MGMAT Flashcard question

[vak3e]

Dear all,

I'm confused about the following two questions that are along the same lines.

Q1

(p-1)p(p+1)<0

How do you simplify this? The difficulty is that either only p-1 is negative, or all three parts are negative.


Q2

lxl>lyl ?

How do you simplify this using algebra?

Thx

[RonPurewal]

Q1

(p-1)p(p+1)<0

How do you simplify this? The difficulty is that either only p-1 is negative, or all three parts are negative.

As in the case of other such inequalities, you should ...
... find the points where sign changes can occur,
... break the number line up according to those points,
... test the resulting intervals to see whether the statement works.

In this case, sign changes can happen only at -1, 0, and 1 (the three points where the expression equals zero).
These points divide the number line into four different intervals. Test those intervals:

* p < -1:
Here the statement is true; a product of three negatives is negative.

* -1 < p < 0:
False; it's a product of two negatives and a positive, and so is positive.

* 0 < p < 1:
True; it's a product of one negative and two positives, so it's negative overall.

* p > 1:
False; everything is positive, so the product is positive.

So, the solution is p < -1 or 0 < p < 1.

There's no way to "solve this algebraically" past the point of finding the zero points, since algebra is unable to contend with the changing sign of the product.

[RonPurewal]

Q2

lxl>lyl ?

How do you simplify this using algebra?

Well, you don't, really.

This kind of thing is best approached by just thinking about what absolute value means.
Absolute value is, essentially, the "size" of a number. It's how BIG the number is, disregarding the sign. So, this statement says...
... x is BIGGER than y, in terms of magnitude;
... we don't know the sign of either number.

So, the best simplification I can give you is this:
x = ±BIG
y = ±small (or zero)
Yes, this is not proper formal notation, but this is one of those times when formal notation is not intuitive.