The set of real numbers x that satisfy the double inequality
0 < |x + 2| < 4

To solve the double inequality \(0 < |x + 2| < 4\), we will break it down into two separate inequalities:

1. \(0 < |x + 2|\)
2. \(|x + 2| < 4\)

### 1. Solving \(0 < |x + 2|\)
The expression \(0 < |x + 2|\) means that \(x + 2 \neq 0\). This implies:
\[
x \neq -2
\]

### 2. Solving \(|x + 2| < 4\)
The inequality \(|x + 2| < 4\) means that \(x + 2\) lies between \(-4\) and \(4\). So, we can write:
\[
-4 < x + 2 < 4
\]

Subtract 2 from all parts of the inequality:
\[
-4 - 2 < x < 4 - 2
\]
\[
-6 < x < 2
\]

### Combining the Results
From the first part, \(x \neq -2\), and from the second part, \(-6 < x < 2\). Combining these, the solution set is:
\[
-6 < x < -2 \quad \text{or} \quad -2 < x < 2
\]

This is the set of all real numbers \(x\) such that \(x\) is in the open interval \((-6, 2)\) but excluding \(x = -2\).

### Final Answer
The solution set is:
\[
x \in (-6, -2) \cup (-2, 2)
\]

Would you like any more details or have further questions?

Here are five related questions to explore further:

1. How do you solve inequalities involving absolute values with different intervals?
2. What happens when you square both sides of an absolute value inequality?
3. How do you solve a system of inequalities graphically?
4. What are some real-world applications of absolute value inequalities?
5. How can you express compound inequalities in interval notation?

**Tip:** Always break down absolute value inequalities into separate cases to simplify the problem-solving process.

Learn how to solve the double inequality 0 < |x + 2| < 4 step-by-step. This problem covers inequalities involving absolute values and interval notation. Suitable for high school students.

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