https://file.aiquickdraw.com/mathbot/file/17266828660449ituyjr3.jpg

The inequality given in the image is:

\[
e^{2x - 4} < e^{-2x - 5}
\]

### Steps to solve:

1. **Simplify the exponential terms:**

Since both sides of the inequality are exponentials with base \(e\), we can compare their exponents directly. This is due to the property that if \(e^a < e^b\), then \(a < b\).

Thus, we compare the exponents:

\[
2x - 4 < -2x - 5
\]

2. **Solve the inequality:**

To solve for \(x\), first gather the terms involving \(x\) on one side:

\[
2x + 2x < -5 + 4
\]

This simplifies to:

\[
4x < -1
\]

Now, solve for \(x\):

\[
x < -\frac{1}{4}
\]

### Solution:

The solution set is \(x \in \left( -\infty, -\frac{1}{4} \right)\).

Would you like further details or have any questions?

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Here are 5 related questions to extend your understanding:

1. How would the inequality change if the exponent signs were reversed?
2. What happens if the inequality was \(e^{2x - 4} > e^{-2x - 5}\)?
3. Can you graph the functions \(e^{2x - 4}\) and \(e^{-2x - 5}\) to visualize the solution?
4. How would the solution change if a constant term were added to both sides?
5. Can this technique be applied to inequalities involving logarithmic functions?

**Tip:** Always check if you can apply logarithmic or exponential properties when simplifying inequalities involving exponentials.

Determine the set of solutions in ℝ of the inequality e^(2x - 4) < e^(-2x - 5).

This page explains how to solve the exponential inequality e^(2x - 4) < e^(-2x - 5) by simplifying the exponentials and solving the inequality step-by-step. Learn how to work with exponential functions and inequalities, suitable for Grades 10-12.

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