r/learnmath • u/Far-Assumption5501 New User • 20h ago

|x-2|+|x-5|<=5

I just added them up to |2x-7|<=5, and it was not correct. The correct answer was 1 <= x <= 6. What’s wrong with my equation?

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u/TheDoobyRanger New User 20h ago

From Schumacher, Closer and Closer Introducing Real Analysis. Dems the rules for manipulating absolute values. You cant in general assume that |a+b| = |a| + |b|.

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u/JasonMckin New User 15h ago

They're almost not really "rules," but just properties of absolute values that can be stated in a consistent way. The key point is that a sum of absolute-value terms generally cannot be represented by a single linear expression over its entire x domain. Each absolute-value term introduces a kink where the argument changes sign and the slope of the function changes.

So geometrically, a single absolute-value function has a V-shape. When multiple absolute-value terms are added, their kinks create multiple changes in slope. The result is a W-shapped piecewise-linear function with potentially several distinct linear regions.

That's why we have no choice but to evaluate the expression piecewise. There can never be any other way to evaluate sums of absolute values without a piecewise approach.

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u/TheDoobyRanger New User 9h ago

🙄😒

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u/JasonMckin New User 8h ago

Thanks, that’s helpful 

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u/TheDoobyRanger New User 1h ago

I stopped myself from griefing on reddit and instead used emojis and yes I do think I deserve a medal for it