How do you solve $\frac{x}{x - 2} \geq 0$ $x/(x-2)>=0$ ?

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How do you solve $\frac{x}{x - 2} \geq 0$ $x/(x-2)>=0$ ?

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Answer 1 · 2018-03-21T11:00:02

The solution is $x \in (- \infty, 0] \cup (2, + \infty)$ $x in (-oo, 0] uu(2, +oo)$

Explanation:

Let $f (x) = \frac{x}{x - 2}$ $f(x)=x/(x-2)$

Build a sign chart

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a a a a$ $color(white)(aaaaaaa)$ $0$ $0$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $2$ $2$ $a a a a a a$ $color(white)(aaaaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $0$ $0$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$

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

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

Therefore,

$f (x) \geq 0$ $f(x)>=0$ when

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

Answer 2 · 2018-03-21T11:07:14

$(- \infty, 0]$ $(-oo, 0]$ U $(2, + \infty)$ $(2, +oo)$

Explanation:

$\frac{x}{x - 2} \geq 0$ $x /(x - 2)≥0$

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

$x ≥0 and x - 2 > 0$
$x > 2$

$x ≤ 0 and x - 2 < 0$
$x ≤ 0$

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

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