ironsferri wrote:
Thank you Thephoenix. I know x^2 can hide the sign, but when I see this form (x(x-1)>0) I don't see it. I think it's more a basic math issue at this point, but still want to understand it. why in x(x-1)=0 I have x=0,1 and in x(x-1)>0 is x>0 and x<0 ? Would you mind elaborate on how to reach that result?
Thank you
Hello ironsferri,
You're right to rewrite x^2 > x as (x)(x-1)>0. For two values to have a positive product, they must have the same sign, so either:
1) they're both positive: x >0 and
x-1 >0 --> x>1
In this case, x>0 and x>1 can just be written as
x>1 (because this is where both inequalities are true)
2) they're both negative: x<0 and
x-1>0 --> x>1
Because in this case x is negative, we can ignore the 2nd inequality (x cannot be bigger than 1) and write this case as x<0
Thus when you include both cases in a single statement, you get:
x>1 or x<0. That's the proof for my rephrase above.
The way I rephrased however was to simply think about which values of x would make "is x^2>x?" true.
- I know that for any negative x, x^2 will always be greater since it will be positive.
- I also know that for "regular" numbers (2, 3, 4, 5...), x^2 is greater.
- However because positive fractions get smaller when you square them, "is x^2>x?" would not be true for positive fractions.
- Finally I know that for x=1 or 0, x^2 would not be greater than x
In short the only way to make the answer be a NO is to make x a positive fraction. So either of the rephrases below would work:
"Is x<0 or x>1?" OR
"Is 0<= x <= 1?"
Good luck,
-Patrick
