Question
The solution set of the inequation $${\log _{\frac{1}{3}}}\left( {{x^2} + x + 1} \right) + 1 > 0$$ is
A.
$$\left( { - \infty , - 2} \right) \cup \left( {1, + \infty } \right)$$
B.
$$[- 1, 2]$$
C.
$$\left( { - 2 , 1 } \right)$$
D.
$$\left( { - \infty , + \infty } \right)$$
Answer :
$$\left( { - 2 , 1 } \right)$$
Solution :
$$\eqalign{
& {\log _{\frac{1}{3}}}\left( {{x^2} + x + 1} \right) > - 1 = {\log _{\frac{1}{3}}}{\left( {\frac{1}{3}} \right)^3} \cr
& \Rightarrow \,\,{x^2} + x + 1 < {\left( {\frac{1}{3}} \right)^{ - 1}} \cr
& \Rightarrow \,\,{x^2} + x - 2 < 0. \cr} $$
Use sign scheme.