We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \). In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as $$ b+wx_1+w_2x_2+\dots +w_px_p=0. $$ If we define a matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \) of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \), $$ \boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}. $$ If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have $$ b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0, $$ if our output \( y_i=1 \). In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if $$ b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0, $$ for the class of observations \( y_i=-1 \), then \( \boldsymbol{x}_i \) lies on the other side.
Equivalently, for the two classes of observations we have $$ y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0. $$
When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.