SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Functions, EXT1 F1 2022 HSC 11f

Solve  `(x)/(2-x) >= 5`.  (3 marks)

Show Answers Only

`5/3<=x<2`

Show Worked Solution

`(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)

Show Answers Only

`x < -2 or x > -1`

Show Worked Solution

`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 EQ-Bank 17

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

Show Answers Only

`x < 2\ ∪\ x > 4`

Show Worked Solution

`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)

Show Answers Only

`:. x < −1\ \ text(or)\ \ x > 1`

Show Worked Solution

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

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

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

 
`=> x > 1\ \text{(satisfies both)}`
 

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

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

 
`=> x < −1\ \text{(satisfies both)}`

`:. x < −1\ \ text(or)\ \ x > 1`

Functions, EXT1 F1 2016 HSC 11e

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

Show Answers Only

`x < -3,\ \-5/2 < x < 1/2`

Show Worked Solution

`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`

 

ext1-hsc-2016-11ei

`x < -3,\ \ \-5/2 < x < 1/2`

Functions, EXT1 F1 2007 HSC 3b

  1. 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)
  2. Hence, or otherwise, find the values of `x` for which
  3. `qquad qquad (x-2)/(x-4) ≤ 3`.   (2 marks)

Show Answers Only

a.    `text(See Worked Solutions.)`

b.    `x < 4\ text(and)\ x ≥ 5`

Show Worked Solution

a.    `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`

 

Geometry and Calculus, EXT1 2007 HSC 3b Answer

 

b.  `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\ \ text{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)

Show Answers Only

`x < −1\ \ text(or)\ \ x > 1/3`

Show Worked Solution

`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`
 

Algebra, EXT1 2004 HSC 1b Answer

`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)

Show Answers Only

`−3 < x ≤ 1, \ \ x ≠ −3`

Show Worked Solution

`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`

 

Real Functions, EXT1 2015 HSC 11c Answer

`:.−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)

Show Answers Only

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

Show Worked Solution
`(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)

Show Answers Only

`0 < x < 3`

Show Worked Solution

`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)

Show Answers Only

 `x < -2\ \ text(or)\ \ x > -5/4`

Show Worked Solution

`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`

Algebra, EXT1 2010 HSC 1d Answer

`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) 

Show Answers Only

 `x<3\ \ text(or)\ \ x>6`

Show Worked Solution

`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`

 

 Algebra, EXT1 2012 HSC 11c Answer

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

Functions, EXT1 F1 2011 HSC 1c

Solve  `(4-x)/x <1`.  (3 marks)

Show Answers Only

 `x<0\ \ text(or)\ \ x>2`

Show Worked Solution

`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`

 

 EXT1 2011 1c

`text(From graph)`

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