The inequality in the image is:

$||x + 4| - 2| < 1$

Step-by-step solution:

Step 1: Break it down into cases

The double absolute value requires breaking into separate cases to solve.

1. $|x + 4| - 2 < 1$ and $|x + 4| - 2 > -1$

This simplifies to: $-1 < |x + 4| - 2 < 1$

Step 2: Solve the compound inequality

Add $2$ to all sides: $1 < |x + 4| < 3$

Now solve $|x + 4| < 3$ and $|x + 4| > 1$ separately.

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Solve $|x + 4| < 3$:

The definition of absolute value gives: $-3 < x + 4 < 3$

Subtract $4$ from all sides: $-7 < x < -1$

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Solve $|x + 4| > 1$:

Similarly, absolute value inequality gives: $x + 4 > 1 \quad \text{or} \quad x + 4 < -1$

Subtract $4$ in both cases: $x > -3 \quad \text{or} \quad x < -5$

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Combine the solutions

The solution is the overlap of: $-7 < x < -1 \quad \text{and} \quad (x > -3 \, \text{or} \, x < -5)$

This gives:

1. From $x > -3$: $-3 < x < -1$
2. From $x < -5$: $-7 < x < -5$

Final Answer:

$x \in (-7, -5) \cup (-3, -1)$

Let me know if you want a detailed graph or further clarification!

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Related Questions

1. What happens if the inequality includes $\leq$ or $\geq$ instead of $<$?
2. How do we graph this solution on a number line?
3. How can this inequality be solved using a different method (e.g., intervals)?
4. What are the critical points of the inner and outer absolute values?
5. How do we verify the solution using test points?

Tip:

For absolute value inequalities, always consider the definition and break them into cases based on positive and negative values.