Find the real values of $x$ that satisfy the equation $7[x]+23\{x\}=191$

For any real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ (i.e. floor function) and $ \{x\}=x-[x]$ .Then, the number of real solutions of the equation $$7[x]+23\{x\}=191$$ are.

My Attempt:

I used $ \{x\}=x-[x]$

This gives us,

$$23x-16[x]=191$$

as,$16[x]$ is an integer, $23x$ also must be an integer.

How to proceed next?

Hints:

• $23x-16[x]=191$ is a nice idea, but $7x+16\{x\}=191$ may be more useful

• Using $0 \le \{x\} \lt 1$, can you put an upper bound on $7x$? A lower bound?

• If you knew $[x]$, could you find $\{x\}$ and so $x$?

• How many possible values of $[x]$ are there? Do they all give a value of $x$ which works?