How do you solve $x-10/(x-1)>=4$ using a sign chart?

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Answer 1 · 2016-11-18T13:18:49

The answer is $x in ]-oo,-1 ]uu [6, +oo[$

Let's do some simplification

$x-10/(x-1)>=4$

Multiply by $(x-1)$

$x(x-1)-10>=4(x-1)$

$x^2-x-10>=4x-4$

$x^2-5x-6>=0$

Factorising

$(x+1)(x-6)>=0$

Let $f(x)=(x+1)(x-6)$

let's do a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaaaa)$ $-oo$ $color(white)(aaaa)$ $-1$ $color(white)(aaaa)$ $6$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $(x+1)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $+$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $(x-6)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $-$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaaaa)$ $+$ $color(white)(aaa)$ $-$ $color(white)(aaa)$ $+$

$f(x)>=0$ , when $x in ]-oo,-1 ]uu [6, +oo[$

graph{(x+1)(x-6) [-17.02, 15.02, -12.81, 3.21]}

How do you solve x-10/(x-1)>=4x-10/(x-1)>=4 using a sign chart?