algebra question

Question

Is the following statement always true, sometimes true, or never true if we assume x is a real

number greater than 0: “The value of x^2 is at least as large as the value of x.” Show that your

answer is correct with mathematical evidence.

Answer 1

Question: Is the following statement always true, sometimes true, or never true?

If we assume x is a real number greater than 0: “The value of x² is at least as large as the value of x.” Show that your answer is correct with mathematical evidence.

Answer: Let's play with a few numbers. Make sure they're not zero.

-If you pick x to be 0.1, which is a decimal between 0 and 1, then 0.1 * 0.1 is 0.01, which is smaller than 0.1. (So much for "always true." Liars, every last one of 'em!)

(A note on fractions: You can also call 0.1 by its fraction form, which is 1/10. A proper fraction where the numerator is smaller than the denominator will always be equal to a decimal number less than 1. An improper fraction will always be equivalent to a decimal number bigger than 1. And if numerator and denominator are the same (as long as they aren't both 0), then that is equal to 1.)

-If you pick x to be 1, then 1² is 1. This is equal to 1. (Here's the "at least as large" of the original statement.)

-If you pick x to be 2, then 2² is 4. This is greater than 2. (Definitely bigger than your choice of x.)

I picked numbers that were easy to work with, so you could technically pick any numbers you want in order to test the statement. What you should pick as test values, however, should be a smart mixture of numbers greater than zero but less than 1, and numbers bigger than 1. And as you saw here, if you pick x=1, then 1² is exactly 1, so this is a special number to throw into the mix.

The original statement is sometimes true.

Answer 2

Gin

Let's assume that x² >= x

Since we are told x is positive , divide both sides by x

x >= 1

These two statements are equivalent

So x² >= x when x>= 1

When 0 <= x < 1, what happens ? Investigate.

Then write down a conclusion

0<=x < 1, x^2 >= x ??

x>=1. x^2 >= x ??

There's your answer

Mike