Solve for x when |x-1|<|2x+3|

A graph is always helpful when dealing with this kind of problems. The first thing to do when sketching is to find the intersection with the axis. Ignore the modulus function for the moment. Then flip what's under x axis to include the modulus function. For |x-1| to be less than |2x+3|, its graph needs to be under the other function. We can see from the graph which section that is.The next step is to work out the intersection of two functions. If a section of the function is flipped, it means the sign has been changed, i.e. x-1 becomes 1-x. The first intersection can be found by solving 1-x=-2x-3, since they are both flipped. The second one can be found by solving 1-x=2x+3. They give x=-4 and x=-2/3. From the graph, it can be seem that x needs to have a range of x<-4 or x>-2/3.