Final answer:
The claim that there exists a number x where |x| is less than x is false, as the absolute value of a number is always non-negative and thus cannot be less than the number itself.
Step-by-step explanation:
The statement 'there exists x such that |x| < x' is false. In mathematics, the absolute value of a number x, denoted by |x|, is the non-negative value of x without regard to its sign. Therefore, for any real number x, |x| is always either greater than or equal to x, making the statement |x| < x impossible.
For positive numbers, |x| equals x, and for negative numbers, |x| equals -x which is greater than x (since x is negative in this case). Absolute value reflects the distance from zero on a number line, which is always a non-negative number. For x=0, |x|=x=0, but even there, |x| is not less than x.