Two people play a game with the following rules: one of them guesses a set of integers \((x_1, x_2, \dots , x_n)\) which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum \(a_1x_1 + \dots + a_nx_n\), where \((a_1, \dots ,a_n)\) is any set. What is the smallest number of questions for which the guesser recognizes the intended set?
Solve the equation: \(|x-2005| + |2005-x|=2006\).
Solve the inequality: \(|x + 2000| <|x - 2001|\).
For any real number \(x\), the absolute value of \(x\), written \(\left| x \right|\), is defined to be \(x\) if \(x>0\) and \(-x\) if \(x \leq 0\). What are \(\left| 3 \right|\), \(\left| -4.3 \right|\) and \(\left| 0 \right|\)?
Let \(x\) and \(y\) be real numbers. Prove that \(x \leq \left| x \right|\) and \(0 \leq \left| x \right|\). Then prove that the following inequality holds \(\left| x+y \right| \leq \left| x \right|+\left| y \right|\).