Solve the inequality `3 - x > 1/|x - 4|` for `x`, expressing your answer in interval notation. (3 marks)
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`3 – x > 1/|x – 4|`
`|x – 4| (3 – x) > 1`
`text(If)\ \ x – 4 > 0, x > 4`
| `(x – 4) (3 – x)` | `> 1` |
| `3x – x^2 – 12 + 4x` | `> 1` |
| `-x^2 + 7x – 13` | `> 0` |
`Delta = 7^2 – 4 ⋅ 1 ⋅ 13 = -3 < 0`
`=>\ text(No Solutions)`
`text(If)\ \ x – 4 < 0, x < 4`
| `-(x – 4) (3 – x)` | `> 1` |
| `x^2 – 7x + 12` | `> 1` |
| `x^2 – 7x + 11` | `> 0` |
| `x` | `= (7 +- sqrt(7^2 – 4 ⋅ 1 ⋅ 11))/2` |
| `= (7 +- sqrt 5)/2` |
`text(Combining solutions)`
`(x < (7 – sqrt 5)/2 ∪ x > (7 + sqrt 5)/2) nn x < 4`
`x ∈ (– oo, (7 – sqrt 5)/2)`