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Functions, EXT1 F1 EQ-Bank 24

Solve the inequality  `x^2<=|\ 2x-1\ |`  for `x`.

Express your answer in interval notation.   (3 marks)

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`:. x in [-1-sqrt2, -1 + sqrt2]\ \ ∪\ \ x in [1]`

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`text(Case 1:)`

`x^2` `<=2x-1`
`x^2-2x+1` `<=0`
`(x-1)^2` `<=0`

 
`x=1`
 

`text(Case 2:)`

`x^2` `<=-(2x-1)`
`x^2+2x-1` `<=0`

 

`x` `=(-2+-sqrt(4+4*1*1))/2`  
  `=-1+-sqrt2`  

 
`text(Test)\ \ x=0\ \ =>\ \ 0<=-(-1)\ \ text{(correct)}`

`:. x in [-1-sqrt2, -1 + sqrt2]\ \ ∪\ \ x in [1]`

Functions, EXT1 F1 2020 SPEC1 4

Solve the inequality  `3-x > 1/|x-4|`  for `x`, expressing your answer in interval notation.   (3 marks)

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`x ∈ (-oo, (7-sqrt 5)/2)`

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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)`