Marginal rate of substitution notation: I am having a dumb doubt in writing some slides for an undergraduate class. 
I want to be consistent with the use in microeconomics but this easy thing is really bugging me:


Mas-colell pag. 54


$ MRS_{lk} = \frac{\frac{\partial u}{\partial x_l}}{\frac{\partial u}{\partial x_k}} $ 

"tells us the amount of good k that the consumer must be given to compensate her for a one-unit marginal reduction in her consumption of good l"


Reny advanced micro pag. 18


$MRS_{ij}(x) ≡ \frac{\frac{\partial u}{\partial x_i}}{\frac{\partial u}{\partial x_j}}$

" ..MRS_{ij}(x) is again a positive number, and it tells us the rate at which good j can be exchanged per unit of good i with no change in the consumer’s utility"

I know also how to derive this result with the total differentiation but one thing really confuses me: 

$du = \frac{\partial u}{\partial x_i}*dx_i+\frac{\partial u}{\partial x_j}*dx_j$

$ 0 = \frac{\partial u}{\partial x_i}*dx_i+\frac{\partial u}{\partial x_j}*dx_j $

$ \frac{\partial u}{\partial x_i}*dx_i = -\frac{\partial u}{\partial x_j}*dx_j$

$ \frac{dx_j}{dx_i} = - \frac{\frac{\partial u}{\partial x_i}}{\frac{\partial u}{\partial x_j}}$

The last equation expresses the differential increase in $x_j$ correspondent to a marginal increase in $x_i$ just as the two formulas above. Where does the minus go in the two books? Am I neglecting something?
Thanks.
It is implicit in the interpretation: 

Mas-Collel: the amount that must be given (+)  to compensate for a reduction (-). 

Reny: The rate at which good j  can be exchanged (+ &amp; -) for good i. 

The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good $i$ increases, the quantity of good $j$ should decrease.

Another reason why some authors omit the negative sign is that in equilibrium  $|MRS_{ij}|=|\frac{P_i}{P_j}|$ So either you use the derivation from total differentiation and also the negative ratio of prices, or leave both sides as positive and focus on interpretation. Most people prefer the latter.