Let’s solve the inequality step by step.

Problem:

We are solving the inequality:
$\left(\frac{1}{5}\right)^x > \left(\frac{1}{5}\right)^5$

Step 1: Recognize the properties of exponential functions.

The base $\frac{1}{5}$ is less than 1. For bases less than 1, the inequality direction reverses when the exponents are compared.

Step 2: Compare the exponents.

Because $\frac{1}{5} < 1$, the inequality becomes: $x < 5$

Step 3: Final Answer.

The correct solution is $x < 5$.

Correct Option:

The answer is the last one: $x < 5$.

Would you like further explanation on this?
Here are 5 related questions to explore this concept deeper:

1. How do exponential inequalities differ for bases greater than 1 vs. bases less than 1?
2. Can you solve $(\frac{1}{2})^x > (\frac{1}{2})^3$ using the same method?
3. What happens if the inequality were reversed, i.e., $(\frac{1}{5})^x < (\frac{1}{5})^5$?
4. How would you approach solving exponential inequalities with different bases?
5. What role does logarithms play in solving inequalities with the same base?

Tip: Always remember the behavior of exponential functions when the base is less than 1—it decreases as the exponent increases!