VeritasKarishma wrote:
mgan wrote:
(1) Could we simplify: (z+1)(z)(z-1)<0
=> (z^2-1)(z)<0
=> z^3-z<0
=> z^3<z
this way we know that z could be either positive fraction or a negative number more than 1?
(2) |z|<1
z could be positive and negative fractions.
(1) and (2)
from (1) could be positive fraction or a negative number more than 1
from (2) could be positive or negative fractions.
The only thing that overlaps is positive fraction.
Therefore, the answer is C
Could this way of solving work?
Yes you can provided you know and understand the relations between a number and its squares, cubes, square roots, absolute values etc.
How do you arrive at "either a positive fraction or a negative number less than -1"? If you understand these number properties well, it's great!
When I tested cases I tired to cover all numbers:
In (1) we have z^3<z relationship, and we can test 5 cases:
1. When z>1. Any number >1 cubed is larger than itself (example: 2^3=8 => 8>2). Conclusion => z is not >1.
2.
Positive fractions (when 0<z<1). Any positive fraction cubed is smaller than itself (example: 1/2^3=1/8 => 1/8<1/2). Conclusion => z could be a positive fraction.
3.
When z<-1 Any number <-1 cubed is smaller than itself, it does not loose the minus sign (example: -2^3=-8 => -8<-2). Conclusion => z could be a number < -1.
4. Negative fraction (-1<z<0). Any negative fraction cubed is larger than itself, it is closer to 0 and positive numbers (example -1/2^3=-1/8 => -1/8>-1/2). Conclusion => z is not a negative fraction.
5. Zero or 1. 0^3=0, so 0^3 is not less than 0. 1^3= 1, so 1^3 is not less than 1. Conclusion z is not 0 or 1.
In (2) we take an absolute value of z. This means that any value of z becomes positive. Therefore, only fractions work.
From (1) z could be a number <-1 or a positive fraction. From (2) z could be either a negative or positive fraction. The only thing that overlaps is positive fraction.
=> z must be a positive fraction.