Question

How do you solve `(x-6)/(x+1)>0`?

Answer 1

`(-oo,-1)uu(6,+oo)`

Explanation:

`"the zero's of the numerator/denominator are"`

`"numerator"color(white)(x)x=6, "denominator"color(white)(x)x=-1`

These indicate where the rational function may change sign and which value x cannot be on the denominator.

`"the intervals for consideration are"`

`x < -1, -1 < x < 6, x>6`

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

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

`"substitute each test point into the function and consider it's sign"`

`color(red)(x=-2)to(-)/(-)tocolor(red)" positive"`

`color(red)(x=2)to(-)/(+)tocolor(blue)" negative"`

`color(red)(x=8)to(+)/(+)tocolor(red)" positive"`

`rArr(-oo,-1)uu(6,+oo)`
graph{(x-6)/(x+1) [-10, 10, -5, 5]}