How do you solve and write the following in interval notation: (x+1) / (x-1) <=2?

2 Answers

the interval is [3,oo)

Explanation:

the given inequlity is (x+1)/(x-1)<=2 rArr(x+1)<=2(x-1)rArr(x+1)<=2x-2 rArr(x-2x+1+2)<=0rArr(-x+3)<=0 rArr(x-3)>=0rArrx>=3
:.x in [3,oo)

Nov 1, 2017

The solution is x in (-oo,1) uu [3,+oo)

Explanation:

We cannot do crossing over

Let's rewrite the expression

(x+1)/(x-1)<=2

(x+1)/(x-1)-2<=0

((x+1)-2(x-1))/(x-1)<=0

((x+1-2x+2))/(x-1)<=0

((3-x))/(x-1)<=0

Let f(x)=((3-x))/(x-1)

Let's build a sign chart

color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaaaaa)1color(white)(aaaaaaa)3color(white)(aaaaaaa)+oo

color(white)(aaaa)x-1color(white)(aaaaa)-color(white)(aaaa)||color(white)(aa)+color(white)(aaaaaaa)+

color(white)(aaaa)3-xcolor(white)(aaaaa)+color(white)(aaaa)||color(white)(aa)+color(white)(aa)0color(white)(aaaa)-

color(white)(aaaa)f(x)color(white)(aaaaaa)-color(white)(aaaa)||color(white)(aa)+color(white)(aa)0color(white)(aaaa)-

Therefore,

f(x)>=0 when x in (-oo,1) uu [3,+oo)

graph{(x+1)/(x-1)-2 [-12.66, 12.65, -6.33, 6.33]}