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 \(\left| x+y \right| \leq \left| x \right|+\left| y \right|\).
Prove that \(|x|\ge x\). It may be helpful to compare each of \(|3|\), \(|-4.3|\) and \(|0|\) with \(3\), \(-4.3\) and \(0\) respectively.