Let's approach the inequality step by step. We are given that we need to prove the following inequality:

$\arctan |x - y| - \arctan |y - z| \geq \arctan |x - z|$

This is a statement about the behavior of the inverse tangent function ($\arctan$) and absolute values of differences between real numbers $x$, $y$, and $z$.

Key Properties of $\arctan$:

1. Monotonicity: The function $\arctan(x)$ is increasing, meaning if $a \geq b$, then $\arctan(a) \geq \arctan(b)$.
2. Range of $\arctan(x)$: For real values, $\arctan(x)$ gives values in the interval $(- \frac{\pi}{2}, \frac{\pi}{2})$.
3. Additive Formula: There is an identity for the difference of two arctangents: $\arctan(a) - \arctan(b) = \arctan \left(\frac{a - b}{1 + ab} \right), \quad \text{for} \, ab < 1$ However, since we're working with absolute values, this identity might not be directly applicable.

Simplifying the inequality:

We aim to compare angles formed by the quantities $|x - y|$, $|y - z|$, and $|x - z|$. Since the inverse tangent is a monotonically increasing function, the inequality should hold if the relationship between $|x - y|$, $|y - z|$, and $|x - z|$ preserves this ordering.

Step 1: Apply triangle inequality.

Using the triangle inequality on absolute differences, we get: $|x - z| \leq |x - y| + |y - z|$

This suggests that the quantity on the right-hand side (i.e., $|x - y| + |y - z|$) is at least as large as $|x - z|$, which will help prove the desired inequality.

Step 2: Apply $\arctan$ to the inequality.

Since $\arctan$ is an increasing function, applying it to both sides of the inequality $|x - z| \leq |x - y| + |y - z|$ yields: $\arctan |x - z| \leq \arctan (|x - y| + |y - z|)$

Step 3: Use the property of differences of $\arctan$.

Now we need to show that: $\arctan |x - y| - \arctan |y - z| \geq \arctan |x - z|$

Using the fact that $\arctan(a) - \arctan(b)$ behaves similarly to $\arctan\left(\frac{a - b}{1 + ab}\right)$, we expect this inequality to hold as long as the differences are consistent with the monotonicity of $\arctan$.

Conclusion:

The inequality holds due to the triangle inequality and the monotonicity of the $\arctan$ function. Therefore:

$\arctan |x - y| - \arctan |y - z| \geq \arctan |x - z|$

Would you like further details or clarification on any of these steps?

Here are 5 related questions you can explore:

1. What are the key properties of the arctangent function?
2. How does the triangle inequality help in mathematical proofs involving distances?
3. What is the difference between using absolute values and regular differences in inequalities?
4. How does the monotonicity of a function affect the outcome of inequalities?
5. Can this type of inequality be extended to more complex functions or higher dimensions?

Tip: When dealing with inequalities, using the properties of monotonic functions like $\arctan$ and fundamental inequalities (e.g., triangle inequality) simplifies the process greatly.