How do you solve $\frac{x}{x - 1} > 2$ $x/(x-1)>2$ ?

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How do you solve $\frac{x}{x - 1} > 2$ $x/(x-1)>2$ ?

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Answer 1 · 2018-06-19T09:45:09

The solution is $x \in (2, 4)$ $x in (2,4)$

Explanation:

You cannot do crossing over.

The inequality is

$\frac{x}{x - 2} > 2$ $x/(x-2)>2$

$\Rightarrow$ $=>$ , $\frac{x}{x - 2} - 2 > 0$ $x/(x-2)-2>0$

Placing on the same denominator

$\Rightarrow$ $=>$ , $\frac{x - 2 (x - 2)}{x - 2} > 0$ $(x-2(x-2))/(x-2)>0$

$\Rightarrow$ $=>$ , $\frac{x - 2 x + 4}{x - 2} > 0$ $(x-2x+4)/(x-2)>0$

$\Rightarrow$ $=>$ , $\frac{4 - x}{x - 2} > 0$ $(4-x)/(x-2)>0$

Let $f (x) = \frac{4 - x}{x - 2}$ $f(x)=(4-x)/(x-2)$

Let's build a sign chart

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

$a a a a$ $color(white)(aaaa)$ $4 - x$ $4-x$ $a a a a a a$ $color(white)(aaaaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ #color(white)(a)+# $a a a a$ $color(white)(aaaa)$ $-$ $-$

$a a a a$ $color(white)(aaaa)$ $f (x)$ $f(x)$ $a a a a a a a$ $color(white)(aaaaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ #color(white)(a)+# $a a a a$ $color(white)(aaaa)$ $-$ $-$

Therefore,

$f (x) > 0$ $f(x)>0$ when $x \in (2, 4)$ $x in (2,4)$

graph{(4-x)/(x-2) [-20.27, 20.27, -10.14, 10.14]}

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How do you solve $\frac{x}{x - 1} > 2$ $x/(x-1)>2$ ?

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