Question

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

Answer 1

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

Explanation:

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]}