To solve the inequality \( 81^{(2x + 5)} \leq \left(\frac{1}{2187}\right)^{4x} \), let's follow these steps:

1. **Simplify the bases:**
   - Note that \( 81 = 3^4 \).
   - Also, \( 2187 = 3^7 \), so \( \frac{1}{2187} = 3^{-7} \).

   Rewrite the inequality using these simplifications:

   \[
   (3^4)^{(2x + 5)} \leq (3^{-7})^{4x}
   \]

2. **Apply the power rule:** Simplify the expressions by multiplying the exponents:

   \[
   3^{4(2x + 5)} \leq 3^{-7(4x)}
   \]

   Simplifying further:

   \[
   3^{(8x + 20)} \leq 3^{-28x}
   \]

3. **Remove the bases:** Since the bases are the same and \(3\) is positive, we can set the exponents less than or equal to each other:

   \[
   8x + 20 \leq -28x
   \]

4. **Solve for \(x\):** Combine like terms:

   \[
   8x + 28x \leq -20
   \]

   \[
   36x \leq -20
   \]

   Divide by 36:

   \[
   x \leq -\frac{20}{36}
   \]

   Simplify the fraction:

   \[
   x \leq -\frac{5}{9}
   \]

Thus, the solution is:

\[
x \leq -\frac{5}{9}
\]

Would you like further details or have any questions?

Here are some related questions:

1. How can you graph the solution of the inequality on a number line?
2. What would happen if the inequality was strict (i.e., \( < \) instead of \( \leq \))?
3. How does changing the inequality sign affect the solution?
4. Can you solve a similar inequality with different bases?
5. How can you verify the solution of this inequality using a numerical approach?

**Tip:** Always simplify the bases first when dealing with exponential inequalities; this often makes solving much easier!

Learn how to solve the exponential inequality 81^(2x+5) <= (1/2187)^(4x) with a detailed step-by-step solution. This problem covers key algebraic concepts such as simplifying exponents and solving inequalities with same bases. Suitable for students in Grades 9-12.

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