Complete the following statement: Given any real number x, a necessary and sufficient condition for the floor of x to equal x is for

Given any real number \( x \), a necessary and sufficient condition for the floor of \( x \) to equal \( x \) is for **\( x \) to be an integer**.

This means:

\[
\lfloor x \rfloor = x \iff x \text{ is an integer}
\]

Would you like further details or have any questions? Here are 5 related questions to consider:

1. What is the definition of the floor function, and how is it different from the ceiling function?
2. How can you express the floor function mathematically in terms of inequalities?
3. Can you prove that \( \lfloor x \rfloor = x \) if and only if \( x \) is an integer?
4. What happens to the floor function when \( x \) is not an integer, and how is the floor value determined?
5. How does the floor function behave when applied to negative real numbers?

**Tip:** Understanding the properties of the floor function is crucial in number theory and various areas of mathematics, especially in problems involving rounding and integer representations.

Understand the necessary and sufficient condition for the floor of any real number x to equal x. Explore how this condition relates to integer properties and the floor function in mathematics. Suitable for Grades 10-12.

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