Problem

An axis-aligned rectangle has consecutive vertices \((0,0)\), \((x,0)\), \((x,3)\), \((0,3)\) and perimeter \(18\), with no assumed left-right order. Derive the horizontal length, write the absolute-value equation, solve for all real \(x\)-values, and check them.

Source diagram: An axis-aligned rectangle has consecutive vertices (0,0), (x,0), (x,3), and (0,3) and perimeter 18; no left-right order or solved x-values are shown.

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What this problem is really about

First use the perimeter relationship to recover the rectangle’s positive missing width. Then translate that width into an absolute horizontal distance from the fixed coordinate; because distance does not encode left or right, keep both locations unless the diagram imposes an orientation.

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horizontal length = 18/2−3=6; equation = x−0=6; solution set = 6; check = x=6 gives side lengths 6 and 3 and perimeter 18

horizontal length = 3; equation = |x−0|=3; solution set = −3,3; check = either value gives side lengths 3 and 3 and perimeter 12

horizontal length = 18/2=9; equation = |x−0|=9; solution set = −9,9; check = either value gives side lengths 9 and 3 and perimeter 24

horizontal length = 18/2-3=6; equation = |x-0|=6; solution set = −6,6; check = either value gives side lengths 6 and 3 and perimeter 18

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An axis-aligned rectangle has consecutive vertices \((0,0)\), \((x,0)\), \((x,3)\), \((0,3)\) and perimeter \(18\), with no assumed left-right order. Derive the horizontal length, write the absolute-value equation, solve for all real \(x\)-values, and check them.

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