Question

If \([\cdot ]\) represents the greatest integer function of less than or equal to type then the solution set of the equation \([x]^{2}-5[x]+6=0\) is

• A. \([2,4)\)
• B. \([2,3)\)
• C. \([3,4)\)
• D. \(\{2,3\}\)

Hint:

Factorise the quadratic in [x] and convert the integer values of [x] into intervals for x.

Answer 1

The Correct Option is A

Step 1: Factorise
$[x]^{2}-5[x]+6=([x]-2)([x]-3)$.

Step 2: Possible values
So $[x]$ is 2 or 3.

Step 3: Check against options
Values in $[2,3)$ and $[3,4)$ both work, so the set is $[2,4)$, not a single sub-interval or the finite set $\{2,3\}$.

Final Answer:
All x from 2 up to but not including 4 satisfy the equation. \[ \boxed{\text{(A) }[2,4)} \]