Slope of the Budget Line When we try to maximise utility constraint to a budget. We find the utility function or indifference curve which is having the budget line as tangent. So for solving a question we can equate the slope of budget line with the slope of a indifference curve and we will get the maximising utility coordinates.In many of the text books they equate ratio of prices with the tangent slope, but the slope of the budget line is negative where as the relative prices are positive. How does this work ? Why are they not considering the - sign of budget line ?
They do consider it. The precise mathematical formulation is usually
$$
-\frac{p_1}{p_2} = \text{MRS}(x_1,x_2)
$$
or
$$
\frac{p_1}{p_2} = \left| \text{MRS}(x_1,x_2) \right|
$$
in the basic cases, when MRS is a negative number (the slope of the indifference curve at basket $(x_1,x_2)$).