Question

If $[x]^2 - 5[x] + 6 = 0$, where $[.]$ denotes the greatest integer function, then ______.

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

Hint:

If $[x] = n$, the valid interval is ALWAYS $[n, n+1)$. If you have consecutive integer solutions like $n$ and $n+1$, they merge seamlessly into $[n, n+2)$!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Treat $[x]$ as a variable $t$ and solve the quadratic equation. Then use the property of the Greatest Integer Function: if $[x] = n$, then $n \le x < n+1$.

Step 2: Formula Application:
$t^2 - 5t + 6 = 0 \implies (t-2)(t-3) = 0$.

Step 3: Explanation:
So, $[x] = 2$ or $[x] = 3$. Case 1: If $[x] = 2$, then $2 \le x < 3$. Case 2: If $[x] = 3$, then $3 \le x < 4$. Combining both intervals: $2 \le x < 4$, which is $[2, 4)$.

Step 4: Final Answer:
$x \in [2, 4)$.