OG 155 th problem

[email protected] wrote:Is X negative?

1)x^3(1-x^2)<0
2)x^2-1<0

Please help me with best strategy to solve this problem.


Peer

the final post on this thread is quite good.

i wanted to give a cautionary note, inspired by the second post:
DO NOT EXPAND FACTORED EXPRESSIONS UNLESS YOU HAVE A VERY, VERY GOOD REASON FOR DOING SO.
in general, factored expressions are extremely useful. it's a cinch to tell whether a factored expression is positive or negative; all you have to do is determine the sign of each of the factors, and then follow the rules for multiplication (pos x pos = neg x neg = pos; pos x neg = neg). if you have to tell whether an expanded polynomial is positive or negative, though, you face a formidable task of weighing the relative sizes of all the different terms in the polynomial. (unless you factor it, of course!)

in general, factored expressions are better than expanded expressions for just about anything you'd ever want to do.

also, note that, if you want, you could go in the opposite direction: you could factor these expressions further before tackling the problem.

for instance, you could factor statement (2) into
(x - 1)(x + 1) < 0
from which you'd deduce that x + 1 is positive and x - 1 is negative (because that's the only way the product can be negative). this translates to -1 < x < 1.
INSUFFICIENT, because both positive and negative numbers are contained in this interval.

same sort of logic applies to statement (1) as well:
(x^3)(1 - x)(1 + x) < 0
set out a number line showing the 'solutions' of this inequality (i.e., the numbers that would be its solutions if it were an equation instead of an inequality): -1, 0, and 1. then, test the four resulting regions to see if they work in the inequality:
x < -1: doesn't work
-1 < x < 0: works
0 < x < 1: doesn't work
1 < x: works
INSUFFICIENT, because x can be negative (-1 < x < 0) or positive (1 < x).

if you take the two statements together, the only interval that satisfies both is -1 < x < 0
SUFFICIENT

answer = c