Question

How do you solve `(2t+7)/(t-4)>=3` using a sign chart?

Answer 1

The solution is `t in ]4, 19]`

Explanation:

Rewrite the equation

`(2t+7)/(t-4)>=3`

`(2t+7)/(t-4)-3>=0`

`((2t+7)-3(t-4))/(t-4)>=0`

`(2t+7-3t+12)/(t-4)>=0`

`(-t+19)/(t-4)>=0`

Let `f(t)=(-t+19)/(t-4)`

We can construct the sign chart

`color(white)(aaaa)` `t` `color(white)(aaaa)` `-oo` `color(white)(aaaaaaa)` `4` `color(white)(aaaaaaaa)` `19` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `t-4` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `19-t` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `-`

`color(white)(aaaa)` `f(t)` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `-`

Therefore,

`f(t)>=0` when `t in ]4, 19]`