Basic Maths and Logarithm
Logarithmic Inequalities
Premium Question
Grade 11
Question:
The solution set of $\log_{1/3}(x^2 + x + 1) + 1 > 0$ is:
(1) $(-\infty, -2) \cup (1, \infty)$
(2) $(-2, 1)$
(3) $[-2, 1]$
(4) $\mathbb{R}$
Step-by-Step Solution
Key Concept: $\log_{1/3}(x^2 + x + 1) > -1 = \log_{1/3} 3$ (since $\log_{1/3} 3 = -1$). Base $< 1$ reverses the inequality: $x^2 + x + 1 < 3 \Rightarrow x^2 + x - 2 < 0 \Rightarrow (x + 2)(x - 1) < 0 \Rightarrow -2 < x < 1$. Domain $x^2 + x + 1 > 0$ is all of $\mathbb{R}$, so solution is $(-2, 1)$.
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Correct Answer: (2)