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

Sep 8, 2016

$x \in \left\{\begin{matrix}- \infty & 3 \\ 4 & + \infty\end{matrix}\right\}$

#### 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 $\frac{x - 4}{x - 3} > 0$ when $x < 3$ and when $x > 4$

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