Question

How do you solve `(a-1)/a>0` using a sign chart?

Answer 1

The solution is `a in (-oo,0) uu (1,+oo)`

Explanation:

Let `f(a)=(a-1)/a`

The sign chart is

`color(white)(aaaa)` `a` `color(white)(aaaa)` `-oo` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaaaaa)` `1` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `a` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `a-1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaaa)#`-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(a)` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(a)>0`, when `a in (-oo,0) uu (1,+oo)`

graph{(x-1)/x [-16.02, 16.01, -8.01, 8.01]}