How do you solve (x-6)/(x+1)>0x−6x+1>0?
1 Answer
Explanation:
"the zero's of the numerator/denominator are"the zero's of the numerator/denominator are
"numerator"color(white)(x)x=6, "denominator"color(white)(x)x=-1numeratorxx=6,denominatorxx=−1 These indicate where the rational function may change sign and which value x cannot be on the denominator.
"the intervals for consideration are"the intervals for consideration are
x < -1, -1 < x < 6, x>6x<−1,−1<x<6,x>6
"Consider a "color(blue)"test point"" in each interval"Consider a test point in each interval
"we want to find where the function is positive, that is ">0we want to find where the function is positive, that is >0
"substitute each test point into the function and consider it's sign"substitute each test point into the function and consider it's sign
color(red)(x=-2)to(-)/(-)tocolor(red)" positive"x=−2→−−→ positive
color(red)(x=2)to(-)/(+)tocolor(blue)" negative"x=2→−+→ negative
color(red)(x=8)to(+)/(+)tocolor(red)" positive"x=8→++→ positive
rArr(-oo,-1)uu(6,+oo)⇒(−∞,−1)∪(6,+∞)
graph{(x-6)/(x+1) [-10, 10, -5, 5]}
