How do you solve $(2x)/(x-5)>=3$ ?

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How do you solve $(2x)/(x-5)>=3$ ?

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Answer 1 · 2018-08-04T14:02:40

The solution is $x in (5,15]$

Explanation:

You cannot do crossing over in inequalities

$(2x)/(x-5)>=3$

$<=>$ , $(2x)/(x-5)-3>=0$

$<=>$ , $(2x-3(x-5))/(x-5)>=0$

$<=>$ , $(2x-3x+15)/(x-5)>=0$

$<=>$ , $(-x+15)/(x-5)>=0$

Let $f(x)=(-x+15)/(x-5)$

Build a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaaa)$ $5$ $color(white)(aaaaaaa)$ $15$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x-5$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $||$ $color(white)(aa)$ $+$ $color(white)(aaaaaa)$ $+$

$color(white)(aaaa)$ $-x+15$ $color(white)(aaa)$ $+$ $color(white)(aaaaa)$ #color(white)(aa)+# $color(white)(aaa)$ $0$ $color(white)(aa)$ $-$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $0$ $color(white)(aa)$ $-$

Therefore,

$f(x)>=0$ when $x in (5,15]$

graph{(2x)/(x-5)-3 [-25.56, 32.17, -14.65, 14.22]}

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How do you solve $(2x)/(x-5)>=3$ ?

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