Question

How do you solve `(2x-1)/(x+5)>=0` using a sign chart?

Answer 1

The answer is `x in ] -oo,-5 [ uu [1/2, +oo[`

Explanation:

Let `f(x)=(2x-1)/(x+5)`

Let's do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `color(white)(aaaa)` `-5` `color(white)(aaaa)` `color(white)(aaaa)` `1/2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+5` `color(white)(aaaaaa)` `-` `color(white)(aa)` `color(white)(aaa)` `∥` `color(white)(aaaa)` `+` `color(white)(aaa)` ###color(white)(aa)#`+`

`color(white)(aaaa)` `2x-1` `color(white)(aaaaaa)` `-` `color(white)(aa)` `color(white)(aa)` `∥` `color(white)(aaaa)` `-` `color(white)(aaa)` ###color(white)(aa)#`+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaa)` `+` `color(white)(aa)` `color(white)(aa)` `∥` `color(white)(aaaa)` `-` `color(white)(aaa)` ###color(white)(aa)#`+`

Therefore,

`f(x)>=0`, when `x in ] -oo,-5 [ uu [1/2, +oo[`

graph{y-(2x-1)/(x+5)=0 [-52, 52.07, -26, 26]}