Wavy Curve Method to Solve inequalities 🔥 | Anurag Sir | Vedantu Learning Centre #vedantu #vlctelugu

追加: 2026-05-15

[00:00:02]This is the question we are doing now again step by step. Step number one as per the rule what are the rules? The rules of var method. The first rule make your RHS is zero. Check in this is my RSS zero. First of all my RS is zero.

[00:00:19]Rule number two. Sorry. Step number one.

[00:00:23]Make the coefficient of X as positive in each factor. cofficient of x as positive here. What is the cofficient of x?

[00:00:32]>> One. Here also one. Here also one. It's already positive. No need to worry.

[00:00:37]Already positive. Then step number two. Put all factors equals to zero. Put each and every factor equal to zero and get the corresponding value of x. Answer me. If I put this one zero, x - 1 =0. x - 2 =0. x - 3 =0. Answers are x = to 1, x = 2, x = 3. Then I'll draw number line. In the number line, minus infinity, infinity.

[00:01:12]Now here there will be two types of questions. One very basic, very easy, which is this. Next little bit complicated will come that again we'll take one more example. I'm plotting these numbers first. 1 2 3 clear how many intervals you got here intervals means how many intervals I'm getting this is my first interval second third four how many values of x three values are giving me how many intervals four intervals okay now we all are looking from this side now who is the rightmost interval 3 to infinity. I'll always put plus symbol in this interval. Always, always, always.

[00:02:00]If I have taken care of all the steps till now, what was the first step?

[00:02:05]>> RHS is zero. Making all the >> coefficients of X as positive. If I have taken care of both the steps, then I will always put the rightmost interval as plus.

[00:02:18]Okay. Now there will be two types of questions after this. First type of question in which it will always follow this pattern. If one interval is plus, other interval will be >> minus. If this is minus, then this will be >> plus. If this is plus, then this will be >> minus. This is one type of questions.

[00:02:38]That means in every interval there will be a change of sign.

[00:02:44]When that change of sign will happen, when that change of sign will not happen. Now listen that part. Can you answer me one by one?

[00:02:52]This is my one factor.

[00:02:55]What is the degree of this? Degree of this means >> degree of this means sir, what is the power on this bracket? Simple.

[00:03:03]>> Power is one. What is the power of this bracket?

[00:03:06]>> One. What is the power of this bracket?

[00:03:08]>> One. Again answer me. What is the power on this bracket? Even or odd?

[00:03:15]>> Odd. Power on this bracket.

[00:03:17]>> Power of this bracket.

[00:03:19]>> Odd. Keep this in mind. I'm getting 1 2 3 going to three. This always positive.

[00:03:27]No doubt in this. You started a journey from infinity. You reached to the first number. You reached three. Always positive. Now you ask a question to yourself. This three. This three is coming from which bracket?

[00:03:40]>> Third.

[00:03:41]>> Third bracket. What is the power of this bracket? Even or odd? Odd. If there is a odd power then always there will be change of sign.

[00:03:52]Again I'm repeating if there is a odd power. Odd power where?

[00:03:58]Odd power on the factor from which I'm getting three?

[00:04:03]I'm getting three from which factor?

[00:04:05]From this factor of this factor. What is the power? Odd. If the power is odd, I'll always change my sign. So plus becomes minus. Then at which number I reached?

[00:04:17]>> Two. Two is coming from which bracket?

[00:04:19]Second bracket. What is the power on two? Top power on this bracket.

[00:04:24]>> Even or odd?

[00:04:25]>> Odd. Again change of sign. So minus becomes plus.

[00:04:30]Again I reach that where?

[00:04:33]>> One. One is coming from which bracket?

[00:04:36]>> Odd. Power odd. Again change of sign plus becomes >> minus. This is how you will plot this plus - plus minus >> the power is >> then change of sign will not happen. We will do one more question that time. And now concluding the answer for this inequality was x - one. I'm coming going back to my question. This was my question. X -1, x - 2, x - 3. And I needed answer of this inequality to be more than zero. More than zero means what? Positive or negative?

[00:05:12]>> This answer this all this part should be more than zero. more than zero means positive. Just highlight positive positive intervals. That means my answer will be this interval and this interval. Tell me in terms of brackets x belongs to one with the help of brackets open close why open this is no equal 1 2 then union >> 3 to infinity this is the answer of this question clear let's do one more question again I am trying to give you a question which should be complicated. So the question is question is let's say x + 1 x + 2 x - 3.

[00:06:20]Let's take these three only. Here the power is 2. Here the power is 3. Here the power is 4.

[00:06:27]less than equals to zero.

[00:06:31]Rule number one, first rule says, first rule says RS >> RS should be zero already zero. No need to worry. Step number one says coefficient of variable. Who is the variable? X coefficient of variable must be must be positive. This is positive.

[00:06:57]Positive positive. No worries. Rule number next. Next step says sir put this factor zero.

[00:07:05]This is my first factor. Factor number one. Factor number two. Factor number three. I will put these factors equal to zero. X + 1 0. Then x + 2 0. Then x - 3 0. Answer x = -1. x = -2. x = 3. Next step says I'll draw a number line I'll draw number line. In number line minus infinity infinity plot these numbers on the number line. Which number will come first?

[00:07:42]>> -2 then minus1 then three. How many values of x?

[00:07:50]Three. How many intervals? Four.

[00:07:53]>> Four.

[00:07:55]Okay. Now, next step says >> sir, I will always start with my rightmost interval. Who is the rightmost interval?

[00:08:04]>> Rightmost interval is after three. And what what symbol should I put here?

[00:08:08]>> Plus, in which condition I can put plus?

[00:08:12]If I have followed all the above steps, which steps? The coefficient of x has to be positive in each factor.

[00:08:22]Then I can make it plus.

[00:08:25]Now with the change or no change, you will help me. You started a journey from infinity. You reach till three.

[00:08:32]Three. Three is coming from which bracket?

[00:08:36]>> This bracket.

[00:08:38]Correct. What is the power of this bracket? Even >> even >> power even >> sign will not change power odd >> sign will change power even sign will not change that means plus will remain >> plus again I'm saying you reach to three check three is coming from which bracket check the power even sign remains same again which number you reached >> minus one minus one is coming From which bracket?

[00:09:12]>> First bracket. And the power is >> even.

[00:09:14]>> Even again the sign will remain >> same.

[00:09:19]Then you reach to where? Minus2. Minus2 is coming from which bracket? This bracket. This bracket is a power or sign will change.

[00:09:28]Now again I'll come back to my question.

[00:09:30]My question in my question I need I need my LHS to be more than zero less than zero.

[00:09:35]>> Less than zero means positive or negative?

[00:09:38]>> Negative that means which interval should be the answer?

[00:09:40]Only this interval. Help me with the interval. X belongs to >> minus infinity >> 2us 2. Open or close?

[00:09:52]>> Why close? Because there's a equals to.

[00:09:57]This is little bit complicated question of V curve method.

[00:10:03]Got it? One more question. We'll do.

[00:10:06]>> So if they will not give zero, >> we have to make zero. We anyhow whatever the applications need to use LCM fractions whatever you know we will make zero that is possible I'll tell you one more question then I will give you time to write down also let's do this also x - 1 let's take some other number x - 3 x - So 5 - x 2 - x 1 - x power 1 power 2 power 3 power 4 power 5 let's try this what you want more than zero less than zero >> more than zero Should I involve equal also not >> equal? Again think of each step first.

[00:11:13]>> First rule is RHS should be zero. RS is the zeroicient.

[00:11:18]>> Second is what?

[00:11:19]>> Coefficient of X must be >> positive. Some brackets it is already >> positive but some brackets it is what?

[00:11:28]Negative. I need to change that.

[00:11:31]First bracket is okay or not? Yes sir.

[00:11:33]>> Okay. Second bracket third bracket not okay in third bracket I need to change this correct that means I need to take minus common basically taking minus common if I take minus common should it go just as minus no if I take what is the meaning of minus common meaning of minus common means minus1 common take minus1 common but minus1 will not go as minus1 minus1 will 2 answer carefully x - 3 power 1 x - 4 power 2 Now -1 common from this so -1 power 3 -1 will come common with the same power so minus1 common now this bracket is sorted now this bracket will become x - 5 whole cube again in this bracket also there's a problem again -1 But minus1 will not come alone.

[00:12:36]>> -1 will come with the power four. Then it will become x - 2 power >> four. Again minus common >> that means x -1 but - 1 sorry -1 common that means -1 to the power of 5 and x -1 power of 5 greater than equals to zero.

[00:13:01]till here all okay this step is okay now let's deal with this minus one brackets therefore I could have done this in short also but first you understand the main method -1^ 3us >> minus one you know I'm I'm writing here this is x - 3 power 1 this is x - 4 I need attention on the board this is -1^ 3us >> minus one let me write minus one here any and all okay Anyways everything is multiplication writing minus one here.

[00:13:35]This minus one is coming because of this. Now this is done. Next bracket power three. Next bracket >> - 1 power 4 >> 1 >> positive 1 does not matter positive 1.

[00:13:50]What is there next?

[00:13:52]>> X - 2^ 4. Next - 1^ 5us >> again minus one. I'm bringing it here.

[00:14:00]What is remaining?

[00:14:02]X - 1^ 5 greater than equals to 0. This is not a problem -1 into - 1 >> + 1. No problem at all. This became + 1.

[00:14:14]That means my final inequal this question will become >> x - 3 x - 4² x - 5 cub x - 2^ 4 x -1 power 5. Should I change the sign or something? No, nothing is changed because that side is already + one greater than equals to zero. If by chance if by chance here I'm not giving power four here I'm giving power suppose one only then this minus1 will become -1 power 1 -1 that means one more minus one will come hereus >> then it will become automatically >> negative and then to cancel this negative I need to again multiply by minus both the sides and because of that inequality will get >> changed but as of now nothing is there so it remains like this is done by two steps till now first step RH as zero second step I have made coefficient of X in each factor as positive but this I am explaining so it should take this much time otherwise this is just 30 seconds question it's it's nothing much hard next step by keeping all the factors equal to >> zero. Now I am not doing it one by one and directly I am thinking that you must have understood till now. Tell me the factors value >> 1 2 3 >> 3 >> 4 5 How many numbers of x? How many values of x? Five. 1 2 3 4 5 How many intervals?

[00:16:04]>> Six. Rightmost interval.

[00:16:07]>> Rightmost interval means which interval?

[00:16:10]This interval will be again. Now let's play the change of rule. Change of sign. From infinity I'll start. From infinity I reach to five.

[00:16:21]Then I'll go to the bracket.

[00:16:23]From where? Five is coming. Five is coming from which bracket?

[00:16:28]>> This bracket.

[00:16:30]>> Power odd. Odd means >> odd means change of sign. Plus becomes >> minus.

[00:16:38]Clear. Next. Next. Next.

[00:16:42]Four. Four is coming from which bracket?

[00:16:45]Four is coming from >> this bracket.

[00:16:49]>> Even even power. Sine remains >> same. That means minus will remain >> sign will remain same.

[00:17:06]Next four is coming from which bracket? This bracket >> change of sign. Okay. Here it's change of sign and then here >> same.

[00:17:23]Now we reach two. Three. Three is coming from which this bracket. This bracket >> change.

[00:17:31]Next two.

[00:17:35]>> Even same.

[00:17:37]>> Then one change.

[00:17:42]Done. What do I need in the answer? Plus symbols or minus symbols?

[00:17:47]>> Plus symbols because >> greater than zero. If it is greater than zero then my overall which which intervals are allowed. This interval is allowed. This interval is allowed and this interval is allowed. Now here only I want to understand one more thing.

[00:18:07]Let's understand the concept of hollow circle dark circles is 1 2 3 4 5 all the values are hollow or dark?

[00:18:17]>> Dark. Why dark?

[00:18:20]Because here this equals 2 is also there. So 1 is also allowed. 2 is also allowed. Three is also allowed. Four is also allowed. Five is also allowed.

[00:18:33]Okay? Because there is a equal symbol and all of them are in the numerator. So my final answer should be >> x belongs to >> there go carefully 1 2 3 1 2 2 1 2 2 is also allowed 2 to 3.

[00:18:55]>> So that means from 1 2 3 everything is okay now.

[00:18:57]>> 1.

[00:18:58]>> So from 1 2 3 closed union 5 to infinity open.

[00:19:08]where >> but four is not allowed now it's coming in the minus part >> yes >> because of minus intervals I am excluding this I'm excluding this but four is allowed now because of the equal that means union >> curly brackets four this is one of the most good example to understand v curve method.