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 ?

Think of it more like a "trick" than a rule.
Everything we do in mathematics should serve a purpose and it just so happens that here, the purpose of what Sal has done was to "get rid" of the $\frac{8}{19}$ that multiplies the x. We say that $\frac{19}{8}$ "cancels" $\frac{8}{19}$.
$$\frac{19}{8} \times \frac{8}{19}x = \frac{19 \times 8}{8 \times 19}x = \frac{19 \times 8}{19 \times 8}x = 1 \times x = x $$
The "price" he pays is that, according to the rules of mathematics, if one alters one side of an equation, one is required to apply the same alteration to both sides, but that's easily fixed with some simplification.
$$\frac{19}{8} \times \frac{8}{19}x \ge \frac{19}{8} \times (-1)$$
$$\implies x \ge -\frac{19}{8}$$
