Find whether the points $\left(2, -3\right)$ and $\left(3, -2\right)$ are on the same side or on opposite sides of the line $5x + 6y = 4$.
Equation of given line: $\;\;$ $5x + 6y = 4$ $\;$ i.e. $\;$ $5x + 6y -4 = 0$ $\;\;\; \cdots \; (1)$
Given points: $\;\;$ $A \left(x_1, y_1\right) = \left(2, -3\right)$; $\;$ $B \left(x_2, y_2\right) = \left(3, -2\right)$
Substituting $\;$ $x = 2, \; y = -3$ $\;$ in $\;$ $5x + 6y - 4$, $\;$ it becomes
$5 \times 2 + 6 \times \left(-3\right) - 4 = -12$ $\;$ i.e. $\;$ it is negative.
Substituting $\;$ $x = 3, \; y = -2$ $\;$ in $\;$ $5x + 6y - 4$, $\;$ it becomes
$5 \times 3 + 6 \times \left(-2\right) - 4 = -1$ $\;$ i.e. $\;$ it is negative.
Hence the points lie on the same side of the given line.