Breaking up inequalities?

[yo4561]

I am confused on breaking up inequalities that have multiple variables to solve for answer.

Let's say, you have the following inequality: -1 < 9 - (a/b) < 3 and then you are given statements to test sufficiency.

I have seen MP split apart inequalities to--> -1<9 - (a/b) and then 9- (a/b) <3 , but why are you allowed to break this up when you do not have an "and" or an "or" in the equation stem?

[esledge]

...why are you allowed to break this up when you do not have an "and" or an "or" in the equation stem?

There is an implied "and," just not in word form.

For example, in 3 < x < 7, both inequality signs have to be telling the truth, so you can read this in words as "x is greater than 3 AND x is less than 7." This is the same as the "3<x AND x<7" that you'd write down.

In contrast, think of a scenario with a true "or," such as "x is either less than -2 or greater than 7." That would be written as "x<-2 OR x>7" where the "or" means that the two parts can't be simultaneously true (if a number is greater than 7, it can't be less than -2, and vice versa). Such an example would never be given in combined form (like this: 7<x<-2....which is illogical).

Thus, if you see an inequality with more than one > or < sign, it is always an AND situation; OR situations just couldn't be given that way.