Question

Is it always, sometimes, or never true that \(|x|=|-x| ?\)

Short answer

The statement \(|x|\) = \(|-x|\) is always true.

Step-by-step solution

Define the Absolute Value Function

The absolute value of a real number x, denoted by |x|, is defined as: \(|x| = x, \text{ if } x\geq0 \); \(-x, \text{ if }\ x<0\)

Analyze Cases

For \(x\geq0\) we can rewrite \(|x|\) simply as \(x\). It follows also the absolute value of \(-x\): \(|-x|\) = \(-(-x)\) = \(x\). Therefore, \(|x|\) = \(|-x|\) for all \(x\geq0\). If \(x < 0\), \(|x| = -x\) and \(|-x| = -(-x) = x\). Therefore, \(|x|\) = \(|-x|\) also holds true for all \(x<0\)

Conclude

Therefore, \(|x|\) = \(|-x|\) is always true for all real number x.