Solve the following equations
`(i) sgn({[x]})=0` `(ii) sgn(x^(2)-2x-8)=-1` `(iii)sgn((x^(2)-5x+4)/({x}))=-1`

Answer

Let's solve the equations step by step.

### (i) Solve `sgn([x]) = 0`

1. **Understanding the Signum Function**: The signum function, `sgn(x)`, is defined as:
- `sgn(x) = 1` if `x > 0`
- `sgn(x) = -1` if `x < 0`
- `sgn(x) = 0` if `x = 0`

2. **Setting up the equation**: We need to find when `sgn([x]) = 0`. This means that `[x]` (the greatest integer function) must equal `0`.

3. **Finding the value of `[x]`**: The greatest integer function `[x]` equals `0` when `0 ≤ x < 1`.

4. **Conclusion**: Therefore, the solution for this equation is:
\[ x \in [0, 1) \]