Solve for x in the floor function equation

[@Gremriel]

I watched this, and understood nothing. But I keep watching these videos, for some unfathomable reason.

[@learntnew6969]

same man.

[@AdrianHernandez-ul4in]

his calming voice?

[@surendrakverma555]

Very good explanation Sir. Thanks 🙏

[@JonibekSharipov-pz6ez]

Wow cool Thanks a lot from Uzbekistan

[@NulliosG]

Very interesting. Extrapolating from this logic, would brackets with lines missing on the bottom signify a ‘ceiling’ function with the left operator being ‘less than’ and the right operator being ‘less than or equal to’ in the drawn-out equation? Or would the operators be flipped to ‘greater…?’ I wonder. Apologies if this is answered in the video, I can only watch it muted.

[@PrimeNewtons]

It will be k-1 < x

[@m.h.6470]

Solution: (15x - 7)/5 = ⌊(6x + 5)/8⌋ Add d with 0 ≤ d < 1 to the left side to remove floor on the right: (15x - 7)/5 + d = (6x + 5)/8 |*40 120x - 56 + 40d = 30x + 25 |-30x +56 -40d 90x = 81 - 40d |:90 x = (81 - 40d)/90 Replacing d with 0 to get first boundary: x = (81 - 40d * 0)/90 x = 81/90 Replacing d with 1 to get second boundary (not included): x = (81 - 40 * 1)/90 x = 41/90 So we know that 41/90 < x ≤ 81/90. Since the right side of the original equation is always integer, we need to see, when the left side is integer. Using the known boundaries, we get: (15 * 41/90 - 7)/5 = (41/6 - 42/6)/5 = -1/30 and: (15 * 81/90 - 7)/5 = (81/6 - 42/6)/5 = 39/30 The only integers between -1/30 and 39/30 are 0 and 1 (15x₁ - 7)/5 = 0 |*5 15x₁ - 7 = 0 |+7 15x₁ = 7 |:15 x₁ = 7/15 (15x₂ - 7)/5 = 1 |*5 15x₂ - 7 = 5 |+7 15x₂ = 12 |:15 x₂ = 12/15 = 4/5 x ∈ {7/15, 4/5}

[@BartBuzz]

You do love your floor functions 😊

[@joaobatistaoliveira4007]

Linda solução, Senhor.

[@ronaldreagan3538]

You said (6x+5)/8 is "less than its own ceiling, ... k+1". If (6x+1)/8 is a whole #, both floor & ceiling will be = k. Inequality you wrote is fine, wording is not perfect. Thanks for the videos.

[@gbshaun]

I jumped to the end, but why is the answer not simply X=0.9 ?

[@Gman6365]

Because he is not solving (15x-7)/5 = (6x+5)/8 Look at the right side of the equation. The (6x+5)/8 is between what looks a bit like an "L" and a backwards "L". This indicates the floor function is being used. Applying the floor function to an integer returns that integer; however, applying it to a non-integer returns the largest integer less than the original number. [So, it rounds down to the next integer.] Examples: floor(9)=9, floor(8.999)=8, floor(8.001)=8 Setting x=0.9, the left side is 1.3 while the right side is floor(10.4/8)=floor(1.3)=1

[@AndDiracisHisProphet]

you should have checked yourself :D k = 15x MINUS 7 over 5, not plus....

[@dirklutz2818]

NO!

[@dirklutz2818]

I found the solution through trial and error. But now I know why!