$$ \left(0.5\right)^{x-1}>\left(0.25\right)^{-x-2} $$

We are tasked with solving the inequality:

\[
(0.5)^{x-1} > (0.25)^{-x-2}
\]

### Step 1: Rewrite in terms of powers of 2

We know that:

- $ 0.5 = 2^{-1} $
- $ 0.25 = 2^{-2} $

Substitute these into the inequality:

\[
(2^{-1})^{x-1} > (2^{-2})^{-x-2}
\]

### Step 2: Simplify the exponents

Using the property $ (a^m)^n = a^{m \cdot n} $, we simplify both sides:

- Left-hand side: 
  \[
  (2^{-1})^{x-1} = 2^{-1 \cdot (x-1)} = 2^{-(x-1)} = 2^{-x+1}
  \]

- Right-hand side: 
  \[
  (2^{-2})^{-x-2} = 2^{-2 \cdot (-x-2)} = 2^{2x + 4}
  \]

Now the inequality becomes:

\[
2^{-x+1} > 2^{2x+4}
\]

### Step 3: Compare exponents

Since the inequality is in terms of the same base (2), we can compare the exponents directly. This yields the inequality:

\[
-x+1 > 2x+4
\]

### Step 4: Solve for $ x $

Now solve this inequality step by step:

\[
-x + 1 > 2x + 4
\]

Add $ x $ to both sides:

\[
1 > 3x + 4
\]

Subtract 4 from both sides:

\[
-3 > 3x
\]

Divide by 3:

\[
x < -1
\]

### Final Answer:

The solution to the inequality is:

\[
x < -1
\]

Let me know if you want further details or clarifications!

Here are 5 related questions to explore:
1. How do you solve inequalities involving logarithms?
2. What are the general properties of exponential functions?
3. How do you solve exponential equations with different bases?
4. Can this inequality be solved graphically? How?
5. How do transformations affect the graph of an exponential function?

**Tip:** When comparing exponents with the same base, you can directly compare the exponents if the base is positive.

Learn how to solve the exponential inequality (0.5)^{x-1} > (0.25)^{-x-2} step by step. This problem involves simplifying exponents and comparing them to solve inequalities, suitable for Grades 10-12.

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