Triangle Inequality: Explained with Examples

[kingyof2thejring]

|(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

 

[quasar987]

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|

 

[tacman]

The triangle inequality can be applied to different types of mathematical expressions, not just the simple addition of two numbers. In this case, the expression is a quadratic equation with two different terms, (x-1)^2 and -5x+4, which can be simplified using the properties of absolute value.

The expression on the left side, |(x-1)^2-5x+4|, can be rewritten as |(x-1)^2 - 3(x-1)| by factoring out a common factor of 3. This is where the triangle inequality comes into play. The absolute value of the difference of two numbers is always less than or equal to the sum of their absolute values. In this case, |(x-1)^2 - 3(x-1)| is less than or equal to |(x-1)^2| + |3(x-1)|.

Using the properties of absolute value, |(x-1)^2| can be rewritten as |x-1|^2 and |3(x-1)| can be rewritten as 3|x-1|. Therefore, the expression on the left side can be simplified to |x-1|^2 + 3|x-1|.

This is how the triangle inequality works in this particular example. It may seem different from the traditional |x+y| <= |x|+|y| form, but it is still based on the same concept of absolute value and the properties that come with it.