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 given 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) = 0` if `x = 0`
- `sgn(x) = 1` if `x > 0`

2. **Applying to the Equation**: We need to find when `sgn([x]) = 0`. This means that `[x]` (the greatest integer less than or equal to `x`) must equal `0`.

3. **Finding the Solution**: The greatest integer function `[x] = 0` when `0 ≤ x < 1`. Therefore, the solution for this equation is:
\[ x \in [0, 1) \]

### (ii) Solve `sgn(x^2 - 2x - 8) = -1`

1. **Setting Up the Inequality**: For `sgn(x^2 - 2x - 8) = -1`, we need:
\[ x^2 - 2x - 8 < 0 \]

2. **Factoring the Quadratic**: We can factor the quadratic:
\[ x^2 - 2x - 8 = (x - 4)(x + 2) \]

3. **Finding the Roots**: The roots of the equation are `x = 4` and `x = -2`.

4. **Testing Intervals**: We test the intervals determined by the roots:
- For `x < -2`: Choose `x = -3` → `(-3 - 4)(-3 + 2) = (-7)(-1) = 7` (positive)
- For `-2 < x < 4`: Choose `x = 0` → `(0 - 4)(0 + 2) = (-4)(2) = -8` (negative)
- For `x > 4`: Choose `x = 5` → `(5 - 4)(5 + 2) = (1)(7) = 7` (positive)

5. **Conclusion**: The solution to the inequality is:
\[ x \in (-2, 4) \]

### (iii) Solve `sgn((x^2 - 5x + 4)/[x]) = -1`

1. **Setting Up the Inequality**: For `sgn((x^2 - 5x + 4)/[x]) = -1`, we need:
\[ \frac{x^2 - 5x + 4}{[x]} < 0 \]

2. **Factoring the Numerator**: The numerator can be factored as:
\[ x^2 - 5x + 4 = (x - 4)(x - 1) \]

3. **Finding the Roots**: The roots of the numerator are `x = 4` and `x = 1`.

4. **Considering the Denominator**: The denominator `[x]` must be positive for the fraction to be negative. This occurs when `x > 1`.

5. **Testing Intervals**: We analyze the intervals:
- For `1 < x < 4`: Choose `x = 2` → `(2 - 4)(2 - 1) = (-2)(1) = -2` (negative)
- For `x > 4`: Choose `x = 5` → `(5 - 4)(5 - 1) = (1)(4) = 4` (positive)

6. **Conclusion**: The solution to the inequality is:
\[ x \in (1, 4) \]

### Final Solutions

1. **(i)**: \( x \in [0, 1) \)
2. **(ii)**: \( x \in (-2, 4) \)
3. **(iii)**: \( x \in (1, 4) \)

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