Question

How do you solve `3( \frac { w } { 6} ) ^ { 2} - 8( \frac { w } { 6} ) + 4> 0`?

Answer 1

Either `w>12` or `w<4`

Explanation:

We can write `3(w/6)^2-8(w/6)+4>0`

as `3(w/6)^2-6(w/6)-2(w/6)+4>0`

or `3(w/6)(w/6-2)-2(w/6-2)>0`

or `((3w)/6-2)(w/6-2)>0`

Hence either both `(3w)/6-2>0` and `w/6-2>0` i.e. `3w>12` and `w>12` i.e. `w>4` and `w>12` i.e. `w>12`

or both `(3w)/6-2<0` and `w/6-2<0` i.e. `3w<12` and `w<12` i.e. `w<4` and `w<12` i.e. `w<12`

Hence either `w>12` or `w<4`

This may also be seen from following graph which shows that the function is positive, when either `w>12` or `w<4`

graph{3(x/6)^2-8(x/6)+4 [-2.25, 17.75, -3.88, 6.12]}