Math problem for advanced students

[@Fasalytch]

2^x is increasing, x is incresing, their sum (2^x + x) is an increasing function. 11 is a constant function, so there is no more than 1 solution. Pick some simple examples and find x = 3

[@TheDavidlloydjones]

That's the long, digression-based, method. Normal people just see that it's three.

[@Fasalytch]

@@TheDavidlloydjones "normal people" should prove that 3 is the only solution

[@아무개-u2i]

X=3

[@kuerzebap]

Im reellen Zahlenraum R für x fällt die einfachste Möglichkeit x=3 direkt ins Auge. Da sowohl f(x)=2^x und g(x)=x streng monoton wachsend sind, kann es keine weiteren Lösungen in R geben.

[@wolframjahn2068]

Main problem here is how to ignore the obvious solution 'x = 3'

[@evgenijstoljarov2276]

Make a replacement x=3+y. Then 2^(3+y)+3+y=11, and 8*2^y+3+y=11. We get: (8*2^y)+y=8. Since 8*2^y >=8, then y=0 and x=3.

[@Fasalytch]

If y = -1 then 8 * 2^y = 4 < 8

[@evgenijstoljarov2276]

I have outlined my decision briefly. Since questions have arisen, I am providing an addition. We are looking for a solution in two different intervals. First, for y >=0, and then for y =0, the inequality 8*2^y >=8 is true, which allows us to find the root y=0 and, accordingly, x=3. In the region y

[@Marinealver]

Whenever you see an exponential with a linear you know you need to use the substitution method. Now figuring out what to substitute is the tricky part.

[@Roarshark12]

What I appreciate about this channel above others: (a) you spell out every step (b) you give practical advice on how to tackle the problem (c) clear and consistent speech with no visual distractions (d) VERY neat and proper handwriting. My requests: can you please also cover calculus, differential equations, linear algebra and discrete math?

[@RobertoStenger]

I completely agree

[@tonhon]

literally when the teacher said "show your work"

[@eduardionovich4425]

Нужно быть изрядным ослом ,чтобы так долго решать устную задачу: в левой части монотонно возрастающая функция,в правой части - константа, х=3 - единственный действительный корень . Прав один из комментирующих: для предложенного метода нужно было выбрать куда более сложный пример. Впрочем,о простом методе с монотонностью уже тоже писали,так что прошу прощения за повторение.

[@francisvanriet6223]

11=2³+3------x=3...2 sec...

[@فیروزاهنگری]

2^x=11-x taking ln , ln2^x=ln(11-x) , xl n2=ln(11-x) |×3 so 3xln2=3ln(11-x) , xln2^3=3ln(11-x) , xln8=3ln(11-x) result x=3 and 8=11-x so x=3

[@rubinakhan9018]

3 just looked and answered 😅😅 lol

[@FredFouelefack]

Very clear and easy to understand. please can you do the next video by helping me solve this: 9^x + 4x = 65 ?🙏🥺🥺

[@jamesbond_007]

Is there a way to solve this w/o using the Lambert W function? It would be great if you could show solving it with the Lambert W and solving it without using it. Thanks for all you do!

[@brettbuck7362]

Yes. It's a binary conversion problem. You know that X has to be positive and odd by inspection, try 1, 2^x is too small, try 3, 2^3=8, +3= 11 solved. No *16 steps* of obscure unnecessary derivations. Failing that, simply calculating the value to get to about step 14 without knowing anything about the "Lambert W function". These examples are too trivial to be solving this way, if you want to force people down a particular solution path, don't pick a trivially simple example case.

[@jamesbond_007]

@@brettbuck7362 You're right, but that's not a general method of solving these sorts of problems. I'm looking for some form of manipulation of the expression to get it to where the answer is produced w/o appealing to the "magic" W function.

[@Roarshark12]

Very nice solution, Lambert W strikes again!

[@richardslater677]

Most of your videos involve impossibly random steps that prove to be correct but are totally unintuitive. This video is a classic case of that. In an exam situation, your solution would take longer than the entire time allowed to figure out the randomness of this solution.

[@baselinesweb]

Cmon man. If you’re gonna do all that stuff, at least pick a problem that you can’t solve by looking at it for a second or two.

[@stevstro4498]

Ask what time it is and you get a Swiss watch built. Not the best approach in a timed exam.

[@davidhalliwell1565]

Yes. You can solve this problem almost instantly. A more challenging problem using the same principles should be used.

[@cabiria0]

Unsmart comment

[@rhc-weinkontore.k.7118]

amazing!

[@shoutplenty]

7:27 lol nice try if i can spot 2^11 = 8 * 2^8 then i can spot 2^3 + 3 = 11, it’s the exact same thing

[@kriswillems5661]

Indeed. He writes a lot and that uses a trick he could have used on the first line. Stupid exercise. Stupid solution.

[@shoutplenty]

yeah exactly. i didn’t learn about lambert w function for university admissions and remember when i read about it later, i thought “i don’t see how this helps solve anything”. so when i saw this video purport to solve the equation methodically, i guessed there was some sophistry goin on and pored over each step till i spotted the sleight of hand

[@shoutplenty]

this is actually harmful for this level of student cos in uni you learn quickly about tricks and how to think in terms of the obvious thing working, so seeing a trick disguised as obvious (cos the solution is obvious… after you’ve been told it…), it trains students to not approach methodical problem solving properly. like looking for innovation is cool but you can’t be doing that while thinking you’re being methodical, and that you’re dumb/doing it wrong if you don’t see the trick

[@민우김-f4n]

2^x+x=11 2^x=11-x 2^x+x-11=0 11=2*2*2+3=2^3+3 X=3

[@jjed88]

At. 7:33, he got (11-x) ln2 = 2*11 ln2 at 10:08 he got (11-x) = 8 Am i seeing something wrong? Very nice understanding of W. Tks.

[@peterreali3950]

The exposition above uses a lot of non obvious tricks to solve the problem with the Lambert function which most people have never heard of. How about using some basic logic without guessing: so write 2^x +x = 2^y + y= 11 now a power of 2 must be less than 11 so it has to be y = 3 hence x =3.

[@ose09]

This is really WOW!

[@thomasharding1838]

Easy and obvious. X=3 2 to third = 8 8 + 3 = 11

[@gaiatetuya92]

y=2^xとy＝‐ｘ＋11の交点は一つだからx=3を代入して成り立っていたら計算は必要ないのでは。

[@nikolayguzman331]

Very clear as usually. Excellent explanation. 🖐

[@michaelpurtell4741]

Interesting approach, but I would surmise that the examiner was looking for the applicant to realize that 11 can be expressed as two to the 3rd+3

[@daolnwoddipar6408]

I'm completely enjoy of author, bloke try to tell about Labmert W-function explaining at the same time (in details!) multiplying of powers

[@andresrebolledobanquet1924]

2x + x = 11 3x = 11 x = 11/3 x = 3.66666 Demostrate. 2*3.66666 + 3.66666 = 11

[@dumitrudraghia5289]

2^x=11-x, x= 1, 2, 3, 4. 11-x=10, 9, 8, 7. 2^x= 2, 4, 8, 16. OBSERV: x=3. *** REPREZENTARE GRAFICA ==> x=3.

[@abdulhusseinalsultani9222]

اقرب رقم الى 11 هو 8 2^3=8 8+3=11 الحل

[@olafwalter2037]

What is with : 2^x +x = (11 -> 2^3 + 3)

[@mayaq8324]

“And never never use Wolfram Alpha to obtain the right side value” 🙄

[@leszeki3333]

A,2 B,2 C,2 A,B,C=3 2×2×2=8 8+3="11"

[@leszeki3333]

X = " 3 "

[@DavidRobinson-rj2sp]

Just quickly in my head x=3. I'll watch the video now to find out how to do it the hard way. Well after watching the video I'll stick to mental arithmetic as I got the solution a lot quicker.

[@wge2268]

3. By observation.

[@ابنالرافدينالوفي]

Simply solution:- X^2+x=11 X^2+x=8+3 X^2+x=3^2+3 By comparing tow sides X=3

[@justinstephenson9360]

An answer is obvious, a child of 10 could work it out but proving that is the only solution is hard. 2^x +x =11 => 2^x=11-x From that we can conclude that (a) x is an odd number because 2^x must always be an even number.; (b) x must be a positive number because if x was negative the right side of the equation would be greater than 11 and the left hand side would be some fraction of 2 which is not possible (c) the maximum value of 2^x

[@brettbuck7362]

Not quite, X=0, 2^X = 1, but that is obviously not going to add up to 11.

[@GulibleKarma20]

the function always has a positive derivative over its entire domain thus it is one-one and only one solution exist

[@brettbuck7362]

@@GulibleKarma20 True, but this is probably asking a bit much of a high school algebra student. So is knowing about the apparently now sainted "Lambert W function" which I also never heard of until about a week ago, despite having a degree in mathematics and having worked in engineering solving all sorts of math equations far more complex than this for most of a century. It is absolutely no surprise that math proficiency is falling like a rock, if this is what math teachers consider important.

[@boywonder6659]

I thought he was going to run out of paper.

[@korakhaing9548]

2^x+x=11 2^x+x=8+3 2^x+x=2³+3 x=3

[@robertgross578]

2 to power 3=8. 8+3=11

[@s.a.r.lcyclopharm2917]

When you decompose 11 into 3+8, you seem to know the solution, rather to use inspection, and after few videos, you are always enforcing W function to resolve equations!!

[@AsadbekRoʻziboyev-u2f]

Very good

[@THESTUDYHUB168]

Can you solve this equation please? 2-Minute Equation Hack for Busy Students - Solve the Equation 3x+5=3(x+2) Fast!

[@pentasquare]

Trial and error mothed while being slow _never fails_

[@kabivose]

Surely only true for integer solutions. How about a question whose answer is (pi)^2 ?

[@bkkboy-cm3eb]

2^x+x=11 x≡1(mod2) → x=2k+1 (k≧1) 2^(2k+1)=11-2k-1=2(5-k) 2^(2k)=5-k≧4 → k=1 ∴x=3

[@anjajohansson258]

Is there a good way to solve this type of equation, only instead of 11 you have a not-so-evident value like 10? Tried it, came as far as trying to figure out the proportion of 2^10 that had to go into the e-exponent to make "a" the same. I ended up with a function identical to the original, so that was upsetting. I thought I was on to something. 😅

[@thewolfdoctor761]

I think you are obsessed with the Lambert W function.

[@dumitrudraghia5289]

Solve 2^x+x=1 - Lambert W-function.

[@abdulhusseinalsultani9222]

x=0 answer

[@jaspervantwout5583]

3, in split second.

[@cephotographic]

Me too.

[@mgrths2009]

Why do soooo difficult?? Just draw y=2^x en y=11-x : only 1 intersection. Try some x-values, and you soon (very soon!) find x=3

[@RealQinnMalloryu4]

{2x+2x ➖ }+{x+x ➖ }= {4x^4+x^2}=4x^6 2^2x^3^2 1^1x^3^2 x^3^2 (x ➖ 3x+2).

[@louisdoumerc2693]

Very good video thank you

[@kanguru_]

x=3 by inspection.

[@thakurfamily9669]

Uhmmm actually.... 2^x + x = 2^3 + 3 The 2nd part of the equation satisfies x... so x = 3 😐

[@bautistagizirian3123]

7:56 were you going to do that you could've just said in the first equation 2^x+x=8+3=2^3+3

[@lapichfamily7595]

Solution simple : 3 est solution évidente, 2^x+x est strictement croissante donc injective. Il y a ainsi au plus une solution. Comme on en a trouvé une, c'est la solution.

[@marleybone6615]

Lambert and nature lock? This has reached the level of absurdity.

[@Blessed-s4b]

He meant natural log.

[@Monchy_10]

bro be like: Solve for x 1 + x = 2

[@donuteater900]

literally solved it in 2 seconds x = 3

[@nigelrafferty2545]

I'm wondering if this video was uploaded on 1st April for a laugh? Seriously, if you didn't realise the answer x = 3 in ten seconds or less you should probably focus on learning skills other than mathematics. Learn to think. Solving problems by adopting laborious conventional approaches will be a useless life skill.

[@cosimo7770]

Maths sites should teach mathematical insight, not the absurd application of rediculous rules.

[@alext8828]

It's 3. We learned this is the 6th grade.

[@valentinapaxley4221]

Just looking at it even my 12 YO brother could say it was 3.

[@josephcamba9370]

why the hell are you torturing this equation so much. its a simple answer.

[@VijayaraghaveluSundarara-ed4xz]

3 is the answer

[@pspprabhat]

2^×=11-×.....( 1:03 .1=(11-×)2^-× 2^11=(11-×)2^11 =>2^3)(2^8)=(11-×).2^( 3:23 3:26 11-×) =>w{8ln2.e^8ln2} =W(11-×)ln2.e^(11-×)ln2} =>8ln2=(11-×)ln2 =>8=11-×&×=11-8=3

[@dardoburgos3179]

X= 3.

[@hungcao4687]

😂 A high school student in Vietnam can solve problems like this easily.

[@stevenjohnson1143]

Simple 3 2^3=8+3=11

[@KartikGupta-cf7ov]

Bro your examination time is over

[@tensor131]

I'm not familiar with the W function. I guess I would be very impressed if the same equation using 10 instead of 11 could also be solved using the W function.... Anyone?

[@HirschElla]

x = 3

[@ramenomori]

x

[@thebasisti2482]

Lambert was a genius

[@sagabarde2296]

3

[@sundareshvenugopal6575]

8 + 3 = 11.

[@RobertoStenger]

"A lot of students might be saying I can easily solve this challenge by inspection, but the main question: What about the solution?" Very clear

[@DeinonychusTV]

No hubiera sido fácil tantear valores.y ya. A mí eso me salió en 5 segundos

[@kishorravichandran]

but i solved and found the answer in 1 mintue itself

[@stefan11804]

Hard Problem but I suppose it is about 3

[@JucLansegers]

Am I right to say that math is a supportive science aimed to describe natural phenomenons in other sciences like physics, chemistry, astronomy, biology, statistics...I can not imagine where such equation :2^x+x =11 could be found as a description of a phenomenon/ process. Can you?

[@robertgross578]

Your presumption is wrong.

[@JucLansegers]

@@robertgross578 Sir, can you prove it's wrong?

[@robertgross578]

@@JucLansegers Prove what is wrong?

[@JucLansegers]

@@robertgross578 My presumption, dear Robert

[@petergomes8788]

3.

[@ВладимирЮвчик]

Х=3

[@artakmelkonyan5166]

х=2.1095............between/////........x=2.10955

[@johannestan8681]

How can this be when 1 + 1 can't be even 2😅

[@richardduggans1358]

For advanced students? What rot. A five year old can solve it just by looking at it. x = 3

[@SaidSalmaoui-l9j]

2^x+x=11, 2^x -2^3=3-x , e^(2^x-2^3)=e^3-x , e^2^x /e^2^3=e^3-x ,e^2x/e^3.2=e^3-x , e^x^2/e^3^2=e^3-x , e^(x^2-3^2)=e^3-x, (x^2-3^2)=3-x, (x-3)(x+3)=-(x-3), (x-3)[(x+3)+1]=0, (x-3)(x+4)=0, x=3 ou [x=-4 impossible ] d'où x =3

[@puppy969]

1 =3, 2 = 6, 3 = 11

[@bjozef1961]

Fejben számolva: x=3 😂

[@liwicarfixingadventuresph-4389]

YOUR POWER IS TO LONG QUIT

[@lvluf4s4]

Mentalmente 3.

[@philmarmanillo1354]

What language is he speaking?

[@Голиб2007Тоиров]

Bunga o'xshash yechgandikku mana buni ishlagandik 2^x+x=5

[@Ariel-g8s]

La cajeteaste mucho

[@moonmissionpassagetototali1952]

The way you make your “x” is really irritating and distracting, especially when you leave a gap between the two halves and then you “bridge the gap.” Please practice making a proper “x”. Then practice making a proper 1 that doesn’t look like a 7. While you’re at it, make a 2 that doesn’t look like an S in reverse.

[@Blessed-s4b]

I like the way he writes the “x”. However, I agree with what you’re saying about how he writes “1” though.😅

[@NitenBarman-hf1qm]

X পাওয়ার 3

[@liwicarfixingadventuresph-4389]

TO LONG I QUIT SO MUCH BLA BLA

[@ellaouihenia7506]

Obviously 3 IS solution

[@miguelgorordofeijoo2997]

Amistad😂😅

[@KjysosooSoso]

😊