We are given the inequality:

$2x^3 + 9x^2 - 18x \geq 81$

Step 1: Move all terms to one side of the inequality.

First, subtract 81 from both sides:

$2x^3 + 9x^2 - 18x - 81 \geq 0$

Step 2: Factor the left-hand side.

We can factor out the greatest common factor (GCF), which is 3:

$3(2x^3 + 3x^2 - 6x - 27) \geq 0$

Now, we factor the cubic expression inside the parentheses. We'll first factor by grouping:

$3[(2x^3 + 3x^2) - (6x + 27)] \geq 0$

Factor each group:

$3[x^2(2x + 3) - 3(2x + 3)] \geq 0$

Now, factor out $(2x + 3)$:

$3(x^2 - 3)(2x + 3) \geq 0$

So, the inequality becomes:

$3(x - \sqrt{3})(x + \sqrt{3})(2x + 3) \geq 0$

Step 3: Solve for the critical points.

Set each factor equal to zero to find the critical points:

1. $x - \sqrt{3} = 0 \implies x = \sqrt{3}$
2. $x + \sqrt{3} = 0 \implies x = -\sqrt{3}$
3. $2x + 3 = 0 \implies x = -\frac{3}{2}$

Thus, the critical points are $x = \sqrt{3}, x = -\sqrt{3},$ and $x = -\frac{3}{2}$.

Step 4: Test intervals between critical points.

Now, we test the sign of $3(x - \sqrt{3})(x + \sqrt{3})(2x + 3)$ in each interval:

1. Interval 1: $(-\infty, -\frac{3}{2})$
2. Interval 2: $\left(-\frac{3}{2}, -\sqrt{3}\right)$
3. Interval 3: $\left(-\sqrt{3}, \sqrt{3}\right)$
4. Interval 4: $(\sqrt{3}, \infty)$

For each interval, we plug in test values into the inequality and determine the sign of the expression.

Step 5: Write the solution in interval notation.

Once we've tested each interval, we identify where the expression is non-negative (since we want $\geq 0$).

The solution is:

$x \in [-\frac{3}{2}, -\sqrt{3}] \cup [\sqrt{3}, \infty)$

Final Answer:

$[-\frac{3}{2}, -\sqrt{3}] \cup [\sqrt{3}, \infty)$

Would you like more details or further clarification on any steps?

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Here are 5 related questions to explore further:

1. How do you factor cubic polynomials?
2. What are the methods for solving inequalities with multiple critical points?
3. How do you perform interval testing in inequalities?
4. What is the significance of each critical point in the context of inequalities?
5. How do you determine whether to include endpoints in interval notation?

Tip: Always check whether your inequality includes equality (≥ or ≤), as this affects whether you use brackets [ ] or parentheses ( ) in the solution.