https://file.aiquickdraw.com/mathbot/file/172738236969253vn5xsq.jpg

The problem in the image is asking to prove the following implication:

\[
\forall (x, y) \in \mathbb{R}^2, \quad |x - y| \leq 2 \implies |x + 2| \leq 2
\]

To prove this statement, we can proceed by manipulating the inequality and analyzing the conditions under which the implication holds.

### Step-by-Step Proof:

1. **Given condition**: 
   \[
   |x - y| \leq 2
   \]
   This means that \(x\) and \(y\) are at most 2 units apart in the real number line.

2. **Target condition**: 
   We want to show that under this condition:
   \[
   |x + 2| \leq 2
   \]
   That is, the value of \(x\) should be within 2 units of \(-2\).

### Approach:

From the given condition, \( |x - y| \leq 2 \), we know that:

\[
-2 \leq x - y \leq 2
\]

Add 2 to both sides of this inequality:

\[
-2 \leq x + 2 \leq 2
\]

This directly shows that \( |x + 2| \leq 2 \), which is exactly what we needed to prove.

### Conclusion:
Therefore, the implication holds, and we have proven that:

\[
\forall (x, y) \in \mathbb{R}^2, \quad |x - y| \leq 2 \implies |x + 2| \leq 2
\]

Let me know if you need further clarification!

Here are 5 related questions:
1. How does adding or subtracting constants affect inequalities?
2. Can the same approach be used to solve other absolute value inequalities?
3. What would happen if we changed the constant on the right-hand side of the inequalities?
4. How does this proof generalize to higher dimensions or other types of functions?
5. What properties of absolute values allow us to handle inequalities in this way?

**Tip**: When dealing with absolute value inequalities, always try to rewrite them as double inequalities to gain more insight into the solution set.

Prove that for all (x, y) in R^2, |x - y| <= 2 implies |x + 2| <= 2.

This problem involves proving an implication with absolute values in real numbers: |x - y| <= 2 implies |x + 2| <= 2. The proof uses properties of inequalities and absolute values to derive the result.

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