Question

How do you solve `(x-4)/(x-3)>0`?

Answer 1

`x in {(-oo,3),(4,+oo)}`

Explanation:

The "critical" values for `x` are obviously `3` and `4`.

Create a chart for the implied ranges of `x`:
#{: (, " | ",x < 3, " | ",x = 3, " | ", 3 < x < 4, " | ", x=4, " | ",x > 4), ((x-4)/(x-3), " | ",>0, " | ","undefined", " | ",< 0, " | ",=0, " | ",>0) :}#

We can see that `(x-4)/(x-3) > 0` when `x < 3` and when `x > 4`

Here's a graph of `(x-4)/(x-3)` that might help verify this result:
graph{(x-4)/(x-3) [-2.05, 10.434, -2.5, 3.74]}