Question

The solution set of the inequality $\left|9^x-3^{x+1}-15\right|<2.9^x-3^x$ is

• A. $\left(-\infty ,1\right)$
• B. $\left(1,\infty \right)$
• C. $(-\infty ,1]$
• D. None of these

Answer 1

Answer: (B)

Let $3^x=y$, then the inequality is $|y^2-3y-15|<2y^2-y...(i)$
The inequality holds if $2y^2-y>0\Rightarrow y<0$ or $y>\frac{1}{2}$
$\because y=3^x\le 0\Rightarrow y>\frac{1}{2}$
Now the inequality on solving,
$-\left(2y^2-y\right)<y^2-3y-15<2y^2-y$
$\Rightarrow 3y^2-4y-15>0$ and $y^2+2y+15>0$
Solution of first inequality $3y^2-4y-15>0$ is $y<-\frac{5}{3}ory>3$
Solution of second inequality $y^2+2y+15>0$ is $y\in R$
The common solution is
$y>3\Rightarrow 3^x>x\Rightarrow x>1\Rightarrow x\in \left(1,\infty \right)$