Show that ||x| − |y|| ≤ |x − y| without using the
triangle inequality

To prove that \( ||x| - |y|| \leq |x - y| \) without using the triangle inequality, we can proceed as follows:

Let's denote \( x \) and \( y \) as real numbers.

### Case 1: \( x \geq y \geq 0 \) or \( y \geq x \geq 0 \)

Assume \( x \geq y \geq 0 \). Then:

\[
|x| = x, \quad |y| = y, \quad \text{and} \quad |x - y| = x - y
\]

So, the left side of the inequality becomes:

\[
||x| - |y|| = |x - y|
\]

In this case, the inequality \( ||x| - |y|| \leq |x - y| \) holds as an equality.

### Case 2: \( x \geq 0 \) and \( y \leq 0 \) (or vice versa)

Assume \( x \geq 0 \) and \( y \leq 0 \). Then:

\[
|x| = x \quad \text{and} \quad |y| = -y
\]

So, the expression for \( ||x| - |y|| \) becomes:

\[
||x| - |y|| = |x - (-y)| = |x + y|
\]

The right side, \( |x - y| \), is:

\[
|x - y|
\]

Now, since \( x \geq 0 \) and \( y \leq 0 \), it is easy to see that:

\[
|x + y| \leq |x - y|
\]

Thus, the inequality \( ||x| - |y|| \leq |x - y| \) holds.

### Case 3: \( x \leq 0 \) and \( y \leq 0 \)

Assume \( x \leq 0 \) and \( y \leq 0 \). Then:

\[
|x| = -x \quad \text{and} \quad |y| = -y
\]

So:

\[
||x| - |y|| = |-x + y| = |y - x|
\]

And:

\[
|x - y| = |(-x) - (-y)| = |y - x|
\]

Thus, the inequality \( ||x| - |y|| \leq |x - y| \) holds as an equality in this case as well.

### Case 4: \( y \geq x \geq 0 \) (Reverse of Case 1)

This case is symmetric to Case 1 and follows the same logic.

### Conclusion

In all cases, the inequality \( ||x| - |y|| \leq |x - y| \) holds. This concludes the proof.

---

Would you like more details on any part of the proof? 

Here are some related questions you might be interested in:

1. Can you provide an example where \( ||x| - |y|| = |x - y| \)?
2. What is the geometric interpretation of \( ||x| - |y|| \leq |x - y| \)?
3. How does the triangle inequality compare to this inequality?
4. Is the inequality \( ||x| - |y|| \leq |x - y| \) strict or can it be an equality?
5. How would the proof change if \( x \) and \( y \) were vectors instead of real numbers?

**Tip:** When dealing with absolute values, breaking down the problem into cases based on the signs of the variables often simplifies the proof.

Learn how to prove the inequality ||x| - |y|| ≤ |x - y| without relying on the triangle inequality. This problem explores properties of absolute values and inequalities in real numbers. Suitable for advanced high school students and college-level learners.

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