https://file.aiquickdraw.com/mathbot/file/1726086043203p5lc3ey5.jpg

The image contains an inequality and some given conditions. Let's break it down:

### Conditions:
1. \( a, b \in \mathbb{R}^2 \) (This seems to imply that \( a \) and \( b \) are real numbers, possibly in a two-dimensional real space, though this is likely shorthand for \( a, b \in \mathbb{R} \)).
2. \( c > 0 \)

### Given Inequalities:
1. \( |a| < c \) and \( |b| < c \)
   
   This means that the absolute values of both \( a \) and \( b \) are less than \( c \), where \( c \) is positive.

2. Conclusion:

   The inequality then transitions to:
   \[
   \left| \frac{a + b}{2} \right| + \left| \frac{a - b}{2} \right| < c
   \]
   This can be interpreted as a condition that combines the averages of the sums and differences of \( a \) and \( b \), and shows that their magnitudes are bounded by \( c \).

### Interpretation:

This is a form of **triangle inequality**. It essentially states that if the magnitudes (absolute values) of two numbers are less than some positive constant \( c \), then certain combinations of their sums and differences are also bounded by the same \( c \).

Do you want a more detailed explanation or proof of how this inequality is derived? Here are some related questions:

1. What is the triangle inequality, and how does it relate to this form?
2. How does this inequality change if \( a \) or \( b \) were negative?
3. How can this inequality be applied to geometric interpretations in real space?
4. Can we extend this result to higher dimensions (e.g., 3D space)?
5. What happens if \( c \) is allowed to approach zero?

**Tip:** The absolute value function helps simplify and ensure non-negative results when working with real numbers, which is useful when bounding sums or differences.

(a, b) ∈ ℝ², c > 0 |a| < c et |b| < c → |(a + b) / 2| + |(a - b) / 2| < c

This solution addresses an inequality involving two real numbers, absolute values, and a geometric interpretation through the triangle inequality theorem. Suitable for advanced algebra and analysis students.

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