The inequality given is:

$|x^3 + 4| < 5$

To solve this inequality, we can interpret it as two separate inequalities since the absolute value expression implies that the expression inside can be both positive and negative.

1. Split the Inequality: $-5 < x^3 + 4 < 5$

2. Break it into two inequalities:

   • $x^3 + 4 < 5$
   • $x^3 + 4 > -5$

3. Solve each inequality separately:

   • For $x^3 + 4 < 5$: $x^3 < 1$ Taking the cube root of both sides: $x < 1$

   • For $x^3 + 4 > -5$: $x^3 > -9$ Taking the cube root of both sides: $x > -\sqrt[3]{9} \approx -2.08$

4. Combine the Solutions: The solution to the inequality is: $-\sqrt[3]{9} < x < 1$

So, the solution set for $x$ is approximately:

$-2.08 < x < 1$

Let me know if you’d like further explanation on any step!

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Related Questions:

1. How does solving an absolute value inequality differ from solving a standard inequality?
2. What are the applications of absolute value inequalities in real-world contexts?
3. How would the solution change if the inequality was $|x^3 + 4| \leq 5$ instead?
4. Could the inequality $|x^3 + 4| > 5$ have a solution, and how would it be determined?
5. What methods can be used to approximate roots like $-\sqrt[3]{9}$?

Tip: When solving absolute value inequalities, always remember to split the inequality into two cases: one for the positive scenario and one for the negative scenario.