Absolute value of x cube plus 4 less than 5

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.

Learn how to solve the inequality |x^3 + 4| < 5 with a step-by-step guide. This solution involves breaking the inequality into two cases and solving for x using cube roots. Ideal for students in Grades 10-12, covering key algebraic concepts such as absolute value and cubic functions.

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