My method is what
@myininaya suggested.
(2x - 3) / (x + 1) < 1
(2x - 3) / (x + 1) - 1 < 0
(2x-3 - x - 1) / (x + 1) < 0
(x - 4) / (x + 1) < 0
The x values that are of interest to us here are the ones that will make f(x) = (x-4)/(x+1) zero or undefined.
f(x) = 0 when x = 4
f(x) is undefined when x = -1.
So the number line is split into three intervals: (-infinity, -1); (-1, 4), (4, infinity).
Pick a convenient number in each interval and see if the inequality is valid.
f(x) = (x - 4) / (x + 1) < 0
when x = -2, f(x) = (-6) / (-1) = 6 which is greater than 0. So not a solution.
when x = 0, f(x) = -4/1 = -4 which is less than 0. This is a solution.
when x = 5, f(x) = 1 / 6 which is greater than 0. Not a solution.
So the solution is x in the interval (-1, 4).