https://file.aiquickdraw.com/mathbot/file/17298040422159ag6iixv.jpg

Let's solve the inequality given in the image:

\[
\log \left( \frac{\sqrt{36x^2 + 12x + 1}}{(6x + 1)^6} \right) - \log_3 \left( 27 \cdot (6x + 1)^2 \right) \geq -6
\]

### Step-by-step solution:

1. **Simplify the logarithmic terms:**
   - The first part involves the logarithmic expression \( \log \left( \frac{\sqrt{36x^2 + 12x + 1}}{(6x + 1)^6} \right) \).
     Using the logarithmic rule \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \), we split it:
     \[
     \log \left( \sqrt{36x^2 + 12x + 1} \right) - \log \left( (6x + 1)^6 \right)
     \]

   - Now, simplify both parts:
     \[
     \log \left( \sqrt{36x^2 + 12x + 1} \right) = \frac{1}{

Solve the inequality: log(sqrt(36x^2 + 12x + 1)/(6x + 1)^6) - log_3(27 * (6x + 1)^2) ≥ -6

Learn how to solve a logarithmic inequality involving log properties and change of base. The inequality given includes expressions with square roots and exponents, testing algebraic manipulation skills. Suitable for students in Grades 10-12.

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