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!