To solve the inequality $x^3 > -12$, follow these steps:

1. Interpret the inequality: The goal is to find the values of $x$ such that the cube of $x$ is greater than $-12$.

2. Rewrite the inequality: $x^3 > -12$

3. Isolate $x$: Take the cube root on both sides. Since the cube root function is monotonic (it preserves the inequality's direction), we can write: $x > \sqrt[3]{-12}$

4. Simplify $\sqrt[3]{-12}$: The cube root of $-12$ is approximately $-2.29$.

   So, the solution is: $x > -2.29$

Final Answer:

The solution set is: $x \in (-2.29, \infty)$

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Would you like further explanation or help plotting this inequality on a graph?

5 Related Questions:

1. What are the rules for solving inequalities involving exponents?
2. How does the cube root differ from the square root in terms of preserving inequality direction?
3. What happens if the inequality is $x^3 \geq -12$ instead of $x^3 > -12$?
4. How would this inequality change if it was $x^3 < -12$?
5. What are the steps to approximate cube roots of non-perfect cubes?

Tip:

For cube roots of negative numbers, remember that the cube root of a negative number is also negative (e.g., $\sqrt[3]{-8} = -2$).