$\"\"$

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The sides of square $\"\"$ are extended to form rectangle $\"\"$ .

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The perimeter of the rectangle is at least twice the perimeter of the square.

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Find the maximum length of a side of square $\"\"$ .

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Let $\"\"$ be the side length of the square $\"\"$ .

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So, the perimeter of the square is $\"\"$ .

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Therefore, the length of the rectangle is $\"\"$ and the width of the rectangle is $\"\"$ .

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So, the perimeter of the rectangle is $\"\"$ . Since the perimeter of the rectangle is at least twice the perimeter of the square.

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The inequality is $\"\"$ .

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Solve the inequality.

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$\"\"$ (Original inequality)

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$\"\"$ (Apply distributive property: $\"\"$ )

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$\"\"$ (Commutative addition property: $\"\"$ )

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$\"\"$ (Combine like terms)

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$\"\"$ (Subtract $\"\"$ from each side)

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$\"\"$ (Apply additive inverse property: $\"\"$ )

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$\"\"$ (Subtract $\"\"$ from each side)

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$\"\"$ (Apply additive inverse property: $\"\"$ )

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$\"\"$ (Divide each side by $\"\"$ )

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$\"\"$ (Cancel common terms)

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Therefore, the maximum length of the side of square $\"\"$ is $\"\"$ .

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$\"\"$

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The maximum length of the side of square $\"\"$ is $\"\"$ .