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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
Break up the root.
Evaluate the square root of 1 to get 1.
Evaluate the square root of 4 to get 2.
Cross multiply (this will help us determine which side is larger)
Since the inequality is FALSE, this means that
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
is also false.
So we've shown that the inequality
is false for all real numbers.
Note: it turns out that
is only true if