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The inequality provided is:

\[
6x^2 - 54 \leq 0
\]

### Step-by-step solution:

1. **Simplify the inequality:**
   \[
   6x^2 - 54 \leq 0
   \]
   Factor out the common factor of 6:
   \[
   6(x^2 - 9) \leq 0
   \]
   Divide both sides by 6 (since 6 is positive, the inequality direction remains the same):
   \[
   x^2 - 9 \leq 0
   \]

2. **Factor the quadratic:**
   \[
   x^2 - 9 = (x - 3)(x + 3)
   \]
   So, the inequality becomes:
   \[
   (x - 3)(x + 3) \leq 0
   \]

3. **Determine the critical points:**
   The critical points are \( x = 3 \) and \( x = -3 \), which are the solutions when the expression equals zero.

4. **Analyze the sign of the expression in intervals:**
   Test the sign of \((x - 3)(x + 3)\) in three intervals determined by the critical points:
   - For \( x < -3 \), both factors \((x - 3)\) and \((x + 3)\) are negative, so their product is positive.
   - For \( -3 \leq x \leq 3 \), one factor is positive and the other is negative, so their product is negative or zero.
   - For \( x > 3 \), both factors are positive, so their product is positive.

   The inequality \((x - 3)(x + 3) \leq 0\) holds when the product is zero or negative, i.e., in the interval:
   \[
   -3 \leq x \leq 3
   \]

### Conclusion:
The solution to the inequality is:
\[
-3 \leq x \leq 3
\]

The correct answer from the provided options is the third one: 
\[
\boxed{-3 \leq x \leq 3}
\]

Would you like more details or have any questions?

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Here are five related questions to expand your understanding:
1. What happens if the inequality was strict (\( < \)) instead of non-strict (\( \leq \))?
2. How would the solution change if the equation had been \( 6x^2 + 54 \leq 0 \)?
3. Can you solve an inequality of the form \( x^2 - 4x + 3 \leq 0 \)?
4. How would you graph the solution to this inequality on a number line?
5. How does solving inequalities differ from solving equalities?

**Tip:** Always check critical points by substituting them back into the original inequality to verify the boundary conditions.

What is the solution to the inequality 6x^2 - 54 ≤ 0?

This page explains how to solve the quadratic inequality 6x^2 - 54 ≤ 0 by factoring and analyzing intervals. The solution shows that -3 ≤ x ≤ 3, covering key concepts in algebra such as quadratic equations and inequalities.

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