 Irrational in equations. Now we are going to talk about irrational, irrational in equations. Irrational equations are basically those in equations which involve irrational functions. Okay. So we'll talk about different forms of irrational equations and how do we deal with those kind of in equations? We primarily deal with irrational in equations by converting it to a system of equivalent in equations. You'll get an idea about the same once I start discussing with you different, different forms. We'll start with the first type of in equation that you will come across. Let's say I ask you, I ask you that there's an in equation of this form f of x. Okay. Let's say some event root is there and the same event root is on g of x. Okay. And let's say you're trying to solve this. Okay. And here is some natural number. Okay. So let's say f of x and g of x both have been subjected to an event root. Same event root. And one of them is less than the other. Okay. Now, if I ask you, how would you solve this in equation by converting it to a simpler set of in equations? Then what all in equations will you take into consideration while solving this? Can I hear from anybody? I would like to know your opinion. This is equivalent to solving this set of in equation. So what some many people say, sir, first of all, f of x should be greater than equal to zero. Okay. Second of all, g of x also should be greater than equal to zero. Now, why is this condition imposed? Because f of x and g of x will not be real value till this condition summit. And then they say, sir, g of x would be greater than f of x. Is this what all of you are thinking? Anybody who has a different opinion? That will give you the third condition, right? From there only we got the third. From that condition that you are stating, we got the third one, right? Right. Now, this is correct. I'm not denying it. But let me tell you, you are wasting your time. You are wasting your time honoring the second condition over here. That means this condition is a waste. This is a redundant condition. Can you tell me why? Can somebody tell me why this is in a redundant condition? Right. You are already taking care that f of x is positive and you are already saying g of x is greater than f of x. That means automatically f of x is greater than zero by transitive property. Okay. So even if you solve it, that is not going to bring any new information to you. Probably it will have some overlapping zone with the first and the third one already. Getting my point. So in your exam time is an important factor. So don't waste your time and energy entertaining conditions which have been already been taken up. Those are called redundant conditions. Okay. Let's take a question on the same. Let's try to take a question on the same. Let's say you are trying to solve this equation under root of x minus 1 is less than under root of 2x minus 3. Okay. Let's solve this. Done. Very good. So here what will you say number one? First of all for this function to be real valued. Okay. This should be greater than equal to zero. Okay. This should be greater than equal to zero. So x should be greater than equal to 1. Okay. Now both the quantities are positive. So you can take the squares on both the sides. So you can say it is as good as saying this. Okay. That means that means 2 is less than equal to x. That means x is greater than equal to 2. Okay. So one condition is this. One condition is this. What is the overlap of these two conditions? You will definitely say it is greater than equal to 2. So your x will belong to 2 to infinity. 2 to infinity. So this will become your solution. If you want, you can, you know, take an additional fact that 2x minus 3 should also be greater than equal to zero. That means x should be greater than equal to 3 by 2. Let me tell you, even if you take this condition into account, the overlap of all these three, the overlap of all these three will still remain this. And hence this is going to be your answer. Okay. So many times people do not remember the fact that you don't have to entertain redundant conditions. So you can go for the normal approach that you have and still you will find your answer. Okay. Let us move on to the second type. Yes. Yes. That is what I'm going to talk about. If you have an odd power. So let me let me call it as a of the first and let's say I take a B of the first. If you have a function raised to an odd through and this is less than equal to this is less than equal to or less than whatever. I mean, whichever inequality you want to take up. Please any other function raised to the same odd power. Remember the condition that it converts is the same as saying f of x is less than g of x. So we don't have this f of x greater than equal to zero condition coming over in. I'll take an example also for the same. Let us say I want to solve. In fact, you will only solve it. I say three by x plus one seven by x plus two whole race to the power of one fifth is less than six by x minus one race to the power one fifth. Solve this inequality. The difference between the first and the second case OSHA case. In this case we don't entertain the condition that f of x would be greater than zero. That's the difference. I think you must be done with this. This is as good as saying three by x plus one plus seven by x plus two is less than six by x minus one. Okay. Let us bring it to one side. So three by x plus one plus seven by x plus two minus six by x minus one is less than zero. Okay. Let me take the LCM. Let me take the LCM. So x plus one x plus two x minus one on the numerator. If I'm not wrong, you'll end up getting three times x minus one x plus two seven times x square minus one minus six times x plus one x plus two. Okay. Less than zero. Let's see what all terms we get here. So three seven ten ten minus six, which is four x square. You will have three x you will have three x you will have minus 18x you'll have minus 15x. You will have minus six minus six minus seven is minus 13 minus 13 plus minus 13 minus 12 is minus 25. Okay. I think this is factorizable. 100 is I think 20 into minus 20 plus five x. So x minus one x plus one x plus two less than zero. Basically, it reduces to a rational function in equality. So make a wavy curve. Make a wavy curve. So the lowest value here would be minus two. Then minus five by four. Minus one one and a five plus minus plus minus plus minus. Okay. So less than zero. I have to write. I think we'll over here. Less than zero. Your x value will be minus infinity to minus two union minus five by four to one union one to five. This is going to be around. I don't even attempt to write it. It's too big. Actually. Auto written it very, very good. Is this clear? Is this clear everybody? Okay. Now let me take the situation to second type of irrational inequality. Let's say you have a function raised to an even power less than a quantity g of x less than a quantity g of x. Can somebody tell me this is equivalent to solving what all inequalities? Or what all inequalities will together help us to solve this inequality? So the first thing that we need to honor here is that f of x should be greater than equal to zero. Okay. Secondly, g of x also should be greater than zero. Else what will happen? There'll be no solution because this quantity is positive. And if this quantity happens to be negative, then there would be no solution. Because if g of x is negative. Okay. Okay. There would be no solution. Okay. There'll be no solution getting the point. So it can't be even less than equal to zero also. Second thing is your f of x should be less than g of x raised to the power of twin. Are you getting my point here? Now many people ask me, sir, why is this and this not sufficient enough is because what if your f of x is positive? And what if your g of x is negative but raised to an even power, it becomes positive? Are you getting my point? So this condition still will not be fulfilled. Do you get what I'm trying to say? If let's say you only took care of the first and the third case f of x is positive. That's why this function is a real valued function. Let's say g of x is a negative function. But when you raise it to the even power, it becomes positive. Correct. So this will still violate the original equation or original in equation. So as to say that's why even the second condition becomes important in this case. Are you getting my point here? Okay. Let me take a question on this. Let me take a question on this. Let's say I want to solve. I want to solve this question under root of x plus 14 is less than x plus two. Please solve this question. Yes. Anybody? Correct. Correct. Okay. So first of all, we need to honor the fact that x plus 14 should be positive. Second x plus two should be positive. Okay. And third we can say x plus four should be less than x plus two the whole square. Okay. So these three conditions must be simultaneously be true. Okay. Let's work on the last one here. So x plus four is less than x square plus four x plus four. Right. So x square plus three x is greater than zero. You can make a number line for the same minus three zero plus minus plus greater than zero means minus infinity to minus three union zero to infinity. Oh, it was 14. Oh, my bad. My bad. Sorry. Sorry. Sorry. Sorry. I got carried away in the. This is actually type one mistake copying the question wrong. Sorry. Yeah. So this will give you x square plus three x minus 10 greater than zero. I think it is factorizable as x plus five into x minus two greater than zero. So we have minus five and two plus minus plus greater than zero means it should be either be less than minus five. Less than minus five or greater than two. At the same time, this guy says greater than minus 14. The first guy says greater than minus 14. That's minus 14 to infinity. Square brackets. Okay. The second one says greater than minus two again square brackets. Let's take the overlap on the real number line. That's the best way to take the overlap. So we have some values like minus 14 minus five. We have a minus two. We have a two. The first interval says minus 14 and onwards second says minus two onwards. The third one says minus infinity to minus five union two to infinity. Okay. Sorry for writing it above it. So the overlap is only happening in this zone overlap is only happening in this zone. So your answer for this will be x should belong to two to infinity. That's it. This is going to be your answer. Is that clear? Any questions here? Please do let me know. Now the B case of it is when you have f of x raised to an odd root less than a quantity g of x. Now this is as good as solving f of x is less than g of x to the power of two n plus one. That's it. Okay. So this is very easy. So I'll just move forward. The third case that I'm going to take is slightly surprising. It's basically the other way round. What if f of x to the power of one by two n is greater than g of x. Now remember in the previous one, I had taken this. Please be vigilant about what I'm writing here. Here I've taken f of x to the power one by two n is less than g of x. Okay. And now I'm taking f of x to the power one by two n is greater than g of x. Okay. So can somebody tell me it is equivalent to which system solving which system of any equations. Anybody wants to unmute himself or talk? I didn't get that auto. Why don't we have one class for fun just to chill? Actually, it's not a class 10th thing, right? We have in class 10th, you can cover the syllabus in four months and revise for the rest of the eight months. Here, I'm not able to, here sometimes I'm not able to finish the syllabus also. My weekends are not free. I have class even on Sundays. Yeah, quizzes and alls we can keep. That's why in 11th and 12th, you'll realize that there's very less holiday and cancellation of class will be very, very rare. Yeah, we'll have those kind of competitions. Don't worry. As of now, we have to meet our target. I have only completed two chapters. So as to say, guys here, everything depends upon this g of x. See, there were two cases arising. Let me write it as case one and case two. If your g of x is negative, the only thing that you need to honor is your f of x should be positive. That's it. So that means these two conditions will be sufficient enough for you to solve this. But if your g of x is positive, then you have to take care of the fact that f of x should be greater than equal to g of x to the power two n. Now, many people have seen they will take this condition also f of x is greater than equal to zero. But this is redundant. This is redundant because you've already claimed g of x to be greater than zero and any power raised to it will also be greater than zero. So f of x will already is taken as positive. So these are the two conditions that we need to solve simultaneously. Okay, so what do I do after solving these two? You have to take their union because these are two cases. These are two cases. See, I'll explain it once again. See, everything is in terms of variables. So we don't know what is g of x? What is f of x exactly? They're all in terms of unknowns. So two conditions will arise here. One is if your g of x is less than zero and let's say f of x is greater than equal to zero, then what will happen and even truth of anything is always positive. So a positive quantity will always be greater than a negative quantity. Right. Only thing we need to assure is that this function should be real valued. So this condition is what is suggesting the same. Okay. So g of x less than zero and f of x greater than equal to zero will give you all the possible conditions, all the possible solutions for such equation. Second case that may arise is if g of x is positive, if this is positive, then you have to solve f of x greater than in fact not greater than equal to greater than because in the question I've taken greater than so you have to take greater than only. You have to take f of x greater than this raise to the power of 2n. That means you're raising both sides to the power of 2n. Are you getting my point here? Okay. Let's take a question on the same. Then you'll understand it in a better way. Let's say I asked you to solve this question. Everybody please try this out. Very good Rubav. Very good. That's the correct answer. No Oshik, they ask solutions. Don't worry. Oshik, it's an inequality. Well, you'll get a range of values. What only Rubav has answered so far. What about others? Almost. Okay. I think your brackets are not correct. I think the figures are correct. Check your brackets, please. Okay, let's discuss this. See, think from common sense point of view. Think from common sense point of view. If this guy is negative, let's say if this guy is negative, that means this guy is negative. Then only thing you need to take care is that this, the left hand side function must exist, right? Because anyways, this is positive and this is negative. So this will be true for all those values of X for which, for which this is negative and this is a real valued function. Isn't it? Right. So this is case one for you. So let us solve it simultaneously. This means X should be greater than three. Okay. This means if I'm not mistaken, this is less than equal to zero. So it's X minus one, X minus three is less than equal to zero. That means X should lie between one and three. I feel there is no overlap between these two. That means there's a null set. Okay. So first condition doesn't give me any result. Second condition is where you are taking six minus two X as a positive quantity. You can take greater than equal to zero also. No worries about it. Okay. If you're taking this as greater than equal to zero, then both the quantities become positive, right? So it is as good as solving. It is as good as solving this. Yes or no. So this means first one means X should be less than equal to three. For the second one, let's square it. So minus X squared plus four X minus three is greater than 36 X minus 24 X plus four X squared. Okay. So I think if you solve it, you'll end up getting five X squared minus 28 X plus 39 is less than zero. Okay. I think this is factorizable as X minus three and five X minus 13. Check it out. That means X should lie between 13 by five and three. And X is less than equal to three. What is the intersection of this and this? The intersection of these two is this only. Correct. So what is the overlap of these two? What is the overlap of these two overlap of these two? Is this only because this is saying less than equal to three. And this is saying from 13 by five, that is 2.623. So ultimately, this is your going to be your output from the second condition. Now this output and this output null set, you have to take their union. You have to take their union. So if you take their union, you will only end up getting the answer as 13 by five to three. So that's going to be your answer. Okay. So Priyam, absolutely correct Priyam. Who else? Pradyan, three by five or 13 by five? Or three is not included. Gurman, three is not included. Aditi is correct. Rubhava is correct. Awesome. Very good. So now many people ask me, sir, what about the second case when you have an odd throat? So let me call this as an A. Let's take a B scenario. If you have f of x raised to the power an odd throat greater than g of x. This is equivalent to just solving f of x is greater than g of x to the power two n plus one. Odd truths are the most chill people. Okay. So solving this is as good as solving this. Plain and simple. You don't have to worry about it. Okay. Let's take a few questions on whatever concepts or whatever types we have discussed so far because more or less all the questions that you are going to see will fall under these types. I think we have done a similar type. Let's do this question. Think carefully and answer. Don't be in a hurry. Yes. Any success anybody? No, Gurman. That's not correct. Your answer is only half correct. I would say that. Shitage. That's not correct. You can't go to minus one because minus one will make the numerator non-rear in character. Minimum you can go is half. No Pradyan, no Sanjana. That's not the complete answer. Partly it is correct. Partly Gurman's also is correct. Partly Pradyan's also is correct. See guys, most of you are not. Ah, there comes one answer. Oh, no, no. Oshe calls are not correct. First of all, if you want this function to exist, 2x minus 1 should be greater than 0, greater than equal to 0, right? That means x should be greater than equal to half. If this condition is violated, the left hand side function will become non-real in character. So you can't violate this condition. That's correct Parvati. Parvati has given the right answer. Very good Parvati. Bingo. Exactly, your brackets also are absolutely correct. Well done. Very good. Yes, all of you please pay attention here. When x is greater than half, then only this inequality will work. Okay. Now try to understand here. Numerator is under root of something. So this always has to be positive. No doubt about it. Correct. So if I make denominator negative, can I say the whole quantity will become negative and hence it will be less than 1? Okay. So can I say when x minus 2 is negative, right? That means x is less than 2. But still greater than half, but still greater than half, this inequality will always work. That means one interval that I've got is half to 2, right? So for this interval, this inequality will be true. Okay. But if your x minus 2 is positive, let me take one first situation and let me take the second situation. If x minus 2 is positive, then you can rewrite this entire inequality like this because now your denominator is positive. You can very well take it to the other side. In other words, you can multiply with x minus 2 throughout. Now this is a positive quantity. This is a positive quantity. So you're free to square both the sides. That means you end up getting x square minus 6x plus 5 greater than 0, which I believe is factorizable as x minus 1, x minus 5 greater than 0. So this means x could be less than 1 or x could be greater than 5. But less than 1 cannot happen because of this parent condition. Parent is saying you have to be greater than 2 for you to write like this. So this cannot come into your interval. This cannot be a part of your answer. So greater than 2, greater than 5, the overlapping condition is greater than 5. So 5 to infinity comes from it. So this condition and this condition, you have to take the union. So you have to take the union of these two conditions. So your answer is going to become half to 2, union 5 to infinity. Dear all, I think you are taking irrational function inequality very lightly. Don't do that, please. Irrational functions have to be treated very carefully. So the only person who got this correct was Parvati. Well done Parvati, very good. Alright, we'll take more questions. Don't worry, we still have 10 minutes time. So you can take your revenge. Revenge from the question, not from anybody, from the question. Solve this, solve it. No, Shik, not that simple. That's not the right answer. Of course, that condition must be met. But that's not the actual thing. For that, the left hand side functions would be defined. But are they greater than 1? Are the difference greater than 1? That's what we need to check. Yeah, certain all is coming. Correct, Shiddhich. Your guess is, oh my God, Shiddhich has got this right. Well done, Shiddhich. Big applause. Whistles, very good. Don't disclose your answer to anybody. Let others try. Okay, guys, in the interest of time, I have to start solving it because only five minutes for the class remain. See, one prediction that you guys have done is that this is greater than 0 and this should also be greater than 0. That means x should be greater than 6 and at the same time, it should be less than 10. That is fine. So x should be confined to 6 to 10. But this is not the complete answer. Okay, this is just a bare minimum thing to keep these functions real. Okay, so your answer has to be a subset of this. Maybe proper subset also. I don't know. I have to check it out. The next thing we can say is if this inequality is true, that means you're trying to say that this is true. You are basically adding under root of 10 minus x on both the sides. Now nobody can deny the fact that both the terms are positive over here. All of you confirm that both the sides are positive. So when you know both the sides are positive, you can square it and the inequality will remain the same. Correct me if I'm wrong. Any questions yet? That means 2x minus 17 is greater than equal to 2 under root of 10 minus x. Now a solution will only exist when you have 2x minus 17 greater than equal to 0. Else no solution can exist. So this is another condition we need to take care. That means your x should be greater than 17 by 2. So this is further restricted. As you can see 17 by 2 is approximately not approximately exactly 8.5. So the answer has to be between 8.5 to 10. Even 6 and all will not work. Even 6 and between 6 and 8.5 will not work. So this is one more condition which we need to honour. Let me just put a cloud symbol on those conditions which we need to honour. This is already there. This is also there. Now if you are claiming this is positive then you can further square it. So you can further square it now and you can write it like this. But not before you have taken cognizance of this fact. This is going to give you 4x square. 68 plus 4 minus 64x and 289 minus 40. 249 greater than equal to 0. This is a super ugly quadratic equation which I think is not factorisable. We have to use our Sridhar Acharya formula. So x is equal to 64 plus minus under root b square minus 4ac by 2a. One thing good is you can take some things out. For example I can cancel a factor of 4 easily from it. So this will be 16 plus minus under root. When 4 comes out this will be 249 and this will leave 164 square. How much will this be? 16 is coming out. 16 is coming out from this. That means a factor of 4 that will lose. So it will be 16 square. So 256 by 2. So this will be 16 plus minus root 7 by 2. Now if you look at the wavy curve. 16 minus root 7 by 2 is here. 16 plus root 7 by 2 is here. This is plus minus plus. So if you want it to be greater than equal to 0. It has to be either greater than this. That means your x should be either less than this. Or x should be greater than this. Less than equal to greater than equal to. So this is your third condition. Third condition that you need to honor. Now these three conditions must be simultaneously holding true. So let me make a big number line for the same. Let me plot the critical points. If I am not mistaken. 6 will be the least. Okay. Then I will get 16 minus root 7 by 2. Am I right? Am I right? Is my calculation correct? Or 17 by 2 will come first. No, that's correct. Then 17 by 2 will come. And then 16 plus root 7 by 2 will come. And then will come 10. Right. I hope I am writing the exact hierarchy here. So one says between 6 to 10. Other says between greater than 17 by 2. And the next one says less than this guy. And greater than this guy. Where do you think is the overlap happening? I can only see the overlap happening here. Okay. So your answer would be from 16 plus root 7 by 2 to 10. And that's exactly what Siddhas had also given. Well done. This was not an easy question. Given that you're doing this topic for the first time. Okay. Good enough. Very good guys. Very happy with your performance. Next class when we meet we are going to talk about quickly. Quickly. Exponential inequalities. And we'll talk about some trigonometric inequalities also. Because I think one more class we have. Okay. And then we'll start with the next topic. Which topic do you want me to start after inequalities?