Algebra: 4th Edition, Page 96 - Optimization Problems

[alulbrich]

Hello,

I have a question regarding this example: "If -4 ≤ m ≤ 7 and -3 < n < 10, what is the maximum possible integer value for m-n?"

Basically, I would like to understand how to properly operate mathematical functions on expressions that are less than or greater than a particular number
(e.g. (-4) - >(-3) = < (-1)).

How do you know when the inequality sign flips such as in the above example?

I thought you only flip the inequality sign if you multiply or divide both sides of an inequality by a negative number?

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[tommywallach]

Hey Alu,

Well there are two ways to think about this. One of them is to attempt to memorize a set of rules. I've never done this, and I don't recommend it, but it's certainly plausible. This would mean you would think of an example:

-4 - (less than -3) =

And then you would think about it. Well, if something is less than -3, it could be -3.1. So let's do the operation --> -4 - (-3.1) = -.9

And now we can derive the rule. When you subtract a LT negative number from a negative number, the result will be GREATER THAN the result you would've gotten if you just had the original negative number, as opposed to the LESS THAN negative number (i.e. -4 - (-3) = -1).

See the problem? This is a true rule...but it's so wordy and complicated that I wouldn't bother.

Instead, I'd encourage you to just THINK LOGICALLY about the situation. If you want maximize m-n, you want the largest possible n, and the smallest possible n. That leads you to m = 7 and n = -2.9999. The result is 9.999, so you know m-n < 10.

Hope that helps!

-t