Question

For the equation $3^{x+2}>{\left(\frac{1}{9}\right)}^{\frac{1}{x}}$, the solution set contains

• A. All positive numbers.
• B. All negative numbers.
• C. All non-positive numbers.
• D. All non- negative numbers.

Answer 1

The correct option is A All positive numbers.

We have $3^{x+2}>{\left(3^{-2}\right)}^{1/x}$
$\Rightarrow 3^{x+2}>3^{-2/x}Here,base3>1\Rightarrow x+2>-\frac{2}{x}\Rightarrow \frac{x^2+2x+2}{x}>0\Rightarrow \frac{{(x+1)}^2+1}{x}>0$
$\Rightarrow \frac{1}{x}>0\therefore x\in (0,\infty )$