https://file.aiquickdraw.com/mathbot/file/172678853852776ma9uwy.jpg

The solution provided in the image refers to an inequality involving exponential expressions. Let's break down the problem and the reasoning behind the result:

### Problem:
The inequality is:

\[
(5^{1-x} - 1)(2^{-x} - 2^2) < 0
\]

This type of inequality can be solved by considering the signs of the two factors individually and ensuring that their product is negative. This happens when one factor is positive, and the other is negative.

### Step-by-step Solution:
1. **First Factor Analysis**: \( 5^{1-x} - 1 \)
   - The expression \( 5^{1-x} \) is an exponential function of \( x \).
   - For different values of \( x \), the result can be greater or less than 1. 

2. **Second Factor Analysis**: \( 2^{-x} - 2^2 \)
   - Similarly, \( 2^{-x} \) is an exponential function where the exponent is negative.

3. **Inequality Setup**:
   To make the product of these two expressions negative, one must be positive and the other negative. The logic behind solving this type of inequality is to create two cases:
   - Case 1: \( 5^{1-x} - 1 > 0 \) and \( 2^{-x} - 4 < 0 \)
   - Case 2: \( 5^{1-x} - 1 < 0 \) and \( 2^{-x} - 4 > 0 \)

4. **Simplification of Cases**:
   Each case can be simplified using properties of logarithmic and exponential functions:
   - Case 1: \( 1 - x > 0 \Rightarrow x < 1 \) and \( -x < 2 \Rightarrow x > -2 \)
   - Case 2: \( x > 1 \) and \( x < -2 \)

5. **Combining Solutions**:
   The final solution is obtained by combining the intervals from the two cases:
   - \( x \in (-2, 1) \)

This is why the final answer is \( x \in (-2, 1) \), as shown in the solution.

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### Relative Questions:
1. Why do exponential functions behave differently for positive and negative exponents?
2. How does logarithmic transformation help in solving exponential inequalities?
3. What are the typical methods to solve inequalities involving multiple factors?
4. Can this inequality be solved graphically, and what would it look like?
5. How do the intervals for \( x \) relate to the properties of the exponential functions?

**Tip**: When solving inequalities with multiple factors, it's useful to analyze each factor's sign changes and carefully combine their effects to find the correct solution intervals.

Explore how to solve inequalities involving exponential functions with opposite signs. Learn key concepts such as sign analysis and exponential functions to understand why the product of two expressions can be less than zero. Suitable for Grades 10-12.

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