Question

Solution set of the inequality $5^{x+2}>{\left(\frac{1}{25}\right)}^{\frac{1}{x}}$ is

• A. $(-2,0)$
• B. $(-2,2)$
• C. $(-5,5)$
• D. $(0,\infty )$

Answer 1

The correct option is D

$(0,\infty )$

Explanation for the correct answer:

Determining the solution set, for the inequality.

Equating both the sides of the inequality

$$\begin{gathered} 5^{x+2}>{\left(\frac{1}{25}\right)}^{\frac{1}{x}} \\ \Rightarrow 5^{x+2}>5^{-\frac{2}{x}} \\ \Rightarrow x+2>\frac{-2}{x}\left[\because a^m>a^n\Rightarrow m>n\right] \\ \Rightarrow \frac{x^2+2x+2}{x}>0 \\ \Rightarrow \frac{{\left(x+1\right)}^2+1}{x}>0 \\ \Rightarrow \left(\frac{1}{x}\right)>0 \\ \Rightarrow x^2+2x+2>0\sout{V}x\in R \\ \Rightarrow x>0 \end{gathered}$$

$x$ belongs to $(0,\infty )$

Therefore, the correct answer is option (D).