# Why mathematica does not solve the equation $x\left \lfloor x \right \rfloor=10$ for $x$?

The equation $$x\left \lfloor x \right \rfloor=10$$ has a real solution, which is $$x=\frac{10}{3}$$.

But when I type

Solve[x*Floor[x] == 10, x]


a message comes: "Solve::nsmet: This system cannot be solved with the methods available to Solve."

Similarly with NSolve.

I am using version 12.1.0.0. Anything wrong I did?

• A more serious issue with Reduce[Floor[x]^2 - x*Floor[x] + 3 == 0, x, Reals] and Reduce[Floor[x]^2 - x*Floor[x] + 3 <= 0, x, Reals]. – user64494 Jul 10 at 9:31

Solve[x*Floor[x] == 10, x, Reals]
• Unfortunately,Solve[Floor[x]^2 - x*Floor[x] + 3 == 0, x, Reals] fails Solve::nsmet: This system cannot be solved with the methods available to Solve. as well as Reduce. However,NSolve[Floor[x]^2 - x*Floor[x] + 3 == 0 && x >= 0 && x <= 10, x, Reals] works. – user64494 Jul 10 at 9:28
• So does Reduce[Floor[x]^2 - x*Floor[x] + 3 == 0 && x >= 0 && x <= 10, x, Reals], outputting x == 19/4 || x == 28/5 || x == 13/2 || x == 52/7 || x == 67/8 || x == 28/3. – user64494 Jul 10 at 9:42
• @user64494 Solve[{Floor[x]^2 - x*Floor[x] + 3 == 0, 0 < x < 10}, x, Reals] evaluates all solutions in the given interval ! – Ulrich Neumann Jul 10 at 20:08