Question

How do you solve `\frac { x - 8} { x - 4} > 0`?

Answer 1

Here manuplate th numerator to relate with denominator,
`(x-4-4)/(x-8)` ,
now seprating numerator
`(x-4)/(x-4)-(4)/(x-4)>0`
thus,
`1-4/(x-4)>0`
hence
`1>4/(x-4)`

`(x-4)>4`

`x>8`
to satisfy the above question

Answer 2

`(-oo,4)uu(8,+oo)`

Explanation:

`"the zeros of the numerator/denominator are"`

`"numerator "x=8," denominator "x=4`

`"these indicate where the rational function may change"`
`"sign"`

`"the intervals on the domain are"`

`x<4,color(white)(x)4 < x <8,color(white)(x)x>8`

`"consider a "color(blue)"test point " "in each interval"`

`"we want to find where the function is positive " >0`

`"substitute each test point into the function and consider"`
`"its sign"`

`color(magenta)"x = 3"to(-)/(-)tocolor(red)" positive"`

`color(magenta)"x=5"to(-)/(+)tocolor(blue)" negative"`

`color(magenta)"x = 10"to(+)/(+)tocolor(red)" positive"`

`rArr(-oo,4)uu(8,+oo)" is the solution"`
graph{(x-8)/(x-4) [-10, 10, -5, 5]}