Question 206509
{{{sqrt(x)<x}}} Start with the given inequality.



To prove this false with a counter-example, we need to use a small number (that is less than 1). I'm going to use {{{1/4}}} (since {{{1/4<1}}})



{{{sqrt(1/4)<1/4}}} Plug in {{{x=1/4}}}



{{{sqrt(1)/sqrt(4)<1/4}}} Break up the root.



{{{1/sqrt(4)<1/4}}} Evaluate the square root of 1 to get 1.



{{{1/2<1/4}}} Evaluate the square root of 4 to get 2.



{{{1*4<1*2}}} Cross multiply (this will help us determine which side is larger)



{{{4<2}}} Multiply



Since the inequality is FALSE, this means that {{{1/2<1/4}}} is FALSE (it turns out that one-half is actually larger than a quarter...draw out a picture to verify yourself). So this means that {{{sqrt(1/4)<1/4}}} is also false.


So we've shown that the inequality {{{sqrt(x)<x}}} is false for all real numbers.


Note: it turns out that {{{sqrt(x)<x}}} is only true if {{{x>1}}}