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What is the least possible value of the sum | x - 1 | + | x - 1.5 | + | x - 2 | + | x - 2.5 | + | x - 3 |?

 Dec 14, 2021
 #1
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What is the least possible value of the sum | x - 1 | + | x - 1.5 | + | x - 2 | + | x - 2.5 | + | x - 3 |?

Let x = 0.5 => 0.5 + 1.0 + 1.5 + 2.0 + 2.5 = 7.5

Let x = 1.0 => 0 + 0.5 + 1.0 + 1.5 + 2.0    = 5.0

Let x = 1.5 => 0.5 + 0 + 0.5 + 1.0 + 1.5    = 3.5

Let x = 2.0 => 1.0 + 0.5 + 0 + 0.5 + 1.0    = 3.0 (Least Possible Value)

Let x = 2.5 => 1.5 + 1.0 + 0.5 + 0 + 0.5    = 3.5

Let x = 3.0 =>  2.0 + 1.5 + 1.0 + 0.5 + 0   = 5.0

Let x = 3.5 => 2.5 + 2.0 + 1.5 + 1.0 + 0.5 = 7.5

 Dec 14, 2021
 #2
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I checked with a quick sketch and it is reasonably clear that that is correct.

 Dec 15, 2021

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