I found this inequality but I don't know where it comes from

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In summary, the given equation is an identity that follows from the triangle inequality. It may not have a specific name, but it is a commonly used method to rewrite the left hand side.
  • #1
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here is the inequality:
##(\sum\limits_{i=1}^n |x_i-y_i|)^2= \ge \sum\limits_{i=1}^n(x_i-y_i)^2+2\sum\limits_{i \neq j}^n |x_i-y_i|\cdot |x_j-y_j|##
does it have a name/is the consequence of a theorem?
Thank you :)
 
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  • #3
I believe it is actually an equality, assuming you mean the following:
$$\begin{align}
\left(\sum_{i=1}^{n}|x_i - y_i|\right)^2
&= \sum_{i=1}^{n} \sum_{j=1}^{n} |x_i - y_i|\cdot |x_j - y_j| \\
&= \sum_{i=1}^{n} |x_i - y_i|^2 +
\sum_{i=1}^{n} \sum_{j\neq i,j=1}^{n} |x_i - y_i|\cdot |x_j - y_j|\\
&= \sum_{i=1}^{n} |x_i - y_i|^2 +
\sum_{i=1}^{n} \sum_{j=1}^{i-1} |x_i - y_i|\cdot |x_j - y_j| +
\sum_{i=1}^{n} \sum_{j=i+1}^{n} |x_i - y_i|\cdot |x_j - y_j|
\end{align}$$
By symmetry, the second two terms are equal to each other, so we have
$$\left(\sum_{i=1}^{n}|x_i - y_i|\right)^2 = \sum_{i=1}^{n} |x_i - y_i|^2 + 2\sum_{i=1}^{n} \sum_{j=1}^{i-1} |x_i - y_i|\cdot |x_j - y_j|$$
I don't know if this equality has a name, but it's a standard and often useful way to rewrite the left hand side.
 
  • #5
Look at jbunniii's derivation, jedishrfu. It's an identity, not an inequality.
 

1. What is an inequality?

An inequality is a mathematical statement that compares two quantities, expressing that one is less than, greater than, or not equal to the other.

2. How is an inequality different from an equation?

An equation shows that two quantities are equal, while an inequality shows that they are not necessarily equal.

3. How do you solve an inequality?

To solve an inequality, you must isolate the variable on one side of the inequality sign and perform the same operation on both sides of the inequality to maintain the balance.

4. What does it mean when an inequality has no solution?

If an inequality has no solution, it means that there is no value or range of values that satisfies the inequality.

5. Can you provide an example of an inequality and its solution?

One example of an inequality is 2x + 5 > 10. The solution to this inequality is x > 2.5, because when x is greater than 2.5, the inequality is true.

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