Is $\left|a-b\right| + \left|c-d\right|$ equal to $\left|(a+c)-(b+d)\right|$ for $a,b,c,d \geq0$?

Question

I'm a little confused about this concept, as whatever example I'm taking holds true. I would like to know is the given problem is true considering the value of the variables is whole numbers.

Is this true for n number of terms in the series?

Answer 1

Note that $\left|(a+c)-(b+d)\right|=\left|(a-b)+(c-d)\right|$.

Let $a-b=X$ and $c-d=Y$, therefore $X, Y$ can be any real numbers.

Apparently $\left|X\right|+\left|Y\right|$ doesn't always equal $\left|X+Y\right|$. You will get a counter example when $XY<0$.

In other words, for example, if $a-b>0$ while $c-d<0$, the equation won't hold.