topspin360 wrote:hi all,
quick question:
I understand that the statement - is m^3 > m^2? - is really asking if m is positive and greater than 1.
but when i try to solve algebraically, i end up with the following:
m^3 - m^2 > 0?
m^2(m-1) > 0?
m>0 or m<0 or m>1?
What's wrong with the derivation above? Why am I getting m>0 and m<0?
Thanks.
m²(m-1) > 0.
The CRITICAL POINTS are m=0 and m=1.
These are the only values where the lefthand side is EQUAL to 0.
To determine the range(s) where the lefthand side is GREATER than 0, test only value to the left and right of each critical point.
Plug m = -1 into m³ > m²:
(-1)³ > (-1)²
- 1 > 1.
Doesn't work.
Thus, m<0 is not a viable range here.
Plug m = 1/2 into m³ > m²:
(1/2)³ > (1/2)²
1/8 > 1/4.
Doesn't work.
Thus, 0<m<1 is not a viable range here.
Plug m = 2 into m³ > m²:
2³ > 2²
8 > 4.
This works.
Thus, m>1 is a viable range here.
Only one range satisfies m³ > m²:
m>1.
An easier approach:
m³ > m² implies that m does not equal 0.
m² cannot be negative, since the square of a value cannot be negative.
Thus, we can safely divide by m², without having to change the direction of the inequality:
m³/m² > m²/m²
m > 1.
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