Simplifying inequalities question

[yo4561]

I have a specific question about cross multiplying when you have an expression on one side of an inequality...if you have to consider the sign of the expression as a whole?

Let's say x, y, and z are all positive numbers.

Then, you have the expression....

(x+y+z) / (z-y) <0

You can't multiply the (z-y) times 0 to get rid of it because you do not know what the sign of the expression takes even though you know that both variables are positive, z could be less than y... e.g. if (z-y) is negative, you would need to flip the sign versus if (z-y) is positive, you wouldn't need to flip the sign? Is my thinking correct?

Many thanks my friends

[esledge]

Let's say x, y, and z are all positive numbers.

Then, you have the expression....

(x+y+z) / (z-y) <0

You can't multiply the (z-y) times 0 to get rid of it because you do not know what the sign of the expression takes even though you know that both variables are positive, z could be less than y... e.g. if (z-y) is negative, you would need to flip the sign versus if (z-y) is positive, you wouldn't need to flip the sign? Is my thinking correct?

Yes, you are correct!

[yo4561]

Let's say x, y, and z are all positive numbers.

Then, you have the expression....

(x+y+z) / (z-y) <0

You can't multiply the (z-y) times 0 to get rid of it because you do not know what the sign of the expression takes even though you know that both variables are positive, z could be less than y... e.g. if (z-y) is negative, you would need to flip the sign versus if (z-y) is positive, you wouldn't need to flip the sign? Is my thinking correct?

Yes, you are correct!

Thank you Emily for all of your thoughtful responses! You are the best!

[esledge]

Aw, thanks! You are very welcome.