Area Under the Curve
Area bounded by absolute value conditions
Grade 12
Question:
<p>The area of the region of the xy plane defined by the inequality \(|x| + |y| + |x + y| \leq 1\) is</p>
<p>(A) \(\frac{1}{2}\)</p>
<p>(B) \(\frac{3}{4}\)</p>
<p>(C) 1</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use absolute value properties and case analysis based on signs; exploit symmetry to simplify computation.
<p>We need to analyze the region defined by $|x| + |y| + |x + y| \leq 1$ in different cases based on the signs of x, y, and x+y. By symmetry and case analysis in each quadrant, the total area evaluates to $\frac{1}{2}$.</p>
Correct Answer: A