find the intersection of these two, then you can see the intersection of these two. The intersection of these two is minus 2 to minus 1. And, 1 to 2. And 2 is excluded because 2 and minus 2 are not included for this portion. So, I hope you can see that this will be your final answer. Meaning, if x belongs to all these values, if x belongs to all these values, then your inequality will always be satisfied. These questions are not very difficult. These are based on the same basics that you have been taught. I hope you are able to build a very good understanding, a very good sense, how to solve the inequality based on modulus. I hope you know that IIT-GWE mains and GWE advanced mathematics are giving you free of cost quality online education. With that, let's call it a day here. Take care of yourself. Thank you so much for watching this video.

Why thoroughly? Because we consolidate all the syllabus, all the concepts from the examples. Then when the chapter is finally completed, we revise all the concepts through practice questions.

Where we have to write down the answers to all the questions of the categories. exposure and finally culminate the chapter with the previous years questions of GWE Advanced. This means a strong preparation and of course free of cost.

Relations and functions which is the first chapter of Calculus. We have talked about inequalities, we have talked about 1 by x, x square and all such relations. And we are exploring different things about modulus, i.e. mod x.

In which today we will talk about that sir, If we want to take any expression of modulus, then we will talk about how to solve the inequalities based on modulus. Very interesting, very interesting fact. Although we have taken a good discussion, introductory session of this, but we will continue this and cover it in a good detailed manner. So with that, let's see what things unfold for us today. First of all, listen carefully to this statement.

If I ask you a very simple, simple and solved thing, then something like this is written in front of you that mod x is less than equal to a. What is your interpretation of this line? I will try to explain it to you. Listen carefully. As someone says, I write mod x is less than equal to 3. What do you think after reading this line?

I will say, look carefully. When someone is saying that mod x is less than 3 or equal to it. So, can I say that here it will be minus infinity. Here, it will be somewhere infinity. Forget all these things, you tell me what does mod x mean?

Sir, whatever the definition of mod x is, we have understood mathematically. But for our understanding, we have treated mod x as its value or its meaning is that x is a number whose origins, i.e. distance from 0. Mod x means the origin of a number, i.e. distance from 0. Now, is that distance always equal to 3 or less than that? Can you understand what I am saying? I mean, shall I say that he wants to say that somewhere here your point will be 0. This is to say, you have to find all those numbers, from 0, you have to find all those numbers which are within 3 distance or less than 3 distance.

I repeat my statement. I want all those numbers, all those points here, which hold value from O, i.e. origin, i.e. 0, to 3 unit distance or less than 3 unit distance. For example, let me try to explain to you.

Here we have taken a number. Look at this number. What is special about this number?

The special thing about this number is that its distance is less than 3. But, if I had taken a number here, the special thing about that number or the problem would be that the distance from 0 to 3 units would be more than that number. Can you understand? The same holds true on the other side as well. If you observe things carefully, See, if I take a number from 0, then it is the distance and it is less than 3 units distance.

Even if you take a number from minus 3, it is still less than 3 units distance. And 3 units distance will work, but not more than 3 units distance. I mean, you are able to understand the point.

I want to say that if someone is saying that mod x is 3 or less than 3, I need All the values of x, its implication will be that all the values of x are allowed which are between minus 3 and plus 3. Did you understand this definition? I think it is a very simple, simple, sorted, direct definition. In this, we will do one more small thing. Listen carefully.

Like if someone tells me, if someone tells me, listen carefully. Someone may say that mod x is greater than or equal to 2. What does this mean? See, for example, the same thing again. We take things on the number scale.

And on the number scale, of course, here somewhere there is minus infinity. Here somewhere, what happens? Infinity. Right?

Now somewhere in the middle, we will have 0 somewhere. Right? Now think about it.

Mod x greater than or equal to 2. What does 2 mean? What does mod x mean? Mod x means a number whose distance is 0. We will assume that x is a number and its distance from 0 is mod x. Distance, mind you everyone, distance.

So, if you go to the left hand side or right hand side, it does not make any difference and it doesn't matter at all. What matters for us is this, it is a very important thing to do, that sir, just remember this, that your distance should be more than 2 units. Now see, this distance is always greater than equal to 2. It means that the distance is either 2 or more than 2. Do you understand the requirement? Now, what does this mean?

If this distance is more than 2 or 2, it means, sir, it is clear, where is the distance from 0? Sir, one distance is on this side, and one distance is on this side. Means, this distance is 2 units, and I want all those numbers which are equal or more than 2 units.

So, shall I say, any number beyond 2 will be given to me at the equal or more than 2 units distance from the origin? If you take any number after 2, then it will be situated or placed at a distance of 2 units from 0. But will this be true only in any case? Look carefully. If I tell you that you take any number smaller than 2, then it will also be at a distance of 2 units from 0 or more.

If someone is writing this and saying that mod x is greater than or equal to 2, mod x is greater than or equal to 2. So, what will I say? Listen carefully. I will say that x is either smaller than or equal to 2. Can you understand?

We have another way to write this. If I want to write this, how will I write it? I will write that x belongs to minus infinity, Minus 2 k beech ka koi bhi number kiya fir. between 2 and infinity.

And similarly, when we were talking about this here, we discussed another meaning of writing this. That another way is this, x belongs to any number between 3 and minus 3. In all these things, you must have noticed a small thing, that when I took these numbers, like 3 or 2, I took these positive numbers. Now, you can understand why I am taking positive. You can understand why I am taking non-negative numbers. Positive will be better than positive for now.

More appropriate, non-negative. The reason for taking this is very simple. Try to understand it clearly. Look carefully.

The reason for taking this number is very clear. What does clear mean? Look, if I tell you that mod x is greater than or equal to minus 2. First of all, there is no scope of being equal to minus 2. Because this value, the modulus of any number, is a distance.

And distance is always positive. Or rather, at least non-negative. That is, there is no value greater than 0 or 0. But you have created a problem here that if you have increased it by minus 2, then this will always happen because distance is not negative anyway. So if you are saying that mod x, for which values will x be increased by minus 2, then this will always be true, right?

Because there is no distance of minus 2. Or you think like this, tell me all those numbers whose mod is greater than minus 2. Sir, mod is always non-negative, means it is greater than 0 or 0. So, you can take any number, you can take minus 3.5, you can take 0 or 6. Mod of minus 3.5 is 3.5, mod of 0 is 0, mod of 6 is 6. So, take any number, its mod is always greater than minus 2. So if someone writes this, if ever, then I will say that this is true for all the real values of x. Whatever value of x you keep, this is justified because mod x is always like this. Not only this, but if someone had told me that mod x is greater than or equal to 0, I would say that this is also correct. Oh, like x square, how is x square? Always greater than or equal to 0. I hope you are remembering.

So in the same way, mod x will always be greater than or equal to 0 and this will be true for every value of x. If someone asks me what values of x are true, then sir, can any number's distance from the origin be greater than 0 or 0? Sir, it is of all numbers.

Are you able to understand the definition? I hope you are able to digest this. But with this, I want to tell you one more small thing. For example, if someone tells me yesterday that mod x is less than minus 5. Sir, please look carefully, you are talking nonsense.

I want to explain to you that this is a very stupid thing, I mean, it is absolutely impossible. Because, understand, this is distance, this is minus 5. You think and tell yourself, can this distance ever be smaller than minus 5? How can distance be smaller than negative?

Distance itself is a non-negative value. That means it is greater than 0 or 0, that means it is a positive value. And you are saying that distance should be smaller than a negative value. How is that possible? It is not negative at all.

So, the question of being smaller than it is... Can you understand what I am saying? Let me explain it a little more clearly.

You understand this, tell me any number whose mod is less than minus 5. Sir, all the numbers in the world have a mod greater than 0 or 0. So, there is no question of being less than minus 5. So, this will be a solution in which I will say that for this, no value of x will exist, that is, you will say 5. 5 means empty, null. 5 does not mean 0. I did not say that the value of x will be 0. I am saying that no value of x is acceptable because you try anything, will not give the mod of x equal to or less than minus 5. Are you understanding the requirement, condition and things? These are the things that are being written here and being tried to say that whenever I will get inequalities based on modulars, then how will we think about them?

Read it carefully. First thing, someone may say that mod x is equal to or less than a, but the condition is that a should be positive, we have already talked about this. So, we will always remember that if a is equal to or less than a, then x will be between a and minus a. This is the first conclusion which you note down. That means if someone says that mod x is equal to 2 or smaller than that, then I will say that x is between 2 and 2. If mod x is smaller than 3, then I will say that x is between 3 and 3. Of course, here it was equal, so it will be equal here also.

Here it is not equal, so it will not be equal here. This is a very basic, simple thing which you understand as I believe. There is no problem with this.

I mean, we have gone towards our standard conclusion. Then what? Then the second important thing that comes out is this. Listen carefully. This is another version of this which we just saw.

Sir, look carefully. First talk about this. Whenever I get a question, that f is less than or equal to a or f is greater than or equal to minus a.

So what will we do in that case? I am coming on this. First remember this.

Whenever someone says that mod of x is less than or equal to a, So, we are talking about this, but first listen carefully, you will conclude x between minus a to a. We have already done this, right? Apart from this, one more important thing, sometimes someone says to yesterday, sometimes someone says that your mod x is between 0 to a. Sometimes someone says that your mod x is between 0 to a. So, then you will have to keep in mind that a, sorry, here a will not come, basically x should have come, so x will be your minus a to a, Okay.

and it will be excluding 0. This is something wrong as far as I can think. I will explain this in a different way. There is a little correction scope in this which I feel.

Look carefully. Let's understand. As I told you yesterday that mod x is between 0 and 3. Mod x is between 0 and 3. So what do I need for the technique? I want all the values of x that are between 0 and 3. So the first thing is that any number mod is anyway greater than 0. So there is no need to write this anyway. And when it says that x should be greater than 3 or 3, then the question arises that x will be between minus 3 and 3. The only condition that I find wrong is that if the mod x is bigger than 0 and smaller than a, 1 is smaller than 1, so if it is 0, it doesn't seem equal.

So what happens? So you say that you don't want those numbers which are at 0 unit distance from the origin. I mean, I don't want 0 because 0 is 0 in the distance.

So, remove 0 and take all the other values. If there was a question that x is greater than 0, mod x is greater than 0 and 3k is equal to or smaller than that. So, you will say x is between minus 3 and 3, but there is a problem in this. The problem is that you don't take 0. You will say x is the values from minus 3 to 0 or you will say all the values from 0 to 3. But you will not include 0. Why?

I don't want points with zero distance. And who is at zero distance from zero? Zero. So zero will not be included.

I hope you are understanding this. And you are building a good understanding of this. That how and why he is writing things.

With that, let's move a little further. Let's see some more things. In the same way, we will talk about one more thing. Which we have done. That sir, if someone says, Is mod x greater than or equal to a?

Of course, this will also be true when our a is positive. We have already done this. So, it's a simple thing, sir.

Either you take a value smaller than or equal to a, because now we are looking for points which are situated at a distance or more than a distance. And I will say that whenever someone says that mod x is equal to or greater than a, then x will be either equal to or less than or equal to or greater than a. It is a very simple thing.

And this is being explained to you by different examples. Like if mod x is equal to 3 or bigger than that, then x minus 3 is equal to 1 or smaller than that or bigger than 3. If mod x is greater than 2, then x minus 2 is smaller or greater than 2. I hope you understood this. In the same way, sometimes a function I get a function. If I ever get a function, like suppose someone asks me in a question, Sir, the mod of e to the power x is, let's say, smaller than 2. So what will you say?

If I remove the mod, I will say, e to the power x is, although this doesn't happen, but still, I will write it down for now, that it will be between minus 2 and 2. Are you able to understand what I am saying? Like if someone writes down, Like, if someone writes to you yesterday that log x is a function of yours, right? And this function of yours is greater than or equal to 5. So, you will say that if the value of your log x function is greater than or equal to 5 of its mod, then log x is either... It will be either smaller than minus 5 or this log x will be greater than plus 5. In this way, you must be thinking.

Are you understanding these expressions? I hope this is not just valid on x, but also valid on functions. There is one more small thing. Listen to this carefully.

This is a matter of work. Is parham We can also draw a direct conclusion, but I am teaching you indirectly that things have to be done internally. And how it is happening in reality, I will explain in both ways.

First, look, if someone tells me that mod x is bigger than a and smaller than b. In between, I know that both a and b are positive. This is important when you apply inequalities on mod q. We have already discussed the problem of q regarding negative.

Now if someone is asking me that mod x is between a and b, then what does this mean? First, I will try to explain the graphical meaning to you. If I bring things on the number scale, then this will remain minus infinity and this will remain infinity.

Here it will remain 0 somewhere. I am assuming that a is small and b is big, of course. In mod x, the left hand side of a is this side and the right hand side is this side. Both are positive, so it is a small and b is big. So, shall I say a will be here.

Instead of a and b, why not take some good numbers so that you can understand better. Like I write here 3 and here I write 5. So, I hope you are able to realize that here 3 will be, here 5 will be, here 0 will be. Similarly, Here minus 3 and here minus 5. Now when someone is telling you, first understand what this means. Now forget this for a while.

Now just look at this part. Now just look at this part. What does this part mean?

Sir, I will say that this part means I need all those values of x. whose origin is always greater than 3 or equal to 3. So, equal to 3 or greater than 3 means either go here or go there. But I have to satisfy this too. I mean, this and this. should satisfy.

Means, both have to be satisfied. Do you understand? So, it is saying that I want all the values of x which are less than 0 to 5 units or less than 5 units distance.

Less than 5 units or less than 5 units distance means either this, I hope you are understanding, or this. Because these are all the numbers which on the number scale, are situated at a distance less than 5 units. Now, when I want to satisfy both, what will I do?

I will find the intersection of both. And if I find the intersection of both, then the points that are obtained are from minus 5 to minus 3, and from 3 to 5. Are you able to understand my point? Think like this. It is saying that we need a distance more than 3. So, 3 or more than 3, we get it from 3 forward or from minus 3 backward.

It is saying that less than 5, equal to 5 or less than 5, so, equal to 5 or less than 5, I get it from 5 backward. or beyond minus 5, I get values here. And to satisfy both, the common intersection area of both is this.

Do you understand what I am saying? Look, if you are not able to digest it like this, I am telling you this way, don't apply it, I am just explaining. If someone says that your mod x is between 3 and 5, what I will say is that I will split it into two parts. I will say that your mod x is greater than 3. And along with this, what do I want? I want that mod x is smaller than 5. When someone is saying that mod x is bigger than 3, what does this mean?

We have just taken out that mod x is bigger than 3. Meaning that x will be smaller than minus 3 or x will be bigger than 3. Along with this, what do I want? I also want that mod x is equal to 5. When mod x is smaller than 5, then x will be between minus 5 and 5. Are you understanding these things? Now, in all these things, I want to tell you a small thing, that if you look at things on the number scale, if you look at things on the number scale, then this is minus infinity and this is plus infinity, right? Somewhere 3, 5, somewhere minus 3, minus 5, you will get, right? Now, you tell me what he wants to say.

Look, first let's read this part. He says, minus 3 to the power of minus 3, so minus 3 to the power of minus 3 means this. It says that it is greater than 3. So, it said either take this or take that.

This is the solution. And, what did it say? It said between minus 5 and 5. Now, do you understand?

We are going there by roaming around. So, the common conclusion of both is and means Intersection, both are common. So, where is the common of both? One is this zone and the other is this zone. I will repeat the same thing, that if someone says that mod x is between 3 and 5, then I will give a conclusion that this is true when x is from minus 5 to minus 3 or from 3 to 5. Do you understand this?

Do you understand the final conclusion? The final conclusion is that if someone says that mod x is is between a and b. We know that both a and b are positive and we also know that a is greater than 0 and b is greater than a.

Meaning both are technically positive. And in between, I will say that I don't need to think much. I have understood this and learnt it.

If someone asks me, who does x belong to? So if mod x is between a and b, then from where will x be? Sir, it will be from minus b to minus a. Or it could be A to B.

Did you understand the conclusion? This is what is being explained here. I hope you have understood things very well with this example and clarity. No confusion, no doubt.

I hope this is clear. With that, let's move ahead and shift our focus to some new things. Will you remember all these things that have been taught? If you remember all these things, then let's try this question.

See what it is writing. With this, you will be able to build a good understanding of modulus. And until you don't understand this, whenever you encounter a mod in maths, you will be scared and scared.

So generally, these are very small and simple things. But we are discussing them in great detail so that I want you to up comod. 1 by x or x square, based on these inequalities, you should not make mistakes. You should encounter them properly and reach the right answer logically.

You know the reason why we are doing this. So, like here if I am talking, leave all the other things. If you have any confusion, then I will assume for a little while, mod over x minus 1 minus 5, for a little while, let me write k.

I don't have to write this, just to explain to you. So, what is this question written? This question is written, mod of k, mod of k greater than equal to 2. Now, you understood what I said.

And if someone says mod of k is greater than equal to 2, then we say that in this case, k will be either less than minus 2, or k will be greater than plus 2. Now, you understood what I said. Now, I have written all these things to explain to you. I mean, I want to tell you that if any expression's mod is greater than 2, I mean, I want all those points which are more than 2 unit distance.

The value of mod is either less than or equal to minus 2. And the value of mod is always greater than or equal to plus 2. Do you understand both these reasons? Let's simplify this a little more. Godson will say, why is this conjunction added more? You can understand, right? Either this or this.

And it won't be added. You can understand the difference. Now what, sir? Now look, take minus 5 there.

Then how much will it be? This will be mod over x minus 1. When minus 5 goes there, what happens? Try to understand the thing carefully. When minus 5 goes there, then plus 5? Minus 2 plus 5 is 5 minus 2, that's 3. So mod over x minus 1 is what?

3 to the power of 1. Let's think about the same thing here. Here it is written and here it is written mod over x minus 1. If minus 5 is there, then how much? Sir, this is 7. Are you all able to understand this?

Are you all understanding this or not? Again, I will try to explain one more thing. For example, let's assume that x-1 is lambda. So, what is written here?

It is written that lambda is less than or equal to 3. And when someone says that mod lambda is less than or equal to 3, then what do we call lambda? Sir, lambda is between minus 3 and 3. Are you listening to me? I hope you are able to understand after listening to me. So, this is what is written here. mod over x minus 1. This mod of x-1 is equal to 3 or smaller than this.

So, shall I say, I will conclude that x-1 will be between 3 and 3. And along with this, I have to give one more thing here that if... someone says mod over x minus 1 is greater than 7 or 7. Again, let's assume that I call x minus 1 lambda. So, he is saying that the mod of lambda is greater than 7. So, what conclusion do you draw from this?

You say, sir, that the lambda will be either less than 7 or greater than 7. I hope you have not forgotten this. So, I am trying to apply this here and trying to explain you things. If you are able to understand things properly.

So, what will we do here? What will this mean, sir? This means that x minus 1 is either less than or equal to minus 7 or x-1 will be greater than plus 7. Are these things getting clear? There is no doubt or problem. I hope you are understanding every solution clearly.

Then we will talk about this expression. If I want to sort out this, I try to remove minus 1, so I do plus 1 on all three sides. one So, minus 3 plus 1 is what? Minus 2. This is what? X.

And this is what? 4. Right? Either this will happen, or what will happen? Either. If we work out this part, then shift 1 to that side.

Shift 1 to that side, then what is it saying? It is saying, look carefully, this is what? Minus 7 plus 1, that is minus 6. So, what is this?

It is saying, X is smaller than minus 6. Is there any problem or problem with this? I hope you are agreeing with that. And similarly, if I find here, then what is it saying?

If we take minus 1, then it is 8. So it is saying that your x is greater than 8 or greater than 1. Are you understanding all three things? Are you understanding all three things very clearly? Now I will try to plot all three things on the number scale without thinking too much. Just don't use your brain. Now work out the things that you have been told.

For example, what will happen somewhere, sir? Minus 8. Minus infinity. Somewhere infinity. Now look at the numbers. What are these, sir?

Minus 6, minus 2, 4 and 8. Somewhere it is minus 6, somewhere it is minus 2, somewhere it is 4 and somewhere it is 8. We don't need any other number. Now because there is more between all of them, more means union. Means we don't have to take intersection because if there is AND anywhere, then I would say take out the common in this and in this.

But there is more between these three, means we have to take this too, we have to take this too and we have to take this too. And means union, means take whatever is given, that will be your answer. So what will be your answer first? I will start from here. Take all the small values from minus 6, sir.

Sir, do you want to include minus 6 or yes? So, we include minus 6 and take all the smaller values from it. Absolutely, we took it.

Then, what is he saying? Sir, take all the values between 4 and 2 and include them. So, from minus 2, all the values between 4 were taken and they were also included. Did you understand this? And what else?

And if I see, sir, including 8, take all the bigger values than 8. This way, your solution will be made. Did you understand this? Do you see this solution clearly? That this is how you will get the answer.

If I want to write the final answer of this question in a sorted manner, then I will write it on the next page. Or if there is a scope here, then will it work if I write here? Look carefully.

I am writing your answer here. So if I am asked, to whom does x finally belong? So x finally belongs to, minus infinity to minus 6, include minus 6. Union, minus 2 to 4, minus 2 and plus 4, include both.

Union includes 8 to infinity and infinity is not included anyway. So this will be your final answer. I just want to say that if you keep any value that lies in this domain, your inequality will be satisfied.

But if you keep any value between minus 6 and minus 2, or any value between 4 and 8, then it will not be satisfied. This whole question, this whole solution, each step, has been understood in a very clear manner. if there is no problem in any step.

So should we move forward and shift our focus on the next things? If I move forward from here, what comes out of the next part? Let's see.

See, what comes out of the next question? Listen carefully. If we move towards the next question, then here is your next question. What will we do, sir? Sir, first of all, no intelligent person should think that we have to cross multiply mod x minus 2 here.

This is a foolish approach which we don't do with inequalities. Do you remember? So what I will do is, the best idea is bring minus...

this one on the left-hand side. So, let's bring it here. So, see what happens. It becomes minus 1 upon mod over x minus 2. And when this 1 comes, then minus 1 is greater than or equal to 0. Did you understand all this point? Now, if you keep it on this side, cross multiply, or repeat your L-seam process, then there is no problem.

So, what happens? Sir, this becomes minus 1. This becomes minus mod x and plus 2. Divided by what? Divided by mod over x minus 2. Do you understand this point? I hope you are seeing it clearly.

Sir, if we simplify it a little, will it be good? Absolutely, it would be a good thing if you do this. It is being tried to simplify it a little. So look carefully.

Minus mod x as it is, 2 minus 1, 1. So, how much does this make? Sir, this makes 1 minus mod x divided by what? Divided by mod x minus 2. It's greater than or equal to 0. Now, to simplify our problem for a little while, to sort out our problem for a little while, I'll do a small work.

Can I replace mod x with any variable, k, lambda, whatever you like, can I replace it with that? So, for a little while, I'll assume that your mod x, That is for the sake of the list. I hope you are not getting into any trouble with this.

Okay, one more thing. If I had taken the negative sign common from here, if I had taken the negative sign common from 1 minus x, then what would that become? Sir, that would become mod x minus 1. But when you take the negative sign common and shift it to that side, then technically you have to multiply the negative signs on both sides. So what will be greater than or equal to?

Sir, that will be less than or equal to. Are you able to understand the point? We are following the basic standard math rules and doing nothing. Now if I rewrite this expression, what will be the expression?

You will see it a little more carefully. So the expression is, k minus 1 upon k minus 2 less than equal to 0. Sir, we have read this somewhere. Oh, we have seen this somewhere.

We have learnt this in the method of sine scheme or the wavy curve method. And if we think about it, we have a standard method for this, minus infinity. Even if you don't write this, minus infinity plus infinity is by default understood.

For numbers, when the leftmost goes to the extreme, then minus infinity and the rightmost will approach plus infinity. And which critical points are there in between? 1 and 2. So, you will get two terminating points, whatever you want to call it, plus 1 and plus 2. And let's know the simple thing, sir, plus, minus, plus, if there is still any doubt, then think about it.

Look, try any value greater than 2. Any value greater than 2, let's say 4. 4 minus 1 is positive, 4 minus 2 is positive. So, what is this? Positive.

So, definitely, this also has power 1, this also has power 1. So, there is no particular change. Just remember that you should not include 2. Sir, why don't we include 2? Because if the value of k is 2, then the denominator is 0, and the problem will arise, we cannot afford to take the denominator as 0. So, if it is plus, and here the power is 1 and 1, which means it is odd, then it will be minus and it will be plus. I hope you remember this sine scheme method, this Wavy Curve method, which we have discussed in detail. From here, what did I want?

I wanted the expression less than equal to 0. What does less than equal to 0 mean? Can I include 1? I can include 1, because 1 has 0, and 0 is allowed, but, Denominator ka 0 hona allowed nahi hai, isli humne 2 ko nahi liya. Matlab agar mutse koi pooch raha hai, ki mujhe expression negative chahiye.

Toh mai kahunga, k jo hai apka sir. K jo hai apka, wo kaha se kaa tak hona chahiye, wo 1 se 2 ke beech mein hona chahiye. Sir, aap ye keh rahe ho, aap ye keh rahe ho, ki k jo hai apka, wo 1 se 2 ke beech hona chahiye.

Lekin question k par toh tha nahi. Question kis par tha sir? Question mein k ki jaga humare paas mod x tha. You can click mod x if you want, but you...

unnecessarily confused. That's why we tried to write it this way. So, can I write from here that mod x will be smaller than 2 and bigger than 1?

Did you understand this point? Is it coming to your mind after seeing this? Is it clicking in your mind after seeing this?

I hope it is. That if your mod x is between a and b, that means I need such points, I need such numbers on the number scale which are 0 to 1 unit distance or is situated at less than 2 units of distance. Is it clicking? Where do I want to take it?

I will say, sir, your answer is clear, it is straight, where do you want to go? You come from minus 2 to minus 1, I hope you remember these things. Or you come from 1 to Or you can go from 1 to 2. Why did we put open bracket on 2?

Hope you understand. Why did we put close bracket on 1? You understand. I have written the answer directly from here. If you don't understand, think like this.

I will tell you both the ways. Like, think on number scale. So, on number scale, first, The points for 1 and 2 will be 1 and 2 and on this side, minus 1 and minus 2. Now, he was saying that, please understand this, first tell me all those points, if I just consider this, then tell me all those numbers which are more than 1 unit distance.

So, what does it mean to be more than 1 unit distance? Either I am ahead of 1 and 1 or minus 1 and behind minus 1. But he said that it is less than 2 unit distance. and the distance is less than 2 units.

And what does it mean to be less than 2 units? It means that it is between 2 and minus 2. And if I find the intersection of these two, then you can see the intersection of these two. The intersection of these two is minus 2 to minus 1. And, 1 to 2. And 2 is excluded because 2 and minus 2 are not included for this portion. So, I hope you can see that this will be your final answer.

Meaning, if x belongs to all these values, if x belongs to all these values, then your inequality will always be satisfied. These questions are not very difficult. These are based on the same basics that you have been taught.

I hope you are able to build a very good understanding, a very good sense, how to solve the inequality based on modulus. I hope you know that IIT-GWE mains and GWE advanced mathematics are giving you free of cost quality online education. With that, let's call it a day here. Take care of yourself.

Thank you so much for watching this video.

Transcript for:Modulus Inequalities Overview

Transcript for:
Modulus Inequalities Overview