Manipulating inequalities

[cjb]

I hate DS inequalities questions with a passion. At the moment, my only workable, quick way of solving these is to try to sketch the inequalities and look for areas of overlap. I can manage algebraic solutions if I have plenty of time or the question is easy. Generally I don't like algebra with inequalities because the range of possible manipulations seems smaller, and I can't see how to get to my target expression.

Aaaanyway, I was looking at this:

https://www.beatthegmat.com/equality-t30669.html

If r + s > 2t, is r > t ?

(1) t > s

It occurred to me that, given (1), I should be allowed to substitute the "s" in the first equation for "t", since this preserves the inequality, and get

r + s > 2t
=> r + t > 2t
=> r > t

and for the second part

2) r > s

can I substitute:

r + s > 2t
=> 2r > 2t
=> r > t

Sorry if this is an obvious question, but it's not something I remember reading or learning anywhere, but it seems reasonable so long as I preserve the inequality (i.e. make the big side bigger or the small side smaller)

Is this a reliable method of manipulation? Is this well taught in any of the books or on any website?

[Ian Stewart]

I hate DS inequalities questions with a passion. At the moment, my only workable, quick way of solving these is to try to sketch the inequalities and look for areas of overlap. I can manage algebraic solutions if I have plenty of time or the question is easy. Generally I don't like algebra with inequalities because the range of possible manipulations seems smaller, and I can't see how to get to my target expression.

Aaaanyway, I was looking at this:

https://www.beatthegmat.com/equality-t30669.html

If r + s > 2t, is r > t ?

(1) t > s

It occurred to me that, given (1), I should be allowed to substitute the "s" in the first equation for "t", since this preserves the inequality, and get

r + s > 2t
=> r + t > 2t
=> r > t

and for the second part

2) r > s

can I substitute:

r + s > 2t
=> 2r > 2t
=> r > t

Sorry if this is an obvious question, but it's not something I remember reading or learning anywhere, but it seems reasonable so long as I preserve the inequality (i.e. make the big side bigger or the small side smaller)

Is this a reliable method of manipulation? Is this well taught in any of the books or on any website?

Yes, that's a perfectly good way to do such problems. More rigorously, you're simply using the fact that if a > b, and b > c, then a > c must be true. For example, using statement 1:

t > s

then we can surely add r to both sides:

r + t > r + s

and since

r + s > 2t

then it must be true that

r + t > 2t

Nothing wrong with that!

There are two other ways you could look at the problem. You could add inequalities here - for example, using statement 1:

t > s

Multiplying by -1 and reversing the inequality:

-s > -t

Now adding this to the given inequality r+s > 2t:

r + s > 2t
(+) -s > -t
r > t

Or you could relate the information given to concepts from statistics. If we know:

r + s > 2t

then

(r+s)/2 > t

That is, the average of r and s is greater than t. In any set, if the elements are different, at least one element must be above average. So if the average of r and s is greater than t, at least one of r or s must be greater than t. Each statement then guarantees that r must be greater than t.

[cjb]

Thanks Ian. Now I've got a few ways of tackling these other than my current method (trying to psych them out with a good long stare).