I always go with the GPAT approach~ Genre, Purpose, Audience, and Tone. You know the genre is argumentative. Therefore, read to learn who the audience is, then the purpose of the argument, which is your topic. Once you know the purpose for the argument, annotate the development of the argument in the text, highlighting specific evidence (details in the text) that supports that topic. These annotations will help you support your thinking and follow the writer's claim/argument/viewpoint. Whenever in doubt, go back to where you annotated and highlighted and you will find your answer or it will clarify your wondering.
Strong evidence that supports a point of view is only strong if it has back up, which would be a reference to something factual, like something in history or a court case, or other research that supports the claim. Otherwise, it is an opinion that doesn't stand up with credibility. Hence, making that type of evidence weak.