a

Best to translate prompt to
a + 1/a < 2b?
in my opinion.

Plugging in numbers works well here. Whenever there are no limitations on negative numbers I like to start there.

Try these and see what happens:

If a = -5
b = -3, or b = -2

This might be a stupid question but can you illustrate how you re-translated the question stem?

Thanks,

No, it's a good question.

Is 1/a - b < b - a?

When I see this, my immediate thought it to try to combine the a's on one side and the b's on the other side. It's not this is always going to be the best possible thing to do, but I find that in general it helps me see what's going on more clearly.

So, start by adding b to both sides, in order to cancel out the (-b) on the left:
1/a - b + b < b - a + b

Simplify:
1/a < 2b - a

Now, do the same with the (-a) on the right:
1/a + a < 2b - a + a
1/a + a < 2b

Definitely ask if you have further questions.

Ok, this is my fault but I should added the () - that's why I could not get the translation

The question is:

Is 1/(a-b) < b -a?

Sorry about that.

Okay, no worries. For this question, it is critical to recognize that (a-b) and (b-a) are opposites of one another.

In other words, (a-b) = -(b-a)
Or, (b-a) = -(a-b)

From this, we can conclude that one of them is always positive, and one of them is always negative.
(a-b) cannot be zero because we can't have 1/0.

Therefore, in order for
1/(a-b) < b-a

We need (a-b) to be the negative one, and (b-a) to be the positive one.
That way, the inequality will be negative < positive
In other words, we need a < b.

Statement 1:
Tells us exactly what we need. Sufficient.

Statement 2:
1 < |a - b|
Here, either one could be the bigger one, which means that (a-b) could be positive, or could be negative. Insufficient.

Thanks.

That was really helpful.