I'm self-learning math and I would like you to confirm a suspicion of mine relative to a problem I just solved.
Given a point A(-1,3) and a point B(2,5), I'm asked to find the equation for the set of points closer to A than B. After finding the midpoint and the slope of [AB], I found the equation for the perpendicular: $y=-\frac{3}{2}x+\frac{19}{4}$. Now, I'm convinced the inequality for the set of points is $y<-\frac{3}{2}x+\frac{19}{4}$. However, the textbook solution states $y>-\frac{3}{2}x+\frac{19}{4}$. I'm fairly certain the proposed solution is wrong, but since I'm self-learning, I don't want to risk it and move on.
Also, could it be written as follows? $$\left\{y\in \mathbb{R}<l\mid\:l=\left\{\begin{pmatrix}5\\ \:2\end{pmatrix}+t\begin{pmatrix}-6\\ \:1\end{pmatrix}\mid t\in \:\mathbb{R}\right\}\right\}$$
Yes, you are right. For instance, it is trivial that $A$ is closer to $A$ than to $B$. And if you take $x=-1$ and $y=3$, then, indeed, $3=y<-\frac32x+\frac{19}4=\frac{25}4$.
Concerning the other question, the answer is negative. That notation makes no sense.