How do you solve x/(x-2)>=0?

2 Answers
Mar 21, 2018

The solution is x in (-oo, 0] uu(2, +oo)

Explanation:

Let f(x)=x/(x-2)

Build a sign chart

color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaaaaa)0color(white)(aaaaaaaa)2color(white)(aaaaaa)+oo

color(white)(aaaa)xcolor(white)(aaaaaaaa)-color(white)(aaaa)0color(white)(aaaa)+color(white)(aaaaa)+

color(white)(aaaa)x-2color(white)(aaaaa)-color(white)(aaaa)#color(white)(aaaaa)-#color(white)(aa)||color(white)(aa)+

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

Therefore,

f(x)>=0 when

graph{x/(x-2) [-10, 10, -5, 5]}

Mar 21, 2018

(-oo, 0] U (2, +oo)

Explanation:

x /(x - 2)≥0

x /(x - 2)≥0" : is true if" {("either", x ≥0 and x - 2 > 0),("or",x ≤ 0 and x - 2 < 0):}

x ≥0 and x - 2 > 0
x > 2

x ≤ 0 and x - 2 < 0
x ≤ 0

Answer: x ≤ 0 OR x > 2
In interval notation: (-oo, 0] U (2, +oo)