Thread: Quick Proof on real numbers

For all positive real numbers x, (xˆ2)-x≥0.

I don't think this is true since for instance x can equal 2/3 and (2/3)ˆ2-(2/3)=-2/9 which is not greater than or equal to 0. Am i wrong?

Originally Posted by Aquameatwad

For all positive real numbers x, (xˆ2)-x≥0.

I don't think this is true since for instance x can equal 2/3 and (2/3)ˆ2-(2/3)=-2/9 which is not greater than or equal to 0. Am i wrong?

Hi Aquameatwad,

You are correct. When x is less than one the statement is wrong. However,

is a correct statement.

I see, so is the original proof "For all positive real numbers x, (xˆ2)-x≥0. " is only correct if you set the parameter x≥1.
So the original statement is false since 2/3 is a positive real number, and (2/3)ˆ2-(2/3)=-2/9 which is not greater than or equal to 0. Correct? Just want to make sure whats in my head is correct.

Originally Posted by Aquameatwad

I see, so is the original proof "For all positive real numbers x, (xˆ2)-x≥0. " is only correct if you set the parameter x≥1.
So the original statement is false since 2/3 is a positive real number, and (2/3)ˆ2-(2/3)=-2/9 which is not greater than or equal to 0. Correct? Just want to make sure whats in my head is correct.

Yes. Correct.

Originally Posted by Aquameatwad

For all positive real numbers x, (xˆ2)-x≥0.

I don't think this is true since for instance x can equal 2/3 and (2/3)ˆ2-(2/3)=-2/9 which is not greater than or equal to 0. Am i wrong?

If you want to prove this directly, you could complete the square...



So the inequality is only true when or .

S0mewhat simpler, I think:

if and only if both factors on the left have the same sigh. If , x> 0 and , so that is true. If , x< 1 so that is true. But if , x is positive while x- 1 is negative so it is not true.