vards
Hi there,
my doubt may sound a bit off,but here it is..
when we reach at this stage= (a+b)(a-b) / (a-b)^2 < 0 ,
then after deducing,we are left with-
a+b/a-b < 0
finally,
a+b<0 = a< -b ...from here,we can say that in any case, B should be greater than A.. please clear my understanding..
Dear
Vards,
I'm happy to respond here.
The first thing I'll say, my friend, is that it's very important to distinguish the broader category of algebraic/mathematical steps that are
allowed to do, that are mathematically legal to perform, with the much smaller category of those legal steps that are actually
strategically beneficial in solving the problem.
When we get to the stage:
\(\frac{(a+b)(a-b)}{(a-b)^2} < 0\)
You choose to cancel a factor of
(a- b) in the numerator and denominator. That's definitely a legal move to do, but it's doubtful whether that is a particularly strategic thing to do. In fact, in terms of strategy, in terms of getting us closer to the answer, I think it's a not very good move.
By the way, canceling is not called "
deducing," at least in American mathematical terms.
You see, that act of canceling leaves us with:
\(\frac{a+b}{a-b} < 0\)
But the trouble is, with just statement #1 ---for the expressions
a, b, (a + b), and
(a - b): we don't know whether any of them are positive or negative or zero. That means if we multiply both sides by the denominator
(a - b), we don't know whether we are multiplying by a positive or a negative number, so we don't know whether we
reverse the inequality sign. Technically, at that stage, we would have to create two different scenarios, one in which we specified that (a - b) > 0, and the other in which we specified that (a - b) < 0, and the follow out the consequences of each case. We could do all that, but that's a very long, very complicated way to get to answer. That's precisely why that initial canceling move was a not-very-strategic thing to do.
What I did was considerably more elegant in terms of strategy. Noting that the original denominator,
(a - b)^2, is a square and therefore always positive, we can just multiply both sides by that, we are left with the fact that
a^2 - b^2 < 0. That immediate leads to the important statement
a^2 < b^2, which is very helpful in answering the question.
Don't assume, because you
can do something, because a move is allowed by the laws of algebra, that just because of this, the move is a particularly useful, a particularly strategic move. There are always dozens of moves that are perfectly legal and disastrous in terms of strategy.
Does all this make sense?
Mike