Linear Inequalities and Linear Programming

Introduction to Linear Inequalities

A linear inequality in variables $x$ and $y$ involves a linear expression related by inequality symbols such as $<$, $>$, $\le$, or $\ge$. To solve these graphically—a staple short-question area in BISE exams—you first replace the inequality sign with an equals sign to graph the corresponding boundary line. If the inequality is strict ($<$ or $>$), the line is drawn dashed; if non-strict ($\le$ or $\ge$), it is drawn solid. The solution set is the half-plane that satisfies the inequality, determined easily by testing a reference point (usually the origin, $(0,0)$) provided the line does not pass through it.

Systems of Linear Inequalities and Feasible Region

When dealing with a system of linear inequalities, you must graph each inequality simultaneously on the Cartesian plane. The overlapping intersection of all these half-planes forms the feasible region (or solution region). In board examinations, you are frequently asked to find the corner points (or vertices) of this region. These points are critical because, according to fundamental theorems of linear programming, the optimal solution (maximum or minimum) of a linear objective function always occurs at one of these boundary vertices.

Linear Programming and Optimization

Linear Programming (LP) is a mathematical technique used to maximize or minimize a linear objective function $Z = ax + by$ subject to a set of constraints represented by linear inequalities. To solve an LP problem for an exam:

1. Formulate the mathematical model (if given a word problem).
2. Graph the constraints to identify the feasible region.
3. Determine the coordinates of all corner points.
4. Substitute these corner point coordinates into the objective function $Z$.
5. Identify the maximum or minimum value depending on the problem's requirement.

Mastering this step-by-step graphical method ensures full marks in both long and short questions for this chapter.