Finding the Restricted Values for a Rational Expression

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  • เผยแพร่เมื่อ 6 พ.ย. 2024

ความคิดเห็น • 157

  • @Kimcharmxd
    @Kimcharmxd 4 ปีที่แล้ว +94

    I spent 2 hours reading my book again and again and I still dont understand, my brain finally worked when I watched this video.

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +7

      I'm so glad it helped, thanks for watching :)

  • @pof9167
    @pof9167 4 ปีที่แล้ว +19

    I'm a slow learner person... My teacher so bad at teaching math... And my Life is the worst... And then my teacher send to me on Facebook, and then it changed my life! Thank you so much!!

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +3

      I'm so glad the video was helpful. Everyone learns at a different pace, keep working hard and you will get where you want to go!

    • @bobjohnson9806
      @bobjohnson9806 3 ปีที่แล้ว

      how did you just now get youtube lmao

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      @@bobjohnson9806 What do you mean?

  • @005winter
    @005winter 2 ปีที่แล้ว +8

    My math exam is tomorrow and I didn't listen because i always fall asleep when i listen to math. I thought restricted value is really hard. Thanks for making it easy my man

  • @AdrianHernandez-st1vs
    @AdrianHernandez-st1vs 3 ปีที่แล้ว +12

    Man I got to subscribe, after 5 hours trying to understand, you make it plain and simple!
    I’m back to my college homework now! Thanks !

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      Glad I could help!

    • @karlishmaelbborja9569
      @karlishmaelbborja9569 ปีที่แล้ว +1

      Bruh we gonna do this again in college I'm grade 8 right now and it's teaching us this🗿

    • @cheerios5849
      @cheerios5849 11 หลายเดือนก่อน

      @@karlishmaelbborja9569it gets worse

  • @sorenvish
    @sorenvish 3 ปีที่แล้ว +8

    you just saved my algebra grade, thank you kind internet man

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว +1

      I'm glad to hear that, let me know if you have any questions or get stuck on a problem. 😎

  • @camellestavhad8927
    @camellestavhad8927 4 ปีที่แล้ว +6

    My teacher is so bad at teaching so i have to go on youtube for me to learn math. This video helped me a lot thank you!!!!

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +2

      Glad it helped! Good luck with your studies :)

  • @josephkim3085
    @josephkim3085 3 ปีที่แล้ว +6

    This video is so much better than any textbook, THANK YOU SO MUCHH

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      Awesome, glad it was helpful!

  • @josiejackson9941
    @josiejackson9941 4 ปีที่แล้ว +13

    i needed this video for my highschool class thank you

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      You are very welcome! :)

  • @Greenemath
    @Greenemath  4 ปีที่แล้ว +9

    The main idea is to set the denominator equal to zero and solve the resulting equation. Hope that helps with homework.

    • @breadpitt3369
      @breadpitt3369 4 ปีที่แล้ว +1

      1.) 2x-5/5-x
      2.) x-2/2x-7
      3.)x²+3x-4/x+3
      4.)x²+5x+6/4x+5

    • @breadpitt3369
      @breadpitt3369 4 ปีที่แล้ว +1

      How can i answer this? Please help me im confused

    • @breadpitt3369
      @breadpitt3369 4 ปีที่แล้ว +1

      5.)5a²-14/4a-7
      6.)m²+m+50/3m-12
      7.)a² - a - 2/ a² - 13a + 30
      8.)x³ - 2x + 3/x² + 12x + 32

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      @@breadpitt3369 Just set each denominator equal to zero and solve. You are looking for where the denominator is zero.
      Let's work through a few:
      1) (2x - 5) / (5 - x)
      Set the denominator equal to zero and solve the equation:
      5 - x = 0
      -x = -5
      x = 5
      So this tells you that 5 is restricted from the domain. Why? If you plug in a 5 for x, you end up with 0 as the denominator and you can't divide by zero.
      {x | x is not equal to 5} is your domain
      Let's look at #2
      2) (x - 2) / (2x - 7)
      Set 2x - 7 = 0 and solve
      2x - 7 = 0
      2x = 7
      x = 7/2
      {x | x is not equal to 7/2}

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      @@breadpitt3369 See the comment below.

  • @Greenemath
    @Greenemath  4 ปีที่แล้ว +5

    GreeneMath.com
    Good luck with class everyone! :)

  • @breadpitt3369
    @breadpitt3369 4 ปีที่แล้ว +3

    1.) x² + 3x -4/x+3
    2.) x²+5x+6/4x+5
    3.) 5a² - 14/4a - 7

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      Just set each denominator equal to zero and solve to find the restricted value(s). Also, I'm not sure if you are asked to simplify? You didn't specify and you should check the instructions. Some teachers want both?

    • @breadpitt3369
      @breadpitt3369 4 ปีที่แล้ว

      @@Greenemath and solution

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +3

      @@breadpitt3369 The answer to what???????????????? You aren't giving instructions. How can you put down an answer with no instructions. Are you finding the restricted values or simplifying? Do you event know? It should state what to do at the top of the homework. If you are finding the restricted values, just set those denominators equal to zero and solve the equations you are producing. Then state the domain by restricted those values.
      x + 3 = 0
      x = -3
      4x + 5 = 0
      x = -5/4
      4a - 7 = 0
      a = 7/4
      So on and so forth, but you don't list those as the solutions, those are the restricted values and are restricted from the domain.

    • @breadpitt3369
      @breadpitt3369 4 ปีที่แล้ว +1

      @@Greenemath restricted values

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      @@breadpitt3369 Alright then just restrict those values I gave you from each domain and you are done.

  • @shukk5613
    @shukk5613 ปีที่แล้ว +1

    wow , what a life saver. My head was going to explode before i came across this video LOL
    thanks for saving me time and confusion

  • @simonranque9514
    @simonranque9514 4 ปีที่แล้ว +5

    Tandewww berrymuddd... I got all my questions answered!😍😍😍

  • @IDK-lz5kh
    @IDK-lz5kh 2 ปีที่แล้ว +1

    How did the restricted value of the last given became -1/2 ? I Substitute it and it did not because undefined

    • @Greenemath
      @Greenemath  2 ปีที่แล้ว +1

      2(-1/2)^2 - 9 (-1/2) - 5 = 0, This means the denominator is zero and the expression is undefined.

  • @arobloxuser4744
    @arobloxuser4744 3 ปีที่แล้ว +1

    I get this video and thanks for this, but what if the thing is a^2/b ??? thank you

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      What is the exact problem you are working on?

  • @yzzak.7025
    @yzzak.7025 4 ปีที่แล้ว +2

    I understand this much better than the other videos tbh😲

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +2

      Perfect, glad it was helpful! :)

  • @choutzuyu9826
    @choutzuyu9826 4 ปีที่แล้ว +2

    What if the factor is in parenthesis? Like 2(2x-4) how will u equal it to 0?

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +2

      Same deal, just set the part with the variable equal to zero.
      2x - 4 = 0
      2x = 4
      x = 2
      If x was 2, that whole expression would be zero, since you'd be multiplying by zero.
      2(2(2) - 4)
      2(4 - 4)
      2(0) = 0
      So the denominator would become zero if x was 2. In general, just set the factors with a variable equal to zero and solve.

  • @danielprct3176
    @danielprct3176 2 ปีที่แล้ว +1

    Thank you so much,i don't understand nothing of this before watch this video

    • @Greenemath
      @Greenemath  2 ปีที่แล้ว

      You are welcome!

  • @tessy9672
    @tessy9672 3 ปีที่แล้ว +2

    can a restriction be a decimal number
    for example
    (2x+3) (2x-3) R=-1.5,1.5

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว +1

      Yes, in your case you could write the restricted values as domain: {x | x ≠ 3/2, -3/2} or you can do domain: {x | x ≠ 1.5, -1.5}

    • @tessy9672
      @tessy9672 3 ปีที่แล้ว +1

      @@Greenemath thanks a lot you really helped me understand this :)

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      @@tessy9672 I'm glad to hear that, let me know if you have any other questions. Happy to try and help if I can. 😎

  • @cloudfone1885
    @cloudfone1885 4 ปีที่แล้ว +2

    How do you find your x,like you did in 9x-1
    3x-4
    5:40

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      Time marker please? I have no idea what you are referring to.

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      If you edit a comment, it doesn't send a notification. You should just reply instead. Your original comment asked where we got 3/4, which I couldn't find. The problem you are looking at is (9x - 1) / (3x - 4) and we are looking for the restricted value. So what can't we plug in for x? Remember you can't divide by zero. So what value makes the denominator zero? Just set up an equation with the denominator equal to zero:
      3x - 4 = 0
      3x = 4
      x = 4/3
      So 4/3 is a restricted value, notice if you plug in a 4/3 for x the denominator becomes zero and that is not defined in math. In general for these problems you are setting the denominator equal to zero and then solving the resulting equation.

  • @w0oahae__
    @w0oahae__ 4 ปีที่แล้ว +1

    how do you do find the restricted value of
    p squared
    ----
    (p+4) (p+5)

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +8

      Just set the denominator equal to zero and solve the equation. Here they made it easy because the denominator is already factored.
      (p + 4)(p + 5) = 0
      p + 4 = 0
      p = -4
      p + 5 = 0
      p = -5
      So p can't be equal to -4 or -5 since both of those values would make the denominator zero.

  • @Plasmaaa
    @Plasmaaa 4 ปีที่แล้ว +3

    I have a question, what's the point in finding the restricted values and the domains. In other words, why would I want to know what x is not, versus finding what x actually is?

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      With a rational expression, it's not an equation. The only value for x is when you give it one. We need to find out what values of x are allowed or not allowed. This will have applications for solving rational equations, graphing, and so forth.

    • @Plasmaaa
      @Plasmaaa 4 ปีที่แล้ว +2

      @@Greenemath ah okay, I thought there was a deeper meaning. thankk you

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      @@Plasmaaa You are very welcome, good luck with your studies :)

  • @cheerios5849
    @cheerios5849 11 หลายเดือนก่อน +1

    my teacher js didnt explain how to find restrictions besides in basic denominators like 3-4x

    • @Greenemath
      @Greenemath  11 หลายเดือนก่อน

      Set the denominator equal to zero and solve.

    • @cheerios5849
      @cheerios5849 11 หลายเดือนก่อน +1

      @@Greenemath tysm u and organic chemistry tutor are saving my grades

  • @quixiee_
    @quixiee_ 2 ปีที่แล้ว +2

    thank you! this video is so clear and helpful ^^

    • @Greenemath
      @Greenemath  2 ปีที่แล้ว +1

      Glad it was helpful! Good luck with class!

  • @satorugojo8786
    @satorugojo8786 3 ปีที่แล้ว +1

    How i can get the restricted value for 5x(x+1) and (x-5)(x+4)??

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      That's not a rational expression. Are those denominators or something? Can you post the full problem.

    • @satorugojo8786
      @satorugojo8786 3 ปีที่แล้ว +1

      @@Greenemath yes they are denominators and i need to find their non permissible values can you help me please?

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      @@satorugojo8786 Yes, in the future please post the full problem though. I can't understand why you would just type half of a problem???? Depending on what's in the numerator, the answer could change. Assuming there's nothing fancy in the numerators.
      If 5x(x + 1) is your denominator then you just need to find what x-values make the denominator zero. You can't divide by zero.
      5x(x+1) = 0
      Solve by setting each factor equal to zero.
      5x = 0 or (x + 1) = 0
      5x = 0
      x = 0
      x + 1 = 0
      x = -1
      So for the first example, x can't be 0 or -1. You can see if you plug in a zero or -1 you would get zero. And you can't have zero as a denominator.
      5(0)(0 + 1) = 0(1) = 0
      5(-1)(-1 + 1) = -5(0) = 0
      For the second example, we would follow the same logic
      (x - 5)(x + 4) = 0
      x - 5= 0 or x + 4 = 0
      x - 5 = 0
      x = 5
      x + 4 = 0
      x = -4
      So for the second example x can't be 5 or -4
      Again, I don't know what's in the numerators because you didn't want to type out the full problem. There could be more restrictions needed, but there's no way for me to know.

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      @@satorugojo8786 Please type the full problem. What is the numerator? I can't understand why you aren't typing the full problem. Do you realize you can get the wrong answer by typing half of the problem?

    • @satorugojo8786
      @satorugojo8786 3 ปีที่แล้ว +1

      @@Greenemath
      Find their non permissible value
      1.numerator 6x+5
      denaminator 3x+4
      2.numerator x^2 - 1
      Denaminator x^2 +4x - 45
      Thankssss

  • @fionavillareal226
    @fionavillareal226 3 ปีที่แล้ว +1

    *Watched it half way*
    *got confused*
    *panicked*
    *cried*
    *ate some chips*
    *recuperated*
    *got back at **8:29*
    *happly* ah record timing!

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      Basically just set the denominator equal to zero, solve the resulting equation, and the solutions are your restricted values. Also let us know what kind of chips. 🥔😋

  • @lendelllappay1347
    @lendelllappay1347 4 ปีที่แล้ว +1

    How about this problem 3x
    6x2
    I don't understand

    • @Plasmaaa
      @Plasmaaa 4 ปีที่แล้ว +2

      6x * 2 ≠ 0
      12x ≠ 0
      x ≠ 0/12

    • @Plasmaaa
      @Plasmaaa 4 ปีที่แล้ว +2

      6x - 2 ≠ 0
      6x ≠ 2
      x ≠ 1/3

    • @Plasmaaa
      @Plasmaaa 4 ปีที่แล้ว +2

      6x + 2 ≠ 0
      6x ≠ -2
      x ≠ -1/3

    • @Plasmaaa
      @Plasmaaa 4 ปีที่แล้ว +2

      I didn't know if you meant 6x times 2, or if you just forgot the operation (-,+,*) so I did all 3.

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +2

      Is it (3x)/(6x^2)? In this case x can't be zero.

  • @AllNickiMinajSongs
    @AllNickiMinajSongs 3 ปีที่แล้ว +1

    Do you still help comments? I need help

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว +1

      What's the problem you are working on?

  • @sylvirette
    @sylvirette 4 ปีที่แล้ว +1

    How to find the restricted value of x²-12

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      Is that your denominator? Is it 1/(x^2 - 12) your problem? If you just have x^2 - 12, there isn't a restricted value since your denominator is 1. If x^2 - 12 is in the denominator, then set it equal to zero and solve the resulting equation.
      x^2 - 12 = 0
      x = 2 * sqrt(3), -2 * sqrt(3)
      So those two values would need to be restricted from the domain. Again this is only if x^2 - 12 is your denominator.

  • @onetamad3804
    @onetamad3804 4 ปีที่แล้ว +1

    How to solve replaceable values of rational expression
    3/2-x what is the answer?

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      I'm not sure what you are trying to do with this expression? When you say replaceable values? You can replace the x, but you are generally given some value to replace it with. There isn't an equal sign, so it's not an equation to solve. The best I can figure on this example is that you want the restricted value? In that case, set your denominator equal to zero and solve.
      Also, I'm going to assume this was your problem as the other was just 3/2 - x.
      3 /(2 - x)
      Since your denominator is 2 - x, we can set this equal to zero and solve the resulting equation.
      2 - x = 0
      2 = x
      This tells you that x can't be 2 since this makes the denominator 0 and dividing by zero is undefined.

    • @onetamad3804
      @onetamad3804 4 ปีที่แล้ว +1

      @@Greenemath yes help me on my questions please

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      @@onetamad3804 I answered above.

  • @jinseiboy5589
    @jinseiboy5589 4 ปีที่แล้ว +1

    I'm confused what's n+7 in domain pleaseeee

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      Is that the whole expression or is it just the denominator?

    • @jinseiboy5589
      @jinseiboy5589 4 ปีที่แล้ว +1

      @@Greenemath denominator thank you😊😊

    • @jinseiboy5589
      @jinseiboy5589 4 ปีที่แล้ว +1

      @@Greenemath wait can you do 2x²-5x+3

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      @@jinseiboy5589 So the whole idea with a domain is that it's the set of allowable values for the variable x or in your case n. So think about what can you plug in? If you have a rational expression, let's say:
      1/(n +7)
      Then we know that we can't divide by zero, this is undefined. So we need to find out what value for n makes the denominator zero. To do this, we just set the denominator equal to zero and solve:
      n + 7 = 0
      n = -7
      So this means -7 is restricted from the domain.
      You could write this as:
      domain: {n | n is not equal to -7}
      So any value can be plugged in for n except -7 since this makes the denominator zero and we can't divide by zero. :)
      I'm also assuming you don't have any radicals in the numerator since I can't see the full problem. As long as the numerator is a simple polynomial that's your answer.

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      @@jinseiboy5589 Again is this the denominator? Why aren't you posting the full problem, you could have something in the numerator with a restriction as well?
      It's the same deal just set the denominator equal to zero and solve:
      2x^2 - 5x + 3 = 0
      x = 3/2, 1
      So x can't be 3/2 or 1 since that involves division by zero. Again this is only if you have a polynomial in the numerator. No way for me to know since you only post part of the problem.
      domain: {x | x is not equal to 3/2, 1}

  • @skimasktheflowgod7208
    @skimasktheflowgod7208 4 ปีที่แล้ว +1

    How do you do. x/x-2 ?

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      Just set the denominator equal to zero and solve.
      x - 2 = 0
      x = 2
      So x can't be 2 since plugging in a 2 for x gives you a denominator of zero which is undefined.

  • @justevenwhy
    @justevenwhy ปีที่แล้ว +1

    3 days before exam
    Will update after my result

  • @heyaheq8837
    @heyaheq8837 4 ปีที่แล้ว +1

    how to know the restricted value of x?

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      Set the denominator equal to zero and solve the equation. The solutions are the restricted values.

  • @miaabeyta7172
    @miaabeyta7172 3 ปีที่แล้ว +1

    How do you simplify it?

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      You would factor the numerator and denominator and then cancel common factors. You can watch this video: th-cam.com/video/n6zn3jflZL4/w-d-xo.html
      If you are still confused leave your example here and I'll show you how to simplify it.

  • @friedegg7994
    @friedegg7994 4 ปีที่แล้ว +1

    what about
    3
    __
    5x ??

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      Set 5x = 0 and solve.
      5x = 0
      x = 0
      This means x can't be zero.

    • @friedegg7994
      @friedegg7994 4 ปีที่แล้ว

      @@Greenemath Thank you! Your video was very helpful, in fact i learn faster by watching your videos! I also aced test so Thank you again!

    • @friedegg7994
      @friedegg7994 4 ปีที่แล้ว +1

      @@Greenemath Thank you! Your video was very helpful, in fact i learn faster by watching your videos! I also aced my test so Thank you again!

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      @@friedegg7994 I'm glad you passed! :)

  • @Yochi-uj4bk
    @Yochi-uj4bk 9 หลายเดือนก่อน

    Thank you 😭math In Philippines is hell

    • @Greenemath
      @Greenemath  9 หลายเดือนก่อน

      It's hard everywhere, math is a difficult subject to learn.

  • @narbinabuda9827
    @narbinabuda9827 3 ปีที่แล้ว +1

    i still don't get it :

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      Just set the denominator equal to zero and solve the resulting equation. That's it. If you have a problem you are stuck on, you can reply and I'll try to help :)

  • @jannatultajrin7516
    @jannatultajrin7516 2 ปีที่แล้ว +1

    Who is watching this in 2022

  • @sharkweek7280
    @sharkweek7280 4 ปีที่แล้ว +1

    Omgg i really get it now tyy

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      That's so awesome to hear!

  • @katenyvin8508
    @katenyvin8508 3 ปีที่แล้ว +1

    I answered all of my schoolworks except for this.. 35x - 35/25x-40 pls help 🙏✨

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      There are a zillion things you can do with that expression. Why not put the instructions in? Instead you just give a random problem that could be anything?

    • @katenyvin8508
      @katenyvin8508 3 ปีที่แล้ว +1

      @@Greenemathi need to find its restricted value :(

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว +1

      @@katenyvin8508 I would start by simplifying. You can factor out a 5 from the top and bottom:
      (35x - 35) / (25x - 40)
      35x - 35 = 5(7x - 7)
      25x - 40 = 5(5x - 8)
      You can cancel the common factor of 5:
      (7x - 7) / (5x - 8)
      is the simplified rational expression.
      A rational expression is not defined where the denominator is zero. This is because division by zero is not defined. What we do is set the denominator equal to zero and solve the resulting equation:
      5x - 8 = 0
      5x = 8
      x = 8/5
      This means that if x is 8/5, the rational expression is not defined since we are dividing by zero.

  • @_vvvvvvvv
    @_vvvvvvvv 3 ปีที่แล้ว +1

    Legend🔥🔥🔥🔥

  • @stannmyself5856
    @stannmyself5856 3 ปีที่แล้ว +1

    Woah thank you

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว +1

      You are welcome!

  • @thesubliminalmaker3951
    @thesubliminalmaker3951 4 ปีที่แล้ว +1

    Thank you

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      You're welcome :)

  • @lean5777
    @lean5777 4 ปีที่แล้ว +1

    I still cabt answer it

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว

      Can't answer what?

    • @lean5777
      @lean5777 4 ปีที่แล้ว

      @@Greenemath i finally answerd it now i need to simplify rational expressions

    • @lean5777
      @lean5777 4 ปีที่แล้ว +1

      @@Greenemath thank u i finally answerd it

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      @@lean5777 It's just factoring and canceling common factors. You can always drop the example in the comments and I'll try to help! :)

    • @Greenemath
      @Greenemath  4 ปีที่แล้ว +1

      @@lean5777 I'm glad to hear that! :)

  • @darion1380
    @darion1380 3 ปีที่แล้ว +2

    wow

  • @johncolombo8437
    @johncolombo8437 3 ปีที่แล้ว +1

    Kl9

    • @Greenemath
      @Greenemath  3 ปีที่แล้ว

      Not sure what that means but thanks for watching and commenting! Good luck with your studies :)