The problem describes two mathematical functions:

1. $x$: The greatest integer less than or equal to $x$, also called the floor function.
2. $(x) = x -$x$$$$, which represents the fractional part of $x$, denoted as $\{x\}$. This gives the decimal part of $x$, meaning $0 \leq \{x\} < 1$.

Now, the equation in question can be understood as:

[ [x] = x - [x] ]

Simplifying this equation:

[ [x] + [x] = x ]

$2[x] = x$

$x = 2[x]$

Since $[x]$ is the greatest integer less than or equal to $x$, this equation implies that $x$ is equal to twice an integer, meaning:

$x = 2n \quad \text{for some integer} \, n.$

However, we must also check that the fractional part of $x$ satisfies $\{x\} = 0$. Since $x = 2n$, $\{x\} = x - [x] = 2n - n = 0$.

Thus, $x$ must be an integer, and specifically even integers.

Conclusion:

The values of $x$ are exactly even integers. Since there are infinitely many such integers, there are more than 2 real values that satisfy the equation.

The correct answer is: More than 2.

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Do you want any further details or have any questions? Here are some related questions to think about:

1. What are the properties of the floor function?
2. How does the fractional part function behave for non-integer values?
3. Can $\{x\}$ ever be negative?
4. What would happen if the equation had been $3[x] = x$ instead of $2[x] = x$?
5. What is the geometric interpretation of the solution to this equation?

Tip: The fractional part function $\{x\}$ is always between 0 (inclusive) and 1 (exclusive).