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The inequality provided is:

\[
1 < \frac{2}{|x + 3|} < 4
\]

### Steps to Solve the Inequality

1. **Break the Inequality into Two Parts**:
   - \(1 < \frac{2}{|x + 3|}\)
   - \(\frac{2}{|x + 3|} < 4\)

2. **Solve the First Inequality** \(1 < \frac{2}{|x + 3|}\):

   - Multiply both sides by \(|x + 3|\) (since \(|x + 3|\) is positive):
     \[
     |x + 3| < 2
     \]
   - This implies:
     \[
     -2 < x + 3 < 2
     \]
   - Subtract 3 from all parts:
     \[
     -5 < x < -1
     \]

3. **Solve the Second Inequality** \(\frac{2}{|x + 3|} < 4\):

   - Multiply both sides by \(|x + 3|\):
     \[
     2 < 4|x + 3|
     \]
   - Divide both sides by 4:
     \[
     \frac{1}{2} < |x + 3|
     \]
   - This implies:
     \[
     |x + 3| > \frac{1}{2}
     \]
   - Therefore:
     \[
     x + 3 > \frac{1}{2} \quad \text{or} \quad x + 3 < -\frac{1}{2}
     \]
   - Subtract 3 from both sides:
     \[
     x > -\frac{5}{2} \quad \text{or} \quad x < -\frac{7}{2}
     \]

4. **Combine the Results**:

   From the first part, \(x\) must satisfy \( -5 < x < -1 \).

   From the second part, \(x\) must satisfy either \(x > -\frac{5}{2}\) or \(x < -\frac{7}{2}\).

   The intersection of these two conditions gives the final solution:

   - For \(x < -\frac{7}{2}\), it must satisfy \( -5 < x < -\frac{7}{2} \).
   - For \(x > -\frac{5}{2}\), it must satisfy \( -\frac{5}{2} < x < -1 \).

   So the solution is:
   \[
   -5 < x < -\frac{7}{2} \quad \text{or} \quad -\frac{5}{2} < x < -1
   \]

Would you like any further details or have any questions?

### Related Questions

1. How would the solution change if the inequality was non-strict (i.e., \( \leq \) or \( \geq \))?
2. What would the solution be if the absolute value was not present?
3. Can you solve a similar inequality involving a different absolute value expression?
4. How can the solution be visualized on a number line?
5. How would you test a specific value of \(x\) to ensure it satisfies the inequality?

### Tip
When dealing with absolute value inequalities, remember to break the problem into different cases based on the definition of absolute value.

Learn how to solve the inequality 1 < 2/|x + 3| < 4 by breaking it into two parts and applying the properties of absolute value and inequalities. Suitable for students in Grades 10-12.

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