Question

How do you solve `1/4<7/(7-x)` using a sign chart?

Answer 1

The solution is `x in (-21,7)`

Explanation:

We cannot do crossing over.

Let's rearrange the equation

`1/4<7/(7-x)`

`7/(7-x)-1/4>0`

`(28-(7-x))/(4(7-x))>0`

`(21+x)/(4(7-x))>0`

Let `f(x)=(21+x)/(4(7-x))`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-21` `color(white)(aaaaaaa)` `7` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `21+x` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `7-x` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-`

Therefore,

`f(x)>0` when `x in (-21,7)`