The inequality given is:

$e^{5x+1} > e^{-4x+3}$

To solve this inequality, we can start by simplifying it by taking the natural logarithm on both sides. Since the exponential function $e^x$ is strictly increasing, applying the natural logarithm will maintain the inequality.

Step 1: Apply the natural logarithm (ln) to both sides:

$\ln(e^{5x+1}) > \ln(e^{-4x+3})$

This simplifies to: $5x + 1 > -4x + 3$

Step 2: Solve the resulting inequality

Now, solve for $x$:

1. Add $4x$ to both sides: $5x + 4x + 1 > 3$ $9x + 1 > 3$

2. Subtract 1 from both sides: $9x > 2$

3. Divide both sides by 9: $x > \frac{2}{9}$

Step 3: Express the solution

The solution set for $x$ is:

$x \in \left( \frac{2}{9}, +\infty \right)$

Thus, the solution set is $\left( \frac{2}{9}, +\infty \right)$.

Would you like a further breakdown of any steps?

5 related questions:

1. How do you solve inequalities involving logarithmic functions?
2. What are the properties of the natural logarithm used in solving exponential inequalities?
3. Can an exponential inequality have no solution? Under what conditions?
4. How would the inequality change if one side were a different base, such as $2^{5x+1} > 3^{-4x+3}$?
5. What are the implications of solving inequalities over the real numbers $\mathbb{R}$ versus other number sets?

Tip: When solving exponential inequalities, applying the natural logarithm is a powerful method because it linearizes the exponents, making the inequality much easier to solve.