Linear Inequalities - Graphical Solution of Linear Inequalities in Two Variables

A linear inequality in two variables is an algebraic expression involving variables $x$ and $y$ related by inequality signs like $<, >, \le,$ or $\ge$. Visually, while a linear equation $ax + by = c$ represents a straight line on a Cartesian plane, a linear inequality represents an entire region or 'half-plane' bounded by that line.

The line $ax + by = c$ acts as the boundary of the solution set. If the inequality is strict ($<$ or $>$), the boundary line is represented as a dashed or dotted line to show that points on the line are not included in the solution. If the inequality is non-strict ($\le$ or $\ge$), the boundary is drawn as a solid line, indicating points on the line are part of the solution set.

A vertical line $x = a$ or a non-vertical line $ax + by = c$ divides the $XY$-plane into two distinct regions called half-planes. A non-vertical line divides the plane into a 'lower half-plane' and an 'upper half-plane', while a vertical line divides it into a 'left half-plane' and a 'right half-plane'.

To determine which half-plane to shade, the 'Test Point Method' is used. Select a point not lying on the boundary line (the origin $(0,0)$ is the simplest choice if the line doesn't pass through it) and substitute its coordinates into the inequality. If the resulting statement is true, the region containing that point is the solution; if false, the opposite half-plane is the solution.

The graphical solution of a system of linear inequalities is the region of the plane that is common to all individual inequalities in the system. Visually, this is represented by the 'overlapping' or 'intersection' of the shaded regions of each inequality.

The solution region of a system of inequalities can be 'bounded' (enclosed by boundary lines on all sides like a polygon) or 'unbounded' (extending infinitely in at least one direction).

Linear inequalities often include 'non-negative constraints' like $x \ge 0$ and $y \ge 0$. Visually, these constraints restrict the solution region to the first quadrant of the coordinate system.