The expression in the image is:

$-3x^3 \geq 81$

Let's solve this inequality step by step:

Step 1: Isolate $x^3$

To get rid of the coefficient $-3$, divide both sides of the inequality by $-3$. Note that when dividing or multiplying by a negative number, the direction of the inequality sign flips:

$x^3 \leq \frac{81}{-3}$

This simplifies to:

$x^3 \leq -27$

Step 2: Solve for $x$

Now, take the cube root of both sides:

$x \leq \sqrt[3]{-27}$

Since the cube root of $-27$ is $-3$, we get:

$x \leq -3$

Final Answer:

The solution to the inequality is:

$x \leq -3$

Would you like more details or further explanations?

Here are five related questions:

1. How do inequalities change when multiplying or dividing by a negative number?
2. What is the cube root, and how does it apply to negative numbers?
3. Can cube roots produce both positive and negative results like square roots?
4. What happens when dealing with higher degree polynomials in inequalities?
5. How do we handle inequalities with even powers like $x^2$?

Tip: Always remember that multiplying or dividing both sides of an inequality by a negative number flips the direction of the inequality sign!