I wonder why we didn't apply the analysis on the first proof, just before multiplying by x. The analysis is done by distinguishing the cases x>=0 or xb a^3 > b^3. This would not work with squaring!
@@thedude882 yes you are right in both statements. It would be “easier” to just apply this analysis you talk about (I even mentioned it in the video), but in order to show visually the correct solution I decided to take a longer explanation. About your second statement, yes with squaring this would not work and the approach would have to be a totally different one
0:36 instead, move -7 to the left side. (1+7x)/x≥0. This clearly leads to two solutions, one is both top and bottom being non-negative and one is both top and bottom being non-positive. So x≠0 && (x ≥ 0 && x ≥ -1/7 || x ≤ 0 && x ≤ -1/7), meaning x > 0 or x ≤ -1/7.
@@Sergonizer yes, you’re right. This solves the problem. I believe there are even other ways to see that there are two solutions (from different arithmetic manipulations), but I just wanted to show visually what these solutions (or conditions for X) mean 😎
Thank you for the video. I think we could make it simple this way: after cubing the two members of the inequation (we can do it with no risk as the power is odd), we can just write (1/x)-1>=-8 then (1/x)+7>=0 then (7x+1)/x>=0 The solution can be obtained quickly using a sign table.
@@benjaminvatovez8823 yes, you are absolutely right. I just chose to solve it “the hard” way because I wanted to explicitly show the interesting Union and intersection relations along the way. Of course, the method you talk about is much more practical in real life scenarios, but I do believe that it obscures the reason why the solution is not as straightforward as it seems 😎
I used the same steps until arriving at the inequality at 0:32, then I essentially solved it graphically: I imagined the hyperbola described by 1/x and figured out which parts of it lie above the line y = -7. That is _much_ faster than both the solutions shown in the video and in other comments here.
Great video!) I love symbols in this video!) And idea for the video is history about solving equations of 2,3,4 degree And in number theory idea is talking about history about prime numbers, like they discovers❤ Love ya and your content❤!)
I wonder why we didn't apply the analysis on the first proof, just before multiplying by x. The analysis is done by distinguishing the cases x>=0 or xb a^3 > b^3. This would not work with squaring!
@@thedude882 yes you are right in both statements. It would be “easier” to just apply this analysis you talk about (I even mentioned it in the video), but in order to show visually the correct solution I decided to take a longer explanation. About your second statement, yes with squaring this would not work and the approach would have to be a totally different one
0:36 instead, move -7 to the left side. (1+7x)/x≥0. This clearly leads to two solutions, one is both top and bottom being non-negative and one is both top and bottom being non-positive. So x≠0 && (x ≥ 0 && x ≥ -1/7 || x ≤ 0 && x ≤ -1/7), meaning x > 0 or x ≤ -1/7.
Typing this before I watched the video so sorry if this is what is said later
@@Sergonizer yes, you’re right. This solves the problem. I believe there are even other ways to see that there are two solutions (from different arithmetic manipulations), but I just wanted to show visually what these solutions (or conditions for X) mean 😎
@@Sergonizer no worries. It nice to see alternative solutions 😁
Why do you write the last inequality as x
Thank you for the video. I think we could make it simple this way: after cubing the two members of the inequation (we can do it with no risk as the power is odd), we can just write (1/x)-1>=-8 then (1/x)+7>=0 then (7x+1)/x>=0 The solution can be obtained quickly using a sign table.
@@benjaminvatovez8823 yes, you are absolutely right. I just chose to solve it “the hard” way because I wanted to explicitly show the interesting Union and intersection relations along the way. Of course, the method you talk about is much more practical in real life scenarios, but I do believe that it obscures the reason why the solution is not as straightforward as it seems 😎
I used the same steps until arriving at the inequality at 0:32, then I essentially solved it graphically: I imagined the hyperbola described by 1/x and figured out which parts of it lie above the line y = -7. That is _much_ faster than both the solutions shown in the video and in other comments here.
@@bjornfeuerbacher5514 wow it’s true! Actually it is even visually easy. Nice comment, thanks!!!
Great video!)
I love symbols in this video!)
And idea for the video is history about solving equations of 2,3,4 degree
And in number theory idea is talking about history about prime numbers, like they discovers❤
Love ya and your content❤!)
cbrt(1/x-1)>=-2
The cube root is monotonically increasing. Therefore,
1/x-1>=-8
1/x>=-7
x>0 or x
@@mathmachine4266 yep, great job! 😎
Thank you for this solution, Professor @Luca
@@sharonk456 you’re welcome!!! Let me know what kind of content you’d like to see in the channel 😎
@@dibeos i like radiologic physics , and optics👀 and acoustics👂
@@sharonk456I just added to our list of video ideas. Stay tuned 😎
Great video like damn holy this is good
Awesome! I am happy you liked it. Let us know what kind of content you'd like to watch in the channel 😎
(-∞, -1/7]
Will edit when shown wrong in video.