Question

If \( f(x) = [x]^2 - 5[x] + 6 = 0 \), where \( [x] \) denotes the greatest integer function, then \( x \in \)

• A. \( (2,4] \)
• B. \( [2,4] \)
• C. \( [2,4) \)
• D. \( (2,4) \)

Hint:

When solving equations involving \( [x] \), always convert the integer solution back into the corresponding interval for \( x \).

Answer 1

The Correct Option is C

Step 1: Substitute \( n = [x] \).
The given equation becomes \[ n^2 - 5n + 6 = 0 \] Step 2: Solve the quadratic equation.
\[ (n-2)(n-3) = 0 \] \[ n = 2 \quad \text{or} \quad n = 3 \] Step 3: Convert back to \( x \).
If \( [x] = 2 \), then \( 2 \le x<3 \).
If \( [x] = 3 \), then \( 3 \le x<4 \).
Step 4: Combine the intervals.
\[ x \in [2,4) \] Step 5: Conclusion.
Hence, the solution set is \( [2,4) \).