From a 'negative taxi' to 'Positive taxi' :D

$1729\displaystyle\sum_{-1729}^{1729} |x+1|-|x-1|$

This is equivalent to :

$1729 \times \left[\bigg[\displaystyle\sum_{r=0}^{1730} r +\displaystyle\sum_{r=0}^{1728} r\bigg] - \bigg[\displaystyle\sum_{r=0}^{1730} r +\displaystyle\sum_{r=0}^{1728} r\bigg]\right]$

$\underline{= 1729(0) = 0}$

$1729\displaystyle\sum_{-1729}^{1729} |x+1|-|x|$

This is equivalent to :

$1729 \times \left[\bigg[\displaystyle\sum_{r=0}^{1730} r +\displaystyle\sum_{r=0}^{1728} r\bigg] - \bigg[\displaystyle\sum_{r=0}^{1728} r +\displaystyle\sum_{r=0}^{1729} r\bigg]\right]$

$\underline{= 1729(1730 - 1729) = 1729(1) = 1729}$

Hence , the answer $=0-1729 = \huge\boxed{\color{#3D99F6}{-1729}}$ .

This way is for those who are not familiar with summation function (like me ) .

We can see that $| x+1| -|x-1|$ is positive for all value x > 0 and negative for all values less than 0 . So we can divide the expression in two parts.

When the expression is positive we can remove modulus sign . Hence $x+1 -x+1 = 2$ which means the value of expression will 2 when x>0 . Similarly the value of expression will be -2 for all values for x<0 .At x= 0 the value will be 0. As we are asked to sum from -1729 to +1720 its value will be 1729[ $1729\times2$ + $1729\times-2 +0$ ]=0

Well the first part of the equation is 0.In the second part

When $x>0$ then $|x+1|-|x| = 1$ At x=0 its value is 1

When $x<0$ then $|x+1|-|x| = -x-1+x = -1$

So the value of second part of expression will be -1729[ $1729\times1 + 1729\times-1 + 1$ ] = -1729 the answer