A) \[(-\infty ,1)\]
B) \[(1,\infty )\]
C) \[(-\infty ,\text{ }1]\]
D) None of these
Correct Answer: B
Solution :
Let \[{{3}^{x}}=y,\] then the inequality is \[\left| {{y}^{2}}-3y-15 \right|<2{{y}^{2}}-y\] ? (1) |
The inequality holds if \[2{{y}^{2}}-y>0\Rightarrow y<0\,\operatorname{or}\,y>\frac{1}{2}\] |
\[\because \]\[y={{3}^{x}}0\Rightarrow y>\frac{1}{2}\] |
Now the inequality on solving, |
\[-(2{{y}^{2}}-y)<{{y}^{2}}-3y-15<2{{y}^{2}}-y\] \[\Rightarrow \]\[3{{y}^{2}}-4y-15>0\]and\[{{y}^{2}}+2y+15>0\] |
Solution of first inequality |
\[3{{y}^{2}}-4y-15>0\operatorname{is}\,y<\frac{5}{3}\operatorname{or}\,y>3\] |
Solution of second inequality |
\[{{y}^{2}}+2y+15>0\] is \[y\in R\] |
The common solution is |
\[y>3\Rightarrow {{3}^{x}}>x\Rightarrow x>1\Rightarrow x\in (1,\infty )\] |
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