How to define the sign of a function $$y=\arctan\frac{x+1}{x-3} + \frac{x}{4}$$
I know that is necessary to put the function $>$ than $0$, but then?
$$\arctan\frac{x+1}{x-3} + \frac{x}{4}>0$$
It's a sum, so I can't set up a "false system" putting the two factors $>0$. In this case which is the rule to study the sign of this function?



Note: These are not homeworks.

You want to determine the sign of $y$ given $x$, correct? In other words, you want to figure out when $y > 0$, which means, as you said, that we have to solve the inequality
$$\arctan \frac{x + 1}{x - 3} + \frac{x}{4} > 0.$$
The first step in solving this inequality are finding its "edges"—where the inequality goes from true to false, or vice versa. It turns out that there are only two places where this can happen: where the equation
$$\arctan \frac{x + 1}{x - 3} + \frac{x}{4} = 0$$
is true, and where the expression
$$\arctan \frac{x + 1}{x - 3} + \frac{x}{4}$$
is undefined. So, begin by making a list of all the points where the equation is true, along with all the points where that expression is undefined. You will use this list to determine the sign of $y$ given $x$ at all points.
