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Home » Set the exponents equal: \(3x = 6\). - Midis

Set the exponents equal: \(3x = 6\). - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Question: An industrial designer is modeling the cross-section of a sustainable packaging shape defined by the region enclosed by the graph of $ |x + y| + |x - y| \leq 8 $. Find the area of this region.

📌 Solution: We analyze the inequality $ |x + y| + |x - y| \leq 8 $. This expression is symmetric and represents a diamond (rhombus) centered at the origin.

📌 Let’s consider the four quadrants formed by the lines $ x + y = 0 $ and $ x - y = 0 $:

📌 In the first quadrant: $ x + y \geq 0, x - y \geq 0 \Rightarrow |x + y| + |x - y| = (x + y) + (x - y) = 2x $

📌 In the second: $ x + y \geq 0, x - y \leq 0 \Rightarrow 2y $

📌 In the third: $ x + y \leq 0, x - y \leq 0 \Rightarrow -2x $

📌 In the fourth: $ x + y \leq 0, x - y \geq 0 \Rightarrow -2y $

📌 $ 2|x| \leq 8 \Rightarrow |x| \leq 4 $

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