smc-1033-10-Algebraic Fractions

Functions, EXT1 F1 2022 HSC 11f

Solve `(x)/(2-x) >= 5`. (3 marks)

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`5/3<=x<2`

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`(x)/(2-x) >= 5`

`text{Multiply b.s. by}\ \ (2-x)^2\ \ (>0):`

`x(2-x)`	`>=5(2-x)^2`
`2x-x^2`	`>=5(x^2-4x+4)`

`6x^2-22x+20`	`<=0`
`2(3x^2-11x+10)`	`<=0`
`(3x-5)(x-2)`	`<=0`

`text{Test}\ \ x=0:`

`(-5)(-2)=10>0`

`:. 5/3<=x<2\ \ (x!=2)`

Functions, EXT1 F1 2019 HSC 11b

For what values of `x` is `x/(x + 1) < 2`? (3 marks)

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`x < -2 or x > -1`

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`x/(x + 1) < 2`

`text(Multiply b.s. by)\ \ (x + 1)^2`

`x(x + 1)`	`< 2(x + 1)^2`
`x^2 + x`	`< 2x^2 + 4x + 2`
`0`	`< x^2 + 3x + 2`
`0`	`< (x + 2)(x + 1)`

`text(From graph:)`

`x < -2 or x > -1`

Functions, EXT1 F1 SM-Bank 2

Solve `3/(|\ x - 3\ |) < 3`. (3 marks)

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`x < 2\ ∪\ x > 4`

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`text(Solution 1)`

`3/(|\ x – 3\ |) < 3`

`\|\ x – 3\ \|`	`> 1`
`(x^2 – 6x + 9)`	`> 1^2`
`x^2 – 6x + 8`	`> 0`
`(x – 4)(x – 2)`	`> 0`

`:. {x: \ x < 2\ ∪\ x > 4}`

`text(Solution 2)`

`|\ x – 3\ | > 1`

`text(If)\ \ (x – 3)`	`> 0,\ text(i.e.)\ x >3`
`x – 3`	`> 1`
`x`	`> 4`

`=> x > 4\ (text(satisfies both))`

`text(If)\ \ (x – 3)`	`< 0,\ text(i.e.)\ x <3`
`−(x – 3)`	`> 1`
`−x + 3`	`> 1`
`x`	`< 2`

`=> x < 2\ (text(satisfies both))`

`:. {x: \ x < 2\ ∪\ x > 4}`

Functions, EXT1 F1 2017 HSC 11c

Solve `(2x)/(x + 1) > 1`. (3 marks)

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`:. x < −1\ \ text(or)\ \ x > 1`

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`(2x)/(x + 1) > 1`

`text(If)\ \ x + 1 > 0,\ \ text(i.e.)\ \ x > −1`

`2x`	`> x + 1`
`x`	`> 1`

`=> x > 1`

`text(If)\ \ x + 1 < 0,\ \ text(i.e.)\ \ x < −1`

`2x`	`< x + 1`
`x`	`< 1`

`=> x < −1`

`:. x < −1\ \ text(or)\ \ x > 1`

Functions, EXT1 F1 2016 HSC 11e

Solve `3/(2x + 5) - x > 0`. (3 marks)

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`x < -3,\ \ – 5/2 < x < 1/2`

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`3/(2x + 5) – x > 0`

`3 (2x + 5) > x (2x + 5)^2,\ \ x != – 5/2`

`3 (2x + 5)-x(2x+5)^2`	`> 0`
`(2x + 5) [3- x(2x + 5)]`	`> 0`
`(2x + 5) (-2x^2 – 5x + 3)`	`> 0`
`(2x + 5) (1-2x) (x + 3)`	`> 0`

`x < -3,\ \ \ – 5/2 < x < 1/2`

Functions, EXT1 F1 2007 HSC 3b

Find the vertical and horizontal asymptotes of the hyperbola `y = (x − 2)/(x − 4)`

and hence sketch the graph of `y = (x − 2)/(x − 4)`. (3 marks)

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Hence, or otherwise, find the values of `x` for which

`qquad qquad (x − 2)/(x − 4) ≤ 3`. (2 marks)

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`text(See Worked Solutions.)`
`x < 4\ text(and)\ x ≥ 5`

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i. `y = (x − 2)/(x − 4)`

`text(Vertical asymptote at)\ x = 4`

`lim_(x → ∞) (x − 2)/(x − 4)`

`= lim_(x → ∞) (1 − 2/x)/(1 − 4/x)=1`

`ytext(–intercept)\ = 1/2`

`xtext(–intercept)\ = 2`

ii. `text(Find)\ \ x\ \ text(so that)\ \ (x − 2)/(x − 4) ≤ 3`

`text(When)\ \ (x − 2)/(x − 4)`	`= 3`
`x − 2`	`= 3x − 12`
`2x`	`= 10`
`x`	`= 5`

`=>(5, 3)\ \ text(is the intersection of)\ \ y = 3\ and\ y = (x − 2)/(x − 4)`

`:. (x − 2)/(x − 4) ≤ 3\ \ text(when)\ \ x < 4\ \ text(and)\ \ x ≥ 5.`

Functions, EXT1 F1 2004 HSC 1b

Solve `4/(x + 1) < 3.` (3 marks)

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`x < −1\ \ text(or)\ \ x > 1/3`

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`4/(x + 1) < 3`

`text(Multiply both sides by)\ (x + 1)^2`

`4(x + 1)`	`< 3(x + 1)^2`
`4x + 4`	`< 3(x^2 + 2x + 1)`
`4x + 4`	`< 3x^2 + 6x + 3`

`3x^2 + 2x − 1`	`> 0`
`(3x − 1)(x + 1)`	`> 0`

`text(LHS) = 0\ \ text(when)\ \ x = 1/3\ \ text(or)\ \ −1`

`text(From the graph,)`

`x < −1\ text(or)\ x > 1/3`

Functions, EXT1 F1 2015 HSC 11c

Solve the inequality `4/(x + 3) ≥ 1`. (3 marks)

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`−3 < x ≤ 1, \ \ x ≠ −3`

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`text(Solution 1)`

`4/(x + 3) ≥ 1`

`4(x + 3)`	`≥ (x + 3)^2`
`4x + 12`	`≥ x^2 + 6x + 9`
`x^2 + 2x − 3`	`≤ 0`
`(x + 3)(x − 1)`	`≤ 0`

`:.−3 < x ≤ 1, \ \ x ≠ −3`

`text(Solution 2)`

`text(If)\ (x + 3)`	`> 0`
`x`	`> −3`

`4/(x + 3)`	`≥ 1`
`4`	`≥ x + 3`
`x`	`≤ 1`

`:. −3 < x ≤ 1`

`text(If)\ (x + 3)`	`< 0`
`x`	`< −3`

`4/(x + 3)`	`≥ 1`
`4`	`≤ x + 3`
`x`	`≥ 1`
`:.\ text(No solution.)`

`:. −3 < x ≤ 1`

Functions, EXT1 F1 2014 HSC 11e

Solve `(x^2 + 5)/x > 6`. (3 marks)

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`0<x<1\ \ text(or)\ \ x>5`

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`(x^2 + 5)/x`	`> 6`
`x^2 ((x^2 + 5)/x)`	`> 6x^2`
`x^3 + 5x`	`> 6x^2`
`x^3 – 6x^2 + 5x`	`> 0`
`x (x^2 – 6x + 5)`	`> 0`
`x (x – 5)(x – 1)`	`> 0`

`:.\ 0 < x < 1\ \ text(or)\ \ x > 5`

Functions, EXT1 F1 2009 HSC 1d

Solve the inequality `(x + 3)/(2x) > 1`. (3 marks)

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`0 < x < 3`

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`text(Solution 1)`

`(2x)^2 xx (x + 3)/(2x)`	`> (2x)^2`
`2x (x + 3)`	`> 4x^2`
`2x^2 + 6x`	`> 4x^2`
`2x^2\ – 6x`	`< 0`
`2x (x\ – 3)`	`< 0`

`:.\ {x: 0 < x < 3}`

`text(Solution 2)`

`text(If)\ \ x > 0,`

`x + 3`	`> 2x`
`x`	`< 3`

`:.\ 0 < x < 3`

`text(If)\ \ x < 0,`

`x + 3`	`< 2x`
`x`	`> 3\ \ \ =>\ text(No solution)`

`:.\ {x: 0 < x < 3}`

Functions, EXT1 F1 2010 HSC 1d

Solve `3/(x+2) < 4`. (3 marks)

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`x < -2\ \ text(or)\ \ x > -5/4`

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`text(Solution 1)`

`3/(x + 2) < 4`

`text(Multiply b.s. by)\ \ (x + 2)^2`

`3(x + 2)`	`< 4(x + 2)^2`
`3x + 6`	`< 4 (x^2 + 4x + 4)`
`3x + 6`	`< 4x^2 + 16x + 16`

`4x^2 + 13x + 10`	`> 0`
`(4x + 5)(x + 2)`	`> 0`

`text(LHS)\ = 0\ \ text(when)\ \ x = -5/4\ \ text(or)\ \ -2`

`text(From graph)`

`x < -2\ \ text(or)\ \ x > -5/4`

`text(Alternate Solution)`

`text(If)\ \ x + 2 > 0\ \ \ \ text{(i.e.}\ \ x > –2 text{)}`

`3`	`< 4(x + 2)`
`3`	`< 4x + 8`
`4x`	`> -5`
`x`	`> -5/4`

`text(If)\ \ x + 2 < 0\ \ \ \ text{(i.e.}\ x < –2 text{)}`

`3`	`> 4 (x + 2)`
`3`	`> 4x + 8`
`4x`	`< -5`
`x`	`< -5/4`
`:. x`	`< –2\ \ \ \ text{(satisfies both)}`

`:.\ x < –2\ \ text(or)\ \ x > –5/4`

Functions, EXT1 F1 2012 HSC 11c

Solve `x/(x - 3) < 2`. (3 marks)

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`x<3\ \ text(or)\ \ x>6`

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`text(Solution 1)`

`x/(x – 3) < 2`

`text(When)\ \ x – 3 > 0,\ \ \ (text(i.e.)\ \ x > 3)`

`x`	`< 2 ( x – 3)`
`x`	`< 2x\ – 6`
`=>x`	`>6\ \ \ text{(satisfies both)}`

`text(When)\ \ x\ – 3 < 0,\ \ \ (text(i.e.)\ \ x<3)`

`x`	`>2 (x – 3)`
`x`	`> 2x – 6`
`x`	`<6`
`=> x`	`<3\ \ \ text{(satisfies both)}`

`:.\ text(Equation is correct for)\ x<3\ text(or)\ x>6`

`text(Solution 2)`

`x/((x\ – 3)) < 2`

`text(Multiply b.s. by)\ \ (x-3)^2`

`x(x – 3)`	`< 2(x^2 – 6x + 9)`
`x^2 – 3x`	`< 2x^2 – 12x + 18`
`x^2 – 9x + 18`	`>0`
`(x – 3)(x – 6)`	`>0`

`:.\ text(Equation correct for)\ x<3\ \ text(or)\ \ x>6`

Functions, EXT1 F1 2011 HSC 1c

Solve `(4-x)/x <1`. (3 marks)

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`x<0\ \ text(or)\ \ x>2`

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`text(Solution 1)`

`(4-x)/x < 1`

`text(If)\ x<0,\ \ \ \ \ 4-x`	`> x`
`2x`	`< 4`
`x`	`<2`

`=> x<0\ \ \ text{(satisfies both)}`

`text(If)\ x>0,\ \ \ \ \ 4-x`	`<x`
`2x`	`>4`
`x`	`>2`

`=> x>2\ \ \ text{(satisfies both)}`

`:.\ x < 0\ \ text(or)\ \ x > 2`

`text(Solution 2)`

`text(Multiply both sides by)\ \ x^2`

`x(4-x)`	`< x^2`
`4x-x^2`	`< x^2`
`2x^2-4x`	`>0`
`2x(x-2)`	`>0`

`text(From graph)`

`x<0\ \ text(or)\ \ x >2`