How do you solve $abs((x+1)/x)>2$ using a sign chart?

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Answer 1 · 2018-07-09T06:06:36

The solution is $x in (-1/3,0) uu (0, 1)$

Start by solving

$(x+1)/x=0$

$=>$ , $x=-1$ and $x!=0$

Let $g(x)=|(x+1)/x|$

The sign chart is as follows

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaa)$ $-1$ $color(white)(aaaaaaaa)$ $0$ $color(white)(aaaaaaa)$ $+oo$

$color(white)(aaaa)$ $g(x)$ $color(white)(aaaa)$ $-(x+1)/x$ $color(white)(aaaa)$ $(x+1)/x$ $color(white)(aaaa)$ $(x+1)/x$ $color(white)(aaaa)$

In the interval $I_1=(-oo,-1)$

$-(x+1)/x-2>0$

$=>$ , $(-x-1-2x)/x>0$

$=>$ , $(-3x-1)/x>0$

$=>$ , x>1/3

In the interval $I_2=(1,+oo)$

$(x+1)/x-2>0$

$=>$ , $(x+1-2x)/x>0$

$=>$ , $(1-x)/x>0$

$=>$ , $x<1$

Therefore ,

The solution is $x in (-1/3,0) uu (0, 1)$

graph{|(x+1)/x|-2 [-10, 10, -5, 5]}

How do you solve abs((x+1)/x)>2abs((x+1)/x)>2 using a sign chart?