Key Concepts and Formulas

1. Absolute Value Inequalities:
   • $|u| \ge k \iff u \ge k \text{ or } u \le -k$ (for $k \ge 0$)
   • $|u| \le k \iff -k \le u \le k$ (for $k \ge 0$)
2. Geometric Interpretation of $|x| + |y| \le k$: This inequality defines a square region centered at the origin with vertices at $(\pm k, 0)$ and $(0, \pm k)$. The region includes its boundary.
3. Set Operations: The intersection of sets ($A \cap B$) consists of elements common to both sets.
4. Coordinate Axes: The x-axis is defined by $y=0$, and the y-axis is defined by $x=0$.

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Step-by-Step Solution

We are given sets A and B, and we need to find set C, which is defined as $C = \{(x, y) \in A \cap B : x=0 \text{ or } y=0\}$. This means C contains points $(x,y)$ that satisfy three conditions simultaneously:

1. $(x, y) \in A \implies |x+y| \ge 3$
2. $(x, y) \in B \implies |x|+|y| \le 3$
3. $x=0$ or $y=0$

We need to find all points satisfying these conditions and then compute $\sum_{(x, y) \in \mathrm{C}}|x+y|$.

Step 1: Analyze the conditions for set C. The third condition, "$x=0$ or $y=0$", implies that the points in C must lie on either the x-axis or the y-axis. This significantly simplifies the problem, as we can analyze these two cases separately.

Step 2: Consider points on the x-axis (where $y=0$). If $y=0$, the conditions become:

1. $|x+0| \ge 3 \implies |x| \ge 3$. This means $x \ge 3$ or $x \le -3$.
2. $|x|+|0| \le 3 \implies |x| \le 3$. This means $-3 \le x \le 3$.

We need to find values of $x$ that satisfy both $|x| \ge 3$ and $|x| \le 3$. The only real numbers that satisfy both these inequalities are $x=3$ and $x=-3$. Therefore, the points on the x-axis that belong to C are $(3,0)$ and $(-3,0)$.

Step 3: Consider points on the y-axis (where $x=0$). If $x=0$, the conditions become:

1. $|0+y| \ge 3 \implies |y| \ge 3$. This means $y \ge 3$ or $y \le -3$.
2. $|0|+|y| \le 3 \implies |y| \le 3$. This means $-3 \le y \le 3$.

We need to find values of $y$ that satisfy both $|y| \ge 3$ and $|y| \le 3$. The only real numbers that satisfy both these inequalities are $y=3$ and $y=-3$. Therefore, the points on the y-axis that belong to C are $(0,3)$ and $(0,-3)$.

Step 4: Determine the set C. Combining the points found in Step 2 and Step 3, the set C is: $\mathrm{C} = \{(3,0), (-3,0), (0,3), (0,-3)\}$

Step 5: Calculate the sum $\sum_{(x, y) \in \mathrm{C}}|x+y|$. We evaluate $|x+y|$ for each point in C:

• For $(3,0)$: $|3+0| = |3| = 3$.
• For $(-3,0)$: $|-3+0| = |-3| = 3$.
• For $(0,3)$: $|0+3| = |3| = 3$.
• For $(0,-3)$: $|0-3| = |-3| = 3$.

The sum is $3 + 3 + 3 + 3 = 12$.

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Common Mistakes and Tips

• Confusing Absolute Value Inequalities: Ensure correct interpretation of $|u| \ge k$ (two separate regions) and $|u| \le k$ (a single bounded interval).
• Geometric Interpretation: Visualizing the region $|x|+|y| \le 3$ as a square with vertices at $(\pm 3, 0)$ and $(0, \pm 3)$ helps understand the constraints.
• Handling "OR" Conditions: The condition "$x=0$ or $y=0$" correctly leads to analyzing points on the x-axis and y-axis separately.

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Summary

The problem required identifying points that satisfy conditions related to absolute value inequalities and lying on the coordinate axes. By applying the definitions of absolute value inequalities and considering the cases where $y=0$ and $x=0$, we found that the set C consists of the four points $(3,0), (-3,0), (0,3), (0,-3)$. Calculating $|x+y|$ for each of these points and summing them yielded the result.

The final answer is $\boxed{\text{12}}$.