Triangle Inequality: Explained with Examples

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The discussion clarifies the triangle inequality, emphasizing that while the initial equality presented is incorrect, the inequality holds true. The triangle inequality states that the absolute value of the sum of two terms is less than or equal to the sum of their absolute values. The example provided illustrates this concept using the expression |(x-1)^2 - 3(x-1)|. Participants seek a deeper understanding of how the inequality applies in different contexts, reinforcing the fundamental principle of the triangle inequality. Overall, the conversation highlights the importance of correctly applying mathematical principles.
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|(x-1)^2-5x+4|=|(x-1)^2 - 3(x-1)| <= |x-1|^2+3|x-1|
how does that work? i thought triangle inequaility was |x+y| <= |x|+|y|
please explain thanks in advance
 
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Well the first equality is false. - 3(x-1) \neq-5x+4. But the inequality is correct. The triangle inequaity says that

|x±y|\leq|x|+|y|
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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