Question

Explain why the equation |x| = -6 has no solutions

Answer 1

The range of
`f(x) = |x|` is
`\left\{f(x) \in \Bbb R ~:~ f(x) \ge 0\right\}`. In other words,
`|x|` is non-negative, so
`|x|=-n` for any positive integer
`n` has no solution.

Just look at the definition of absolute value:

`|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}`

If
`x\ge0`, then
`|x|` doesn't change its value, since
`x` is already non-negative.

But if
`x<0`, then
`|x|` negates the negative and returns a positive number
`-x`.