Why are both of these forms of the CHSH inequality correct? Most of the time, when looking up the CHSH inequality, I find something like
$$E(0,0)+E(0,1)+E(1,0)-E(1,1)\leq 2.$$
But other times I find something along the lines of 
$$\vert E(0,0)+E(0,1)\vert+\vert E(1,0)-E(1,1)\vert\leq2.$$
(Take a look at pages 12 and 13 of this link (the presentation "Bell inequalities made simple(r)", by Dan Browne) for examples of both forms.) 
These seem like very different equations to me, but apparently they're not.
Edited bonus question:
How come $\vert a+b+c-d \vert=\vert a+b \vert + \vert c - d \vert$?
 A: The derivation of the CHSH inequality actually gives
$$
 -2\leq E(0,0)+E(0,1)+E(1,0)-E(1,1)\leq 2.
\tag{1}
$$
This can also be written
$$
 \Big|E(0,0)+E(0,1)+E(1,0)-E(1,1)\Big|\leq 2.
\tag{2}
$$
The arguments of $E(\cdots)$ are just labels indicating two different measurable properties, so other inequalities may be obtained by permuting the labels. Permute the labels $0\leftrightarrow 1$ in the second argument of $E(\cdots)$ to get
$$
 \Big|E(0,1)+E(0,0)+E(1,1)-E(1,0)\Big|\leq 2.
\tag{3}
$$
The pair of inequalities
$$
 |a+b|\leq 2
 \hskip2cm
 |a-b|\leq 2
\tag{4}
$$
is equivalent to the single inequality
$$
 |a|+|b|\leq 2.
\tag{5}
$$
To see that (4) implies (5), first consider the case where $a$ and $b$ have the same sign (and then it's obvious), and then consider the case where $a$ and $b$ have opposite signs (and then it's obvious again). Now set
$$
 a = E(0,0)+E(0,1)
\hskip2cm
 b = E(1,0)-E(1,1)
$$
so that the first and second equations in (4) are equations (2) and (3), respectively, and equation (5) is
$$
 \Big|E(0,0)+E(0,1)\Big|+\Big|E(1,0)-E(1,1)\Big|\leq 2.
\tag{6}
$$
A: The provided presentation relies on two sources, Cirelson B. S. (1980) (in this paper, their name is incorrectly transliterated from Цирельсон to Cirelson instead of Tsirelson) and Popescu S., et. al. (1994). Both of these only support the absolute version, but one of them uses an alternate notation with two inequalities.
Cirelson B. S. (1980) uses the normal version:

(picture contains the CHSH equation with absolute marks)
Popescu S., et. al. (1994) uses the double-inequality version:

(picture contains the CHSH equation without absolute marks, but with another inequality sign showing it needs to be higher than -2.)
It is likely that the provided presentation has incompletely reproduced the second, leaving out the first inequality. Either way, the first inequality on slide 13 in the provided presentation is incomplete based on these sources.
