I am having trouble learning how to work with conditional inequalities (i.e. inequalities within an if statement), especially when it comes to solving for a variable that would make the if statement true. I do not think that my issue is with the inequalities per say, but more so with the logic behind combining an if statement and an inequality. I have been unable to find any good sources to aid me on this matter, and chat gpt has been of little use.
Let me provide some examples:
If $|x-2| < 1$, then $|2x - 4| < A$
from my understanding, we need to make |x-2| resemble |2x-4| by multiplying |x-2| < 1 by 2 which gives |2x-4| < 2. I do not understand why this suggests that the if statement is true for all A >=2.
Also, what would be the difference in ones approach when dealing with the previous problem, versus this one? :
If $|x - 2| < A$, then $|2x-4| < 3$
I am asking this because in this problem, the variable is in the hypothesis/antecedent of the if statement whereas in the last problem the variable was in the consequent of the if statement. I am not able to discern the logical difference between the two situations. I know that for an if statement to be true, the consequent needs to be true if the antecedent is true, but the consequent can be either true or false if the antecedent is false. Yet I am not able to use this knowledge to figure out what is going on here.
To conclude, I am kindly requesting for an explanation behind what the first and second problems really mean mathematically and logically, and how they are different from one another. Additionally, I would appreciate some guidance as to how to approach to solve problems similar to the two examples, and an explanation on why the approach makes sense based on the logic behind the problem would be nice as well. I have many more examples of problems like these so feel free to let me know if you would like more examples. Any input / advice is greatly appreciated. Thanks in advance.