The process of solving the inequality $\frac{8}{19} x\ge -1$ 
Why did he multiply both sides by 19/8 and not 8/19 ? 
Is this a rule when dealing with inequalities that to remove fractions, you have to multiply by the reciprocal ?

If he had multiplied both sides by $8/19$, he would have obtained
\begin{align*}
&\left( \frac{8}{19} \right) \cdot \left( \frac{8}{19} x \right)  \ge \left( \frac{8}{19} \right) \cdot (-1) \\
\iff & \frac{64}{361} x \ge - \frac{8}{19},
\end{align*}
which is true but not helpful.  It doesn't bring us any closer to figuring out what $x$ is.
