The inequality in the image is:

$10.1^x > 10.1^{-5.63}$

Step-by-step solution:

1. The base $10.1$ is greater than 1, so we can apply logarithmic rules without changing the inequality sign.
2. For inequalities involving exponents with the same base, we can directly compare the exponents. Thus, we have:

$x > -5.63$

Correct Answer:

The solution to the inequality is $x > -5.63$.

Thus, the correct choice from the options is: $\boxed{x > -5,63}$

Let me know if you would like further explanation or have any additional questions!

Related Questions:

1. How would the inequality change if the base was less than 1?
2. What happens if the inequality was $10.1^x < 10.1^{-5.63}$?
3. Can you solve exponential inequalities without having the same base?
4. How does logarithmic transformation help in solving exponential inequalities?
5. What are the key properties of exponents when working with inequalities?

Tip:

Always check whether the base of the exponent is greater or less than 1, as it affects the direction of the inequality when solving.