Question
Question: Exhaustive set of values of x satisfying log<sub>\|X\|</sub> (x<sup>2</sup> + x +1) ≥ 0 is...
Exhaustive set of values of x satisfying log|X| (x2 + x +1) ≥ 0 is
A
(-1,0)
B
(-∞, -1) ∪ (1, ∞)
C
(-∞, ∞) ~ {-1, 0, 1}
D
(-∞, -1) ∪ (-1, 0) ∪ (1, ∞)
Answer
(-∞, -1) ∪ (-1, 0) ∪ (1, ∞)
Explanation
Solution
In this case base is variable. Thus we must take two separate cases :
(i) | x | ∈ (0, 1). In this case we have to ensure that 0 < x2 + x + 1 < 1 ⇒ x ∈ [-1, 0]. Common part of |x|∈ (0, 1) and x ∈
(-1, 0) is x ∈ (-1, 0).
(ii) |x|>|. In this case we must have x2 + x + 1 ≥ 1 => x ∈ (-∞, 1) ∪ (0, ∞). Common part of |x| > 1 and x ∈ (-∞, -1) ∪ (0, ∞) is (-∞, -1) ∪ (1, ∞) . Thus final solution is x ∈ (-∞, -1) ∪
(-1, 0) ∪ (1, ∞)