Absolute value of x cube plus 4 less than 5 

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.

Absolute value of x cube plus 4 less than 5

Explore a step-by-step solution to the inequality |x^3 + 4| < 5. This problem involves absolute value inequalities and properties of cubic functions, ideal for Grades 10-12 students. Learn how to split and solve inequalities with practical approximation tips.

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