Let [ x ] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = √((| [x] | - 2)/(| [x] | - 3)) is (- ∞, a) ]∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is :

Solution

Maths sir · 4 steps

Put t = |[x]|, solve an easy sign inequality in t, then turn t back into intervals of x.

1. Substitute. Let t = |[x]|. Then t can only be 0, 1, 2, 3, …. The square root needs (t-2)/(t-3) ≥ 0, and the denominator needs t ≠ 3.
2. Sign analysis. The fraction is zero at t = 2 and undefined at t = 3. It is negative between them and positive outside, so t ≤ 2 or t > 3. Since t is a whole number, t > 3 means t ≥ 4.
3. Back to x. |[x]| ≤ 2 means -2 ≤ [x] ≤ 2, so -2 ≤ x < 3. |[x]| ≥ 4 means [x] ≥ 4, so x ≥ 4, or [x] ≤ -4, so x < -3.

   x ∈ (-∞, -3) ∪ [-2, 3) ∪ [4, ∞)

4. So. a = -3, b = -2, c = 3, and a + b + c = -3 - 2 + 3 = -2, which is option (c).

The idea to keep. When converting [x] ≤ n, go up to the next integer: x < n + 1. When converting [x] ≥ n, write x ≥ n. Most mistakes in these questions happen at the endpoints.

Answer from the NTA final answer key. Solution by PrepWiser.