Final answer:
To solve the inequality (1)/(x+1)+(1)/(x-3)<0, find the values of x for which the expression is negative, considering the critical points x=-1 and x=3. The solution in interval notation is (-1, 3), as this is the interval where the expression is negative but not undefined.
Step-by-step explanation:
To solve the inequality (1)/(x+1)+(1)/(x-3)<0, we must find the values of x for which the expression is negative. We can consider the critical points, which are where the expression is undefined or equal to zero. These points are at x=-1 and x=3. We can test the intervals created by these critical points to determine where the expression is negative.
First, we analyze the sign of each fraction in the intervals (-∞, -1), (-1, 3), and (3, ∞). The sign of the first fraction changes at x=-1 and that of the second fraction changes at x=3. By testing values from each interval, we find that the expression is negative in the interval (-1, 3).
In interval notation, the solution is (-1, 3). We must exclude the points x=-1 and x=3 because at these points the expression is undefined.