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Q.
The solution set of the inequality $\left|9^{x}-3^{x+1}-15\right|< 2.9^{x}-3^{x}$ is
Linear Inequalities
Solution:
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 $y > \frac{1}{2}$
$\because y=3^{x} \le0 \Rightarrow y >\frac{1}{2}$
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 $y<-\frac{5}{3} or y>3$
Solution of second inequality $y^{2}+2y+15>0$ is $y\in R$
The common solution is
$y>3 \Rightarrow 3^{x}> x \Rightarrow x>1 \Rightarrow x\in\left(1, \infty\right)$