To solve the inequality:

$|x^3 + 4| < 5$

we'll break this absolute value inequality into two cases.

Step 1: Set up the cases

The absolute value inequality $|A| < B$ implies: $-B < A < B$ In our case, $A = x^3 + 4$ and $B = 5$. So we rewrite the inequality as: $-5 < x^3 + 4 < 5$

Step 2: Solve each part of the inequality

Part 1: $-5 < x^3 + 4$

1. Subtract 4 from both sides: $-5 - 4 < x^3$ $-9 < x^3$
2. This inequality tells us that $x^3 > -9$.

Part 2: $x^3 + 4 < 5$

1. Subtract 4 from both sides: $x^3 < 5 - 4$ $x^3 < 1$

Step 3: Combine the results

From the two parts, we have: $-9 < x^3 < 1$

Step 4: Take the cube root

To isolate $x$, take the cube root of each part: [ \sqrt[3]{-9} < x < \sqrt[3]{1} ] The cube root of $-9$ is approximately $-2.08$ and the cube root of $1$ is $1$.

Final Answer

$-2.08 < x < 1$

This is the solution for $x$ in the inequality $|x^3 + 4| < 5$.

Would you like more details or have any questions?

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Relative Questions:

1. How does the solution change if we modify the inequality to $|x^3 + 4| \leq 5$?
2. What are the steps if the inequality is $|x^3 + 4| > 5$?
3. How can we solve similar inequalities with even higher powers, like $|x^4 + 4| < 5$?
4. How would this solution change if it involved an equation instead of an inequality?
5. Can we visualize the solution on a number line or using a graph?

Tip: For inequalities involving absolute values, break the problem into two cases to remove the absolute value signs and solve each part separately.