Linear Inequalities - Solution of System of Linear Inequalities in Two Variables

A system of linear inequalities consists of two or more linear inequalities in the same variables, such as $x$ and $y$. The solution is the set of all ordered pairs $(x, y)$ that satisfy all inequalities in the system simultaneously. Visually, this solution is represented by the intersection of the shaded regions of each individual inequality on a Cartesian plane.

The boundary line for an inequality is the equation $ax + by = c$. If the inequality involves $\le$ or $\ge$, the boundary line is drawn as a solid line, indicating points on the line are included in the solution. If the inequality involves $<$ or $>$, the boundary line is drawn as a dashed or dotted line, indicating points on the line are excluded.

Every linear inequality divides the coordinate plane into two half-planes. A non-vertical line divides the plane into an upper half-plane and a lower half-plane, while a vertical line divides it into a left half-plane and a right half-plane. The solution to a single inequality is always one of these half-planes.

To determine which side of the boundary line to shade, use the 'Test Point' method. Choose a point $(x_0, y_0)$ not lying on the boundary line (the origin $(0, 0)$ is often the easiest choice). If the point satisfies the inequality, shade the region containing that point; if it does not, shade the opposite side.

The solution region (or feasible region) of a system of inequalities is the area where the shaded regions of all individual inequalities overlap. In a graph, this is often represented as the most heavily shaded or cross-hatched area. If there is no common region where all inequalities overlap, the system has no solution.

Horizontal and vertical inequalities are common components of systems. An inequality like $x \ge a$ represents the region to the right of the vertical line $x = a$, while $y < b$ represents the region below the horizontal dashed line $y = b$.

Non-negative constraints, specifically $x \ge 0$ and $y \ge 0$, are frequently included in systems. Visually, these constraints restrict the entire solution set to the first quadrant of the Cartesian plane, representing real-world scenarios where quantities cannot be negative.