To solve the inequality \( |x^2 - 6| < 5 \), we follow these steps:

### Step 1: Rewrite the inequality without the absolute value
The expression \( |x^2 - 6| < 5 \) implies:
\[
-5 < x^2 - 6 < 5
\]

### Step 2: Solve the compound inequality
Break this into two parts:
\[
-5 < x^2 - 6 \quad \text{and} \quad x^2 - 6 < 5
\]

#### Part 1: Solve \( -5 < x^2 - 6 \)
Add 6 to both sides:
\[
1 < x^2
\]
This simplifies to:
\[
x^2 > 1
\]

#### Part 2: Solve \( x^2 - 6 < 5 \)
Add 6 to both sides:
\[
x^2 < 11
\]

### Step 3: Combine the results
We now combine \( x^2 > 1 \) and \( x^2 < 11 \). Together, this means:
\[
1 < x^2 < 11
\]

### Step 4: Solve for \( x \)
The inequality \( 1 < x^2 < 11 \) can be solved by taking square roots:
1. From \( x^2 > 1 \): \( x > 1 \) or \( x < -1 \).
2. From \( x^2 < 11 \): \( -\sqrt{11} < x < \sqrt{11} \).

Combining these, we have:
\[
x \in (-\sqrt{11}, -1) \cup (1, \sqrt{11})
\]

### Final Solution:
The solution to the inequality is:
\[
x \in (-\sqrt{11}, -1) \cup (1, \sqrt{11})
\]

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### 5 Related Questions:
1. How do you solve absolute value inequalities in general?
2. What happens when \( |x^2 - 6| > 5 \)?
3. How does squaring affect the inequality when solving \( x^2 < k \)?
4. Can you explain the difference between strict and non-strict inequalities?
5. What is the graphical representation of \( |x^2 - 6| < 5 \)?

### Tip:
When dealing with absolute value inequalities, always remember to split the inequality into two cases—one positive and one negative!

Learn how to solve the absolute value inequality |x^2 - 6| < 5 through a step-by-step guide. This explanation covers essential algebraic concepts, properties of inequalities, and square root methods. Ideal for students in Grades 9-12.

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