• # question_answer The solution set of the inequality $\left| {{9}^{x}}-{{3}^{x+1}}-15 \right|<{{2.9}^{x}}-{{3}^{x}}$is A) $(-\infty ,1)$ B) $(1,\infty )$ C) $(-\infty ,\text{ }1]$ D) None of these

 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 $y>3\Rightarrow {{3}^{x}}>x\Rightarrow x>1\Rightarrow x\in (1,\infty )$