Question

If \( x \) is a real number and \( \lfloor x \rfloor \) is the greatest integer \( \leq x \), then \[ 3\lfloor x \rfloor + 2 - \lfloor x \rfloor = 0 \] Will the above equation have any real root?

• A. Yes
• B. No
• C. Will have real roots for \( x<0 \)
• D. Will have real roots for \( x>0 \)

Hint:

Test floor equations over the range implied by \( \lfloor x \rfloor = k \).

Answer 1

The Correct Option is B

Given: \[ 3\lfloor x \rfloor + 2 - \lfloor x \rfloor = 0 \Rightarrow 2\lfloor x \rfloor + 2 = 0 \Rightarrow \lfloor x \rfloor = -1 \] Now \( \lfloor x \rfloor = -1 \Rightarrow -1 \leq x<0 \).
Check: For any \( x \in [-1, 0) \), \[ 3\lfloor x \rfloor + 2 - \lfloor x \rfloor = 3(-1) + 2 - (-1) = -3 + 2 + 1 = 0 \] So yes, equation holds for all \( x \in [-1, 0) \).
But the question says "Will the equation have any real root?"
The confusion is in the wording: it says “\( = 0 \)” as a single point value, not range. However, it does satisfy over range. So: \[ \boxed{\text{Yes}} \]