I am trying to get a grasp of hard margin SVMs. In the lecture I am watching the professor talks about a classification equation which when a positive sample is input, returns a value of $1$ or more; and when a negative sample is input, returns a value of $-1$ or less. The graph below shows the vector $\overrightarrow{w}$ which is perpendicular to the separating hyperplane and an arbitrary point with unknown class $\overrightarrow{u}$.
In the lecture it says that:
$$\overrightarrow{w} \cdot \overrightarrow{u_+} + b \geq 1 \textrm{ , for positive class samples}$$
$$\overrightarrow{w} \cdot \overrightarrow{u_-} + b \leq -1 \textrm{ , for negative class samples} $$
To me it seems that we should take the projection of $\overrightarrow{u}$ on $\frac{\overrightarrow{w}}{|\overrightarrow{w}|}$, since this would give the component of $\overrightarrow{u}$ in the direction of $\overrightarrow{w}$. If this component is greater than the distance to the decision boundary/hyperplane, $b$, then $\overrightarrow{u}$ is a positive sample, if it is less then it is a negative sample. In math terms
$$\frac{\overrightarrow{w}}{|\overrightarrow{w}|} \cdot \overrightarrow{u_+} - b > 0$$
$$\frac{\overrightarrow{w}}{|\overrightarrow{w}|} \cdot \overrightarrow{u_-} - b < 0$$
If $\overrightarrow{u}$ lies on the decision boundary then the above expressions will be equal to 0.
If $\overrightarrow{u}$ lies on a support vector hyperplane, then the above expressions will be equal to $m$ or $-m$ for positive and negative samples respectively, where $m$ is the margin from the decision boundary to any support vector.
I don't understand the lecturer's equations. Why is the value for positive sample classification $\geq 1$? Why is the value for negative sample classification $\leq -1$? Furthermore, what does $b$ represent in the lecturer's equations?
Basically, I have trouble understanding the proposed equations, but they should be doing the same thing as mine.