if[x Greatest integer less than or equal to x and (x)=x-[x]. How many real values of x satisfy the equation

Pick ONE option

2

0

More than 2

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).

if [x] is the greatest integer less than or equal to x and (x) = x - [x], how many real values of x satisfy the equation?

Learn how to solve an equation involving the floor function and fractional part of a number: x = 2[x]. Explore key mathematical concepts such as floor functions, fractional part functions, and the properties of even integers. Suitable for advanced high school students.

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