Q.The half plane represented by $3x + 2y < 8$ contains the point _______. A) $\left(1, \dfrac{5}{2}\right)$
B) (2, 1)
C) (0, 0)
D) (5, 1)

A linear inequation in two variables $x,y$ is an expression such as $ax+by<c$, $ax+by\le c$, $ax+by>c$ or $ax+by\ge c$, where $a,b,c$ are real numbers and $a,b$ are not both zero. Every linear equation $ax+by=c$ is a straight line that splits the whole coordinate plane into exactly two open half-planes (points strictly on one side or the other) plus the line itself as their common boundary. Graphically solving a linear inequation means shading the correct one of these two half-planes: first draw the boundary line (solid if the inequality includes equality, i.e. $\le$ or $\ge$), then pick a convenient test point NOT on the line — the origin $(0,0)$ whenever the line doesn't pass through it, otherwise any other off-line point such as $(1,0)$ or $(0,1)$ — and substitute it into the inequation. If the test point satisfies the inequality, shade the half-plane containing that point (usually called the 'origin side' when the origin was used); if not, shade the opposite half-plane (the 'non-origin side'). This same idea covers the sim …