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Functions, EXT1 F1 SM-Bank 17

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)`