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)`