(1/2)^5p > (1/2)^(4p+3) complete solutions 

We are tasked with solving the inequality:

\[
\left( \frac{1}{2} \right)^{5p} > \left( \frac{1}{2} \right)^{4p + 3}
\]

### Step 1: Understand the inequality
The base of the exponent on both sides is \(\frac{1}{2}\), which is a number between 0 and 1. When the base is a fraction (between 0 and 1), the inequality is reversed when the exponents are compared directly.

### Step 2: Compare exponents
Since the base is the same, we can directly compare the exponents but reverse the inequality because \(\frac{1}{2}\) is less than 1. So, we get:

\[
5p < 4p + 3
\]

### Step 3: Solve for \(p\)
Now we solve the inequality:

\[
5p - 4p < 3
\]

\[
p < 3
\]

### Step 4: Conclusion
The solution to the inequality is:

\[
p < 3
\]

### Summary:
The inequality \(\left( \frac{1}{2} \right)^{5p} > \left( \frac{1}{2} \right)^{4p + 3}\) simplifies to \(p < 3\).

---

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How would the solution change if the base was greater than 1, like \((2)^{5p} > (2)^{4p + 3}\)?
2. What happens to the inequality if we change the \(>\) to \(\geq\)?
3. Can you solve \(\left( \frac{1}{2} \right)^{5p} = \left( \frac{1}{2} \right)^{4p + 3}\)?
4. What if the base was a negative number instead of \(\frac{1}{2}\)?
5. How would the inequality change if we replaced the exponent with a more complex expression, like \(5p^2\)?

**Tip:** Always pay attention to the base of an exponential expression, as whether the base is greater than or less than 1 affects how inequalities are handled!

(1/2)^5p > (1/2)^(4p+3) complete solutions

This problem guides you through solving the exponential inequality (1/2)^5p > (1/2)^(4p+3). Learn how to compare exponents with the same base, particularly when the base is a fraction less than 1. Suitable for Grades 9-11, this example shows key algebraic techniques for handling exponential inequalities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation